The convex core of quasifuchsian manifolds with particles

Cyril Lecuire ^{∗}and Jean-Marc Schlenker ^{†}

April 2012 (v2)

Abstract

We consider quasifuchsian manifolds with “particles”, i.e., cone singularities of fixed angle less than π going from one connected component of the boundary at infinity to the other. Each connected component of the boundary at infinity is then endowed with a conformal structure marked by the endpoints of the particles. We prove that this defines a homeomorphism from the space of quasifuchsian metrics with n particles (of fixed angle) and the product of two copies of the Teichmüller space of a surface with n marked points. This extends the Bers Double Uniformization theorem to quasifuchsian manifolds with “particles”.

Quasifuchsian manifolds with particles also have a convex core. Its boundary has a hyperbolic induced metric, with cone singularities at the intersection with the particles, and is pleated along a measured geodesic lamination. We prove that any two hyperbolic metrics with cone singularities (of prescribed angle) can be obtained, and also that any two measured bending laminations, satisfying some obviously necessary conditions, can be obtained, as in [BO04] in the non-singular case.

Résumé

On considère des variétés quasifuchsiennes “à particules”, c’est-à-dire ayant des singularités coniques d’angle fixé inférieur à π allant d’une composante connexe à l’infini à l’autre. Chaque composante connexe du bord à l’infini est alors muni d’une structure conforme marquée par les extrémités des particules. On montre que ceci définit un homéomorphisme de l’espace des métriques quasifuchsiennes à n particules (d’angle fixé) vers le produit de deux copies de l’espace de Teichmüller d’une surface à n points marqués. Ceci étend le théorème de double uniformisation de Bers aux variétés quasifuchsiennes à “particules”.

Les variétés quasifuchsiennes à particules ont aussi un coeur convexe. Son bord a une métrique induite hyperbolique, avec des singularités coniques aux intersections avec les particules, et est plissé le long d’une lamination géodésique mesurée. On montre que toute paire de métriques hyperboliques à singularités coniques (d’angle prescrit) peut être obtenu, et aussi que toute paire de laminations de plissages, satisfaisant des conditions clairement nécessaires, peut être obtenus, comme dans le cas non-singulier [BO04].

Contents

1 Introduction 2 1.1 Convex co-compact manifolds with particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 The conformal structure at infinity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 The geometry of the convex core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Prescribing the bending lamination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 The induced metric on the boundary of the convex core. . . . . . . . . . . . . . . . . . . . . . . . 7 1.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.7 Outline of the proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 The geometry of the convex core 9

∗Institut de Mathématiques de Toulouse, UMR CNRS 5219, Université Toulouse III, 31062 Toulouse Cedex 9, France. Partially supported by the ANR programs Groupes and ETTT (ANR-09-BLAN-0116-01).

†Institut de Mathématiques de Toulouse, UMR CNRS 5219, Université Toulouse III, 31062 Toulouse Cedex 9, France. Partially supported by the ANR program ETTT (ANR-09-BLAN-0116-01).

2.1 Surfaces orthogonal to the singular locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 The convex core of a manifold with particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 The geometry of the boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 The distance between the singular curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Compactness statements 12 3.1 Main statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 A finite cover argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.3 Pleated annuli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.4 Long geodesics in M. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.5 Convergence of convex cores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.6 The bending lamination of the convex core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Prescribing the measured bending lamination on the boundary of the convex core 23 4.1 A doubling argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2 Local deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.3 Proof of Theorem 1.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.4 Proof of Theorem 1.13. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.5 The conditions are necessary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Earthquakes estimates 25 5.1 The average curvature of geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.2 The grafted metric and the hyperbolic metric at infinity. . . . . . . . . . . . . . . . . . . . . . . . 27 5.3 An upper bound on the lengths of the curves at infinity . . . . . . . . . . . . . . . . . . . . . . . 29 5.4 A bound on the length of the earthquake lamination . . . . . . . . . . . . . . . . . . . . . . . . . 31

6 The conformal structure at infinity 33 6.1 A topological lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6.2 Compactness relative to the conformal structure at infinity . . . . . . . . . . . . . . . . . . . . . 33 6.3 Proof of Theorem 1.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 6.4 Proof of Theorem 1.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

7 Some questions and remarks 35 7.1 Some questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 7.2 AdS manifolds with particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 7.3 The renormalized volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

A Quasiconformal estimates 36 A.1 Pants decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 A.2 Proof of Proposition 1.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

1 Introduction

1.1 Convex co-compact manifolds with particles

Quasifuchsian manifolds. A quasifuchsian manifold is a complete hyperbolic manifold M diffeomorphic to S×ℝ, where S is a closed, oriented surface of genus at least 2, which contains a non-empty, compact, geodesically convex subset, see [Thu80]. Such a manifold has a boundary at infinity, which is the union of two copies of S. Each of those two copies has a conformal structure, τ_{+} and τ_{−}, induced by the hyperbolic metric on M. A celebrated theorem of Bers [Ber60, AB60] asserts that the map sending a quasifuchsian metric to (τ_{+}, τ_{−}) determines a parameterization of the space of quasifuchsian metrics on a M by the product of two copies of the Teichmüller space of S.

A quasifuchsian manifold M contains a smallest non-empty geodesically convex subset, called its convex core C(M). Here we say that a subset K ⊂ M is geodesically convex if any geodesic segment in M with endpoints in K is contained in K. This implies that the inclusion of K in M is a homotopy equivalence.

The boundary of C(M) is again the union of two copies of S, and each is a pleated surface in M, with a hyperbolic induced metric m_{+}, m_{−} and a measured bending lamination λ_{+}, λ_{−}. (There is a special, “Fuchsian” case, where C(M) is a totally geodesic surface, the two pleated surfaces mentioned here are then the same, m_{+} = m_{−}, and λ_{+} = λ_{−} = 0.)

It is known that any two hyperbolic metrics can be obtained in this way, this follows from [EM86] or from [Lab92], however it is not known whether any couple (m_{+}, m_{−}) can be uniquely obtained. Similarly, any two measured laminations λ_{+}, λ_{−} can be obtained in this manner if the weight of any leaf is less than π and if λ_{+} and λ_{−} fill S [BO04], but it is not known whether uniqueness holds. Recall that λ_{−} and λ_{+} fill S if there exists ϵ > 0 such that, for any closed curve c in S, i(λ_{−}, c) + i(λ_{+}, c) ≥ ϵ.

The main goal here is to extend those results to quasifuchsian manifols with “particles”, that is, cone singularities of a certain type connecting the two connected components of the boundary at infinity, as described below.

Note that all the results mentioned here are actually known in the more general context of convex co-compact hyperbolic manifolds, i.e., interiors of compact manifolds with boundary, with a complete hyperbolic metric, containing a non-empty, compact, geodesically convex subset (or, even more generally for geometrically finite hyperbolic manifolds), the result concerning the measured bending lamination of the boundary can then be found in [Lec06]. We stick here to the quasifuchsian setting for simplicity.

Cone-manifolds. We consider here hyperbolic cone-manifolds of a special kind, which have cone singularities along curves (a more general notion is defined in [Thu80], allowing for singularities along graphs). Let θ ∈ (0, π), we call H_{θ}^{3} the hyperbolic manifold with cone singularities obtained by gluing isometrically the two faces of a hyperbolic wedge of angle θ (the closed domain in H^{3} between two half-planes having the same boundary line). There is a unique such gluing which is the identity on the “axis” of the wedge. We will be using here the following (restrictive) definition.

Definition 1.1. A hyperbolic cone-manifold is a manifold along with a metric for which each point has a neighborhood modeled on H_{θ}^{3} for some θ ∈ (0, π).

Let M be a hyperbolic cone-manifold, it has two kind of points. Those which have a neighborhood isometric to a neighborhood of a point of some H_{θ}^{3} outside the cone singularity are called regular points, while the others are called singular points or cone points. The set of regular points will be denoted by M_{r}, and the set of singular points by M_{s}. By definition, M_{s} is a union of curves, if M is complete then those curves can be either closed curves or infinite lines. To each of those curves is associated an angle θ ∈ (0, π) — such that all points have a neighborhood isometric to a neighborhood of the cone singularity in H_{θ}^{3} — which is called its cone angle or simply its angle.

Recall the usual notion of convexity, which differs from other possible notions (e.g. the local convexity of the boundary of a domain).

Definition 1.2. Let M be a hyperbolic cone-manifold. A subset K ⊂ M is geodesically convex if any locally geodesic segment in M with endpoints in K is contained in K.

A non-empty geodesically convex subset of M is homotopically equivalent to M and contains all closed geodesics of M, see [MS09, Lemma A.12].

Quasifuchsian manifolds with particles. Quasifuchsian manifolds with particles are defined in the same way as non-singular quasifuchsian manifold.

Definition 1.3. A quasifuchsian manifold with particles is a complete hyperbolic cone-manifold M isometric to the product S × ℝ, where S is a closed, orientable surface, endowed with a complete hyperbolic metric with cone singularities of angles in (0, π) on the lines {x_{i}} × ℝ, for x_{1}, ⋯, x_{n_{0}} distinct points in S, which contains a non-empty, compact, geodesically convex subset. We require that n_{0} ≥ 4 if S is a sphere, i.e. that M has at least 4 singularities, and that n_{0} ≥ 1 if S is a torus.

Notice that the definition also makes sense if S is a sphere and n_{0} = 3 but then the metric would be uniquely defined (up to isotopy) by the cone angles. We are not considering this case for technical reasons but also because there is not much to say about it.

Given a non-empty convex subset K of a quasifuchsian manifold with particles, then K contains all closed geodesics of M (see [MS09, Lemma A.12]) and the inclusion of K in M is a homotopy equivalence (this is proved below).

Geometrically, quasifuchsian manifolds with particles can be considered as I-bundles in the category of hyperbolic 3-manifolds with cone singularities. The term “particle” comes from physical motivations. Quasifuchsian manifolds have Lorentzian siblings called Anti-de Sitter (AdS) globally hyperbolic (GH) manifolds which share many of the key properties recalled above, see [Mes07, ABB^{+}07]. From a physics viewpoint, GH AdS 3-manifolds are a 3-dimensional toy model for gravity, as they model an empty space with negative cosmological constant. To go beyond an empty model, massive point particles can be added and modeled as cone singularities along time-like lines, see e.g. [tH96, tH93]. The resulting GH AdS manifolds with particles display some properties which are parallel to those obtained here, see [BS09].

The restrictions on the cone angles — supposed to be in (0, π) — are necessary at several points here, as they were in [MS09]. It seems to be physically relevant, too. We will mention some points where this hypothesis is useful below as they occur. We do not know whether Theorem 1.7, for instance, can be extended to cone angles less than 2π. In the parallel Lorentzian theory concerning globally hyperbolic anti-de Sitter manifolds, new phenomena arise when the cone angles are larger than π, see [BBS11, BBS12].

Quasifuchsian manifolds with particles are always considered here up to isotopies.

Convex co-compact manifolds with particles. The previous definition can be extended to a definition of convex co-compact manifolds with particles.

Definition 1.4. A convex co-compact hyperbolic manifold with particles is a complete hyperbolic cone-manifold M such that:

• M is homeomorphic to the interior of a compact manifold with boundary N, with non-trivial fundamental group,

• the singular locus corresponds under the homeomorphism to a disjoint union of curves in N with endpoints on ∂N,

• the angle at each singular curve is less than π,

• M contains a non-empty compact subset which is convex.

A further extension to geometrically finite manifolds with particles is possible, we leave the details to the interested reader. We consider here only quasifuchsian mainfolds (with particles) although some of the intermediate statements can be extended to convex co-compact manifolds with particles. There is also some hope to extend the main results to this more general setting, however some technical hurdles have to be overcome before this can be achieved.

Note that there is another possible notion of quasifuchsian manifolds with cone singularities: those which are singular along closed curves, as studied in particular by Bromberg [Bro04b, Bro04a]. Although there are similarities between those two kinds cone-manifolds (in particular concerning their rigidity), the questions considered here are quite different from those usually associated to those considered for quasifuchsian conemanifolds with singularities along closed curves (drilling of geodesics, etc).

1.2 The conformal structure at infinity.

Conformal structures and hyperbolic metrics on surfaces. Let’s fix some notations.

Definition 1.5. Let S be a closed orientable surface, let x_{1}, ⋯, x_{n_{0}} ∈ S be distinct points with n_{0} ≥ 4 if S is a sphere and n_{0} ≥ 1 is S is a torus, and let θ_{1}, ⋯, θ_{n_{0}} ∈ (0, π). We then call:

• T_{S,x} the space of conformal structures on S, considered up to isotopies of S fixing the x_{i},

• H_{S,x,θ} the space of hyperbolic metrics on S, with cone singularities at the x_{i} where the angle is θ_{i}, considered up to isotopies fixing the x_{i}.

There is a one-to-one map between T_{S,x} and H_{S,x,θ}, because any conformal structure contains a unique hyperbolic metric with cone singularities at the x_{i} of prescribed angle (see [Tro91]). We keep distinct notations for clarity.

Notice again that these definition also make sense when S is a sphere and n_{0} = 3 provided that ∑_{i=1}^{n_{0}} θ_{i}−2π < 0 . In this case the spaces T_{S,x} and H_{S,x,θ} are points.

The statements considered here are already well understood when n_{0} = 0, so we will focus below on the case n_{0} ≥ 1.

The conformal structure at infinity. Non-singular quasifuchsian manifolds have a natural conformal structure at infinity, which can be defined by considering the action of their fundamental group on their discontinuity domain, see [Thu80]. This definition cannot be used directly for quasifuchsian manifolds with particles, however it is still possible to define a conformal structure at infinity, see [MS09, Section 3.2].

Therefore, to each quasifuchsian metric g ∈ QF_{θ_{1},···,θ_{n_{0}}} are associated two points τ_{+}, τ_{−} ∈ T_{S,n_{0}} corresponding to the conformal structures — marked by the endpoints of the “particles” — on the upper, resp. lower, connected component of the boundary at infinity.

Note that we always implicitly consider conformal structures on the boundary at infinity up to isotopy. (It is therefore not necessary to consider markings.)

A compactness lemma for the conformal structure at infinity. We consider again a closed surface S along with n_{0} distinct points (n_{0} ≥ 1) x_{1}, ⋯, x_{n_{0}} ∈ S and angles θ_{1}, ⋯, θ_{n_{0}} ∈ (0, π) so that

2πχ(S) − ∑_{i=1}^{n_{0}}(2π − θ_{i}) < 0 .

Proposition 1.6. Let (g_{n})_{n∈ℕ} be a sequence of quasifuchsian metrics on S×ℝ, with particles (cone singularities) on the lines {x_{i}} × ℝ, of angle equal to θ_{i}. Suppose that the conformal structures at infinity, τ_{−,n}, τ_{+,n} ∈ T_{S,n_{0}}, converge to conformal structures τ_{−,∞}, τ_{+,∞}. Then (g_{n})_{n∈ℕ} has a subsequence converging to a quasifuchsian metric with particles.

The proof is contained in Section 6.2, it is based on the compactness results described below (in Section 3) relative to the induced metric and bending lamination on the boundary of the convex core.

A Bers-type theorem with particles. Using the previous proposition, along with the main result of [MS09], leads to an extension to quasifuchsian manifolds with particles of a classical result of Bers [Ber60] on “double uniformization”.

Theorem 1.7. The map from QF_{S,x,θ} to T_{S,x} ×T_{S,x} sending a quasifuchsian hyperbolic metric to the conformal structures at +∞ and at −∞ (marked by the endpoints of the particle) is a homeomorphism.

1.3 The geometry of the convex core

Measured laminations. We refer the reader to e.g. [CB88, PH92, Ota96] for the definition and main properties of measured laminations on closed (non-singular) surfaces as well as the topology on the space of measured laminations. There are two possible definitions. One is geometric, in terms of measured geodesic laminations on hyperbolic surfaces, with a transverse measure, while the other definition is topological, and can involve the boundary at infinity of the universal cover of the surface. The two definitions are equivalent, basically because, in a closed (or finite volume) hyperbolic surface, any closed curve can be realized uniquely as a closed geodesic.

Proposition 1.8. Let Σ be a hyperbolic surface with cone singularities, where the angle is less than π. Let λ be a (topological) lamination on Σ. Then λ can be realized uniquely as a geodesic lamination.

The space of measured geodesic laminations on a hyperbolic surface with cone singularities of angles less than π therefore does not depend on the cone angles.

Definition 1.9. We call ML_{S,n_{0}} the space of measured lamination on S with n_{0} marked points.

Thus, for any hyperbolic metric m on S with n cone singularities of angle less than π, ML_{S,n_{0}} can be canonically identified with the space of measured geodesic laminations on (S, m).

The convex core. The following basic proposition can be found in the appendix of [MS09].

Proposition 1.10. Let M be a convex co-compact hyperbolic manifold with particles, and let K and K′ be two non-empty geodesically convex subsets. Then K ∩ K′ is a non-empty geodesically convex subset.

It leads to a natural definition.

Definition 1.11. Let M be a quasifuchsian manifold with particles. Its convex core C(M) is the smallest non-empty geodesically convex subset contained in it.

By construction, C(M) is a “minimal” convex subset of M and it follows from general arguments (see [Thu80]) that its boundary is, outside the singular curves, a pleated surface (a locally convex, ruled surface). It turns out that, under the condition that the cone angles are less than π, the boundary of C(M) intersects the cone singularities orthogonally, and is even totally geodesic in the neighborhood of each such intersection, see [MS09, Lemma A.15].

It follows that there is a well-defined notion of closest-point projection from M to C(M). As a consequence, the inclusion of C(M) in M is a homotopy equivalence. The same holds for any non-empty convex subset of M.

Therefore, given a quasifuchsian metric with particles g ∈ QF_{S,θ}, the induced metrics on the upper and lower boundary components of C(M) (which might coincide in special cases) are two hyperbolic metrics m_{+}, m_{−} ∈ H_{S,x,θ}. Moreover, those two boundary components are pleated along measured bending laminations l_{+}, l_{−} ∈ ML_{S,n_{0}}.

A remark on the hypothesis A well-known fact concerning hyperbolic surfaces with cone singularities is that, as long as the cone angles are less than π, it remains true that any homotopy class of closed curves in the regular part contains a unique geodesic (see e.g. [DP07]). A fairly direct consequence is that, as for closed surfaces, any topological measured lamination (in the complement of the cone singularities) can be uniquely realized as a measured geodesic lamination.

This is one reason — albeit not the only one — why it is relevant to consider here cone singularities of angle less than π, rather than less than 2π. Indeed for cone singularities of angles less than 2π, the induced metric on the boundary of the convex core might also have cone singularities of angle between π and 2π, and for those metrics the one-to-one relation between measured laminations and measured geodesic laminations is lost.

1.4 Prescribing the bending lamination

Results in the non-singular case. For non-singular convex co-compact hyperbolic manifolds an existence and uniqueness theorem for metrics with a given rational measured bending lamination was proved by Bonahon and Otal [BO04]. (Recall that a lamination is rational if its support is a disjoint union of closed curves.) When the lamination is not rational, an existence result was proved in [BO04] for manifolds with incompressible boundary, it was extended in [Lec06] to manifolds with compressible boundary.

Rational laminations with particles. As for quasifuchsian manifolds (without particles), it is possible to give an existence and uniqueness statement concerning the bending lamination on the boundary of the convex core, but only for rational laminations.

Theorem 1.12. Let S be a closed orientable surface, let x_{1}, ⋯, x_{n_{0}} ∈ S be distinct points, and let θ_{1}, ⋯, θ_{n_{0}} be in (0, π). Suppose that n_{0} ≥ 4 if S is a sphere, and that n_{0} ≥ 1 if S is a torus. Let λ_{−}, λ_{+} ∈ ML_{S,x} be measured laminations, each with support a disjoint union of closed curves. Suppose that:

• λ_{−} and λ_{+} fill S,

• each closed curve in the support of λ_{−} (resp. λ_{+}) has weight less than π.

Then there exists a metric g ∈ QF_{S,x,θ} such that the measured bending lamination on the upper (resp. lower) boundary component of the convex core of (S × ℝ, g) is λ_{+} (resp. λ_{−}). Moreover g is unique up to isotopies.

The proof, which is given in Section 4, is based on the rigidity theorem of Hodgson and Kerckhoff [HK98] for closed hyperbolic manifolds with cone singularities. We prove in Lemma 4.3 that the hypothesis are necessary conditions.

General laminations. When considering laminations which are not necessarily rational, we obtain only a weaker result, because we can only claim existence, but not uniqueness (this remains an open problem even in the non-singular case, see [BO04, Lec06]).

Theorem 1.13. Let S be a closed surface, let x_{1}, ⋯, x_{n_{0}} ∈ S be distinct points, and let θ_{1}, ⋯, θ_{n_{0}} be in (0, π). Let λ_{−}, λ_{+} ∈ ML_{S,x}. Suppose that:

• λ_{−} and λ_{+} fill S,

• each closed curve in the support of λ_{−} (resp. λ_{+}) has weight less than π.

Then there exists a metric g ∈ QF_{S,x,θ} such that the measured pleating lamination on the upper (resp. lower) boundary component of the convex core of (S × ℝ, g) is λ_{+} (resp. λ_{−}).

The two conditions on λ_{−}, λ_{+} in this theorem are easily seen to be necessary when g is not fuchsian, see Lemma 4.3. Note that both Theorem 1.12 and Theorem 1.13 are restricted to quasifuchsian manifolds with particles, rather than more general convex co-compact manifolds with particles.

1.5 The induced metric on the boundary of the convex core.

The Bers-type result on the conformal metric at infinity can be used to obtain an existence result concerning the prescription of the induced metric on the boundary of the convex core.

Theorem 1.14. Let m_{−}, m_{+} ∈ H_{S,n_{0},θ}, where θ = (θ_{1}, ⋯, θ_{n_{0}}) ∈ (0, π)^{n}. There exists a quasifuchsian metric with particles on S × ℝ, with particles of angle θ_{i} at the lines {x_{i}} × ℝ, for which the induced metric on the boundary of the two connected components of the convex core are m_{−} and m_{+}.

In the smooth case – i.e. for quasifuchsian hyperbolic manifolds without conical singularities – the corresponding result is well-known, it follows either from results of Labourie [Lab92] or from a partial answer, first given by Epstein and Marden [EM86], to a conjecture of Sullivan. (The conjecture made by Sullivan turned out to be wrong, see [EM05], but the result proved by Epstein and Marden is sufficient to prove Theorem 1.14 in the non-singular context.)

As for the conformal structure at infinity, it might be possible to extend this statement to cover convex co-compact (resp. geometrically finite) manifolds with particles. The uniqueness remains elusive, as in the non-singular case.

1.6 Applications

Quasifuchsian manifolds can be used as tools in Teichmüller theory. By extension, the quasifuchsian manifolds with particles considered here can be used as tools for the study of the Teichmüller space of hyperbolic metrics with cone singularities (of angle less than π) on a surface.

One such application is through the renormalized volume of those quasifuchsian manifolds with particles, as considered in [KS08a, KS12]. In the non-singular case this renormalized volume is equal to the Liouville functional (see [TZ87, TT03, TZ03]), it is a Kähler potential on H_{S,x,θ}. Other applications of closely related tools, in the non-singular context, for the global geometry of the Weil-Petersson metric on Teichmüller space, can be found in [McM00]. Yet other applications, to some properties of the grafting map, are considered in [KS08b], and the manifolds with particles considered here should allow for an extension to the grafting map on H_{S,x,θ}.

1.7 Outline of the proofs

We now turn to a description of the main technical points of the proofs.

Measured bending laminations. Theorem 1.12 is proved by an argument strongly influenced by the proof given by Bonahon and Otal [BO04] for non-singular convex co-compact manifolds. Thanks to a doubling trick, the infinitesimal rigidity of the convex cores of convex co-compact manifolds with particles, with respect to the (rational) measured bending lamination, is reduced to an important infinitesimal rigidity result proved for hyperbolic cone-manifolds by Hodgson and Kerckhoff [HK98]. A deformation argument then provides the proof of the theorem.

The existence result for general laminations on quasifuchsian manifolds with particles (Theorem 1.13) can then be obtained by an approximation argument, as in the non-singular case in [BO04, Lec06]. The key step of the proof is a compactness statement, showing that if the measured bending laminations converge to a limit having good properties, then the quasifuchsian metrics converge after extracting a subsequence. However the arguments developed in [BO04, Lec06] cannot be used in the context of quasifuchsian manifolds with particles, because they rely heavily on the representation of the fundamental group. Different arguments are therefore used here, which are more differential-geometric in nature.

Those arguments are sometimes technically involved because of the added difficulties induced by the particles. However, after stripping the proof of the elements which are needed only because of the particles (for instance the multiple cover argument used in Section 3.4 to find simplicial surfaces with given boundary in the convex core), the compactness proof given here is simpler than the one in [BO04, Lec06].

Prescribing the induced metric on the boundary of the convex core. We give in Section 6 a rather elementary proof of Theorem 1.14, which has two parts. Call t_{−} (resp. t_{+}) the hyperbolic metric in the conformal class τ_{−} (resp. τ_{+}) with cone angles θ_{i} at x_{i}. The first part is an upper bound on the length of the curves in the hyperbolic metric at infinity t_{±}, following [Bri98, BC03].

Now recall that, by Thurston’s Earthquake Theorem [Ker83, Thu86a], there exists a unique right earthquake sending a given hyperbolic metric to another one. This extends to hyperbolic metrics with cone singularities of angle less than π, see [BS09]. In particular there is a unique measured lamination ν_{+} such that the right earthquake along ν_{+}, applied to m_{+}, yields the hyperbolic metric t_{+}. The second part of our proof is a bound on the length of ν_{+} for m_{+} (see Proposition 5.2).

This is then used in Section 6 to prove Theorems 1.7 and 1.14. The proof of Theorem 1.7 also uses another main ingredient, the local rigidity of quasifuchsian manifolds with particles proved in [MS09].

Quasi-conformal estimates. There is another possible way to prove Theorem 1.14, closer to the argument used in the non-singular case (as seen in [EM86, Bri98, BC03]). It uses a bound on the quasi-conformal factor between the conformal structure at infinity τ and the conformal class of the induced metric m on the boundary of the convex core, both understood as elements of T_{S,n_{0}}.

Proposition 1.15. There exists a constant C > 0 (depending only on the topology of M) such that τ is C-quasiconformal to m.

This proposition is not formally necessary to obtain the main results presented here, its proof can be found in Appendix A.

As mentioned above, the proof of Theorem 1.14 through Proposition 1.15 would be much closer to the proof(s) known in the non-singular case. It can be pointed out that the proof given in Section 6 is quite parallel, but in the context of Teichmüller theory understood as the study of hyperbolic rather than complex surfaces. From this viewpoint, Proposition 5.2 is a direct analog of Proposition 1.15, with quasiconformal deformations replaced by earthquakes.

What follows. Section 2 presents the definition of the convex core of a convex co-compact manifold with particles, and some of its simple properties, extending well-known properties with no cone singularity. In Section 3 we state and prove a key compactness statement with respect to the measured bending lamination on the boundary of the convex core. Section 4 contains the proof of Theorem 1.12, using a local rigidity statement of Hodgson and Kerckhoff [HK98] and the compactness Lemma of Section 3. Section 5 contains the proof of Theorem 1.13, and Section 6 contains the proof of Theorem 1.7 and of Theorem 1.14. Section 7 contains some remarks on the analogy with corresponding problems in anti-de Sitter geometry and on applications to the Weil-Petersson metric of the Teichmüller space of hyperbolic metrics with cone singularities of prescribed angles on a closed surface (see [KS08a, KS12, KS08b]). Finally, Appendix A contains the proof of Proposition 1.15, based on the estimates on the length of the earthquake lamination obtained in Section 5.

2 The geometry of the convex core

This section contains some basic statements necessary to understand the geometry of convex co-compact manifolds with particles, concerning in particular the convex core and its boundary. We consider here such a convex co-compact manifold with particles, M, and denote by M_{r} its regular part and by M_{s} its singular part (the union of the singular lines).

We exclude below the simplest case where M is Fuchsian, that is, where it is the warped product of a hyperbolic surfaces with cone singularities (S, h) by ℝ, with the metric dt^{2} + cosh(t)^{2}h. In this Fuchsian case the convex core is a surfaces, corresponding to t = 0, and it is totally geodesic outside the intersection with the particles, and orthogonal to those particles.

2.1 Surfaces orthogonal to the singular locus

We define here a natural notion of pleated surface orthogonal to the singular locus in M. The first step is to define the notion of totally geodesic plane orthogonal to a cone singularity in a hyperbolic cone-manifold. The first condition is that the surface is totally geodesic outside its intersections with the particles. The second condition is local, in the neighborhood of the intersections with the particles; there, the surface should correspond to the image in H_{θ}^{3} of the restriction to the wedge (used to define H_{θ}^{3}) of a plane orthogonal to the axis of the wedge.

Definition 2.1. Let Σ be a pleated surface in M_{r}, and let Σ′ be its closure as a subset of M; suppose that Σ′ \ Σ ⊂ M_{s}. We say that Σ′ is orthogonal to the singular locus if any x ∈ Σ′ \ Σ has a neighborhood in Σ′ which is a totally geodesic surface orthogonal to the singular locus.

This definition can be extended to encompass more general surfaces, i.e., surfaces which are neither pleated nor totally geodesic in the neighborhood of the singular locus. In this more general case the definition can be given in terms of the convergence of the unit normal vector to a vector “tangent” to the singular locus at its intersection with the surface. This will however not be needed here.

2.2 The convex core of a manifold with particles

Among the defining properties of a quasifuchsian cone-manifold M is the fact that it contains a compact subset K which is convex in the (strong) sense that any geodesic segment in M with endpoints in K is contained in K. We have already seen that it is possible to define the convex core of M as the smallest compact subset of M which is convex, denoted by C(M).

Theorem 2.2. Suppose that C(M) is not a totally geodesic surface. Then its boundary is the disjoint union of surfaces which are orthogonal to the singular locus. Each connected component of the singular locus of M intersects C(M) along a segment.

The proof is a consequence of two lemmas, both stated under the hypothesis of the theorem. The second lemma in particular gives more precise informations on the geometry of the convex core, it is taken from [MS09, Lemma A.14].

Lemma 2.3. The boundary of C(M) is a surface orthogonal to the singular locus.

Let x ∈ M, we denote by L_{x} the link of M at x, that is, the space of geodesic rays starting from x (parametrized at speed 1), with its natural angle distance. When x is a regular point of M, L_{x} is isometric to the 2-dimensional sphere S^{2}, with its round metric. When x is contained in a singular line of angle θ, L_{x} can be described as the metric completion of the quotient by a rotation of angle θ of the universal cover of the complement of two antipodal points in S^{2}.

Definition 2.4. Let K ⊂ M be convex, and let x ∈ M. The link of K at x is the set of vectors v ∈ L_{x} such that there is a (small) geodesic ray starting from x in the direction of v which is contained in K. It is denoted by L_{x}(K).

Clearly L_{x}(K) = ∅ when x is not contained in K, while L_{x}(K) = L_{x} when x is contained in the interior of K.

To go further, we define the oriented normal bundle of ∂C(M), denoted by N_{r}^{1}∂C(M), as the set of (x, n) ∈ T M such that x ∈ C(M) is not in the singular locus of M and n is a unit vector such that its orthogonal is a support plane of C(M) at x, and n is oriented towards the exterior of C(M).

Let x ∈ M be a non-singular point, let v ∈ T_{x}M and let t ∈ ℝ_{+}. For t small enough, it is possible to define the image of (x, tv) by the exponential map, it is the point exp(x, tv) := g(t), where g is the geodesic, parametrized at constant speed, such that g(0)x and g′(0) = v. As t grows, exp(x, tv) remains well-defined until g intersects the singular set of M.

Lemma 2.5. The exponential map is a homeomorphism from N_{r}^{1}∂C(M) × (0, ∞) to the complement of C(M) in M, and its restriction to the complement of the points of the form (x, v, t), for x ∈ M_{s} and v a singular point of L_{x}, is a diffeomorphism to complement of C(M) in M_{r}. The map:

exp_{∞} : N_{r}^{1}∂C(M) → ∂_{∞}M
(x, v)  ↦ lim_{t→∞} exp(x, tv)

is a homeomorphism from N_{r}^{1}∂C(M) to the complement in ∂_{∞}M of the endpoints of the singular curves in M.

This follows directly from Lemma A.11 in [MS09].

The proof of Theorem 2.2 clearly follows from Lemma 2.3 and Lemma 2.5, since Lemma 2.5 shows that the cone singularities cannot re-enter the convex core after exiting it.

2.3 The geometry of the boundary

By construction, C(M) is a minimal convex set in M, and it follows as in the non-singular case (see [Thu80]) that its boundary is a “pleated surface” except at its intersections with the singular curves.

Lemma 2.6. The surface ∂C(M) has an induced metric which is hyperbolic (i.e. it has constant curvature −1) with conical singularities at the intersections of ∂C(M) with the singular curves of M, where the total angle is the same as the total angle around the corresponding singular curve. It is “pleated” along a measured lamination λ in the complement of the singular points. Moreover the distance between the support of λ and the intersection of the singular set of M with ∂C(M) is strictly positive.

Proof. Since C(M) is a minimal convex subset, its boundary is locally convex and ruled, therefore developable (see [Spi75] for the Euclidean analog, or [Thu80]) so that its induced metric is hyperbolic. The fact that its intersection points are conical singularities, with a total angle which is the same as the total angle around the corresponding singularities, is a consequence of the fact that ∂C(M) is orthogonal to the singularities.

Similarly, the fact that ∂C(M) is pleated along a measured lamination is a direct consequence of the fact that it is ruled and locally convex, i.e. that each point in ∂C(M) is in either a complete hyperbolic geodesics or a totally geodesic ideal triangle. The support of λ is a disjoint union of embedded maximal geodesics, and it is well-known (see e.g. [DP07]) that (under the hypothesis that the angles at the cone singularities are strictly less than π) embedded geodesics remain at positive distance from the singular locus. So the distance between the support of λ and the singular locus of ∂C(M) is strictly positive.

2.4 The distance between the singular curves

We state and prove here some elementary statements on the distance between singular points in the boundary of C(M) and between singular curves in M. They will be useful at several points below.

Lemma 2.7. Let θ ∈ (0, π). There exists ϵ > 0 and ρ > 0, depending on θ, such that:

1. in a complete hyperbolic surface with cone singularities of angle less than θ (not homeomorphic to a sphere), two cone singularities are at distance at least ϵ,

2. if D is a closed 2-dimensional geodesic disk of radius ϵ centered at a singular point x_{0} of cone angle θ, and if Ω ⊂ D is a convex subset whose closure intersects the boundary of D, then Ω contains all points of D at distance at most ρ from x_{0}.

In particular, it follows from point (1) that no embedded geodesic in D can come within distance less than ρ from the cone singularity.

Proof. The first point is well-known, see e.g. [DP07]. The interested reader can construct an elementary proof based on Dirichlet domains, as in 3-dimensional manifolds in the proof of the second point, below.

For the second point let x_{1} ∈ ∂D ∩ Ω, and let γ be the minimizing geodesic segment from x_{0} to x_{1}. Since D contains no other singular point by the first point, the complement of γ in D is isometric to an angular sector in the disk of radius ϵ in H^{2}. This angular sector has three vertices, one corresponding to x_{0} and the other two corresponding to x_{1}. Since θ < π, it is convex at the vertex corresponding to x_{0}. Let s be the geodesic segment joining the two vertices corresponding to x_{1}. Then Ω, being convex, contains the projection in D of the triangle bounded by s and by the two geodesic segments in the boundary of D joining x_{0} to the two vertices projecting to x_{1}. This proves the statement, with ρ equal to the distance between x_{0} and s.

We now turn to a similar lemma, but concerning 3-dimensional manifolds with particles.

Lemma 2.8. Let θ ∈ (0, π). There exists ϵ > 0 and ρ > 0, depending on θ, such that, if M is a quasifuchsian manifold with particles of angle less than θ, then any two particles in M are at distance at least ϵ.

Proof. We reason by contradiction, that is, we fix θ ∈ (0, π) and for any n > 0 there is a quasifuchsian manifold M_{n} with particles of angles less than θ, with two particles p_{n}, p′_{n} at distance less than 1/n (and no two particles strictly closer than p_{n} and p′_{n}). Let s_{n} be the length-minimizing segment between p_{n} and p′_{n}, and let x_{n} be its midpoint. We call D_{n} the Dirichlet domain in M_{n} centered at x_{n}.

We call (M′_{n}, x_{n}) the pointed cone-manifold obtained by performing on (M_{n}, x_{n}) a homothety of ratio 1/L(s_{n}), so that M′_{n} has constant curvature L(s_{n})^{2} ≤ 1/n^{2}. Let D′_{n} be the Dirichet domain centered at x_{n} in M′_{n}, so that D′_{n} is obtained by performing a homothety of ratio 1/L(s_{n}) on D_{n}.

By construction the cone singularities in M′_{n} are at distance at least 1, so that, after extracting a subsequence, (M′_{n}, x_{n}) converges in the pointed Gromov-Hausdorff topology to a pointed manifold (M′, x). Still by construction, M′ contains at least two cone singularities p and p′, limits respectively of p_{n} and p′_{n}, at distance 1, with x at the midpoint of a geodesic segment of minimal length connecting p to p′. Let D′ be the Dirichlet domain centered at x in M′, then (D′, x) is the limit of the (D′_{n}, x_{n}) in the pointed Gromov-Hausdorff topology. By definition all D′_{n} are unbounded, so D′ is also unbounded.

Let θ_{p} and θ_{p′} be the cone angles at p and p′, respectively, in M, so that θ_{p}, θ_{p′} ≤ θ. We now consider D′ as a convex polyhedron in Euclidean space ℝ^{3}, with two edges e and e′ corresponding to p and p′, respectively. Let H_{1}, H_{2} be the two half-planes bounded by p at angle θ_{p}/2 with s, and let H′_{1}, H′_{2} be the two half-planes bounded by p′ at angle θ_{p′}/2 with p′. Then H_{1}, H_{2}, H′_{1}, H′_{2} are faces of D′, so that D′ is contained in D′′, the intersection of the half-spaces bounded by the four planes containing H_{1}, H_{2}, H′_{1}, H′_{2} and containing s.

Suppose that e and e′ are not parallel. Then D′′ has at most one end, so that D′ has also at most one end. This is clearly impossible since all M_{n} are quasifuchsian manifolds with particles, so that all D_{n} have two ends. Therefore, e and e′ are parallel. For the same reason any other edge of D′ which corresponds or not to a cone singularity of M has to be parallel to e and to e′. So D′ is invariant under translations parallel to e, that it, it is the product by ℝ of a polygon π in a plane orthogonal to e. It follows that M′ is also invariant by translation parallel to p.

Consider the regular part M′_{reg} of M′. Since M′_{reg} is a Euclidean manifold, its holonomy representation Hol is a morphism from π_{1}(M′_{reg}) to Isom(ℝ^{3}) = ℝ^{3} ⋊ O(3). Since M′ is invariant under translations parallel to p, Hol actually takes values in ℝ^{3} ⋊ O(2). We consider the morphism Hol′ : π_{1}(M′_{reg}) → ℝ^{2} ⋊ O(2) = Isom(ℝ^{2}) obtained by projecting the translation component of each element on the plane orthogonal to p.

Then Hol′ is the holonomy representation of a 3-dimensional Euclidean cone-manifold M′_{par}. (We do not discuss whether Hol = Hol′ in all cases.) By construction M′_{par} contains a complete surface orthogonal to the singular locus, say S. This surface has at least two singular points, and each of its singular points has angle equal to the angle of the corresponding cone singularity of M′, so those angles are less than θ. Since θ < π it follows from the Gauss-Bonnet formula that S is homeomorphic to a sphere, and that it has at most three cone singularities.

Still by construction, the fundamental group of M′_{reg} surjects to the fundamental group of M′_{par}. However this surjection is actually an isomorphism, since otherwise an element of π_{1}(M′_{reg}) would act trivially on S, which means that it would act on M′_{reg} by translations parallel to the invariant direction, and this is impossible since M′_{reg} is non-compact.

This shows that M′ is homeomorphic to the product of a sphere by a line, and that it has three cone singularities. Therefore this is also true of all M_{n} for n large enough. But this is impossible since the M_{n} are quasifuchsian manifolds with particles, and the definition explicitly excludes manifolds homeomorphic to the product of a sphere by a line with three singularities.

We now call ϵ_{0} > 0 the number ϵ associated by the previous two lemmas to the maximum of the θ_{i}, and ρ_{0} the corresponding value of ρ.

3 Compactness statements

3.1 Main statement.

The main goal of this section is to prove the following compactness lemma.

Lemma 3.1. Let M_{n} be a sequence of quasifuchsian manifolds with particles with the same topological type and converging angles. Let λ_{n} be the measured bending laminations on the boundary of the convex core of M_{n}. Suppose that λ_{n} → λ_{∞}, where λ_{∞} satisfies the hypothesis of Theorem 1.13. Then, after taking a subsequence, M_{n} converges to a quasifuchsian manifold with particles with the common topological type, the limit particles and measured bending lamination λ_{∞}.

Let us explain the definitions used in this statement. The topological type of a quasifuchsian manifold with particles M has the form (S, x_{1}, ⋯, x_{n_{0}}) where S is a compact surface with genus at least 2 and x_{1}, ⋯, x_{n_{0}} are distinct points on S. A quasifuchsian manifold with particles M has topological type (S, x_{1}, ⋯, x_{n_{0}}) if M is isometric to the product S × ℝ endowed with a complete hyperbolic metric with cone singularities of angles θ^{i} ∈ (0, π) on the lines {x_{i}} × ℝ.

Consider a sequence of quasifuchsian manifolds M_{n} with particles with the same topological type (S, x_{1}, ⋯, x_{n_{0}}). Denote by θ_{n}^{i} ∈ (0, π) the cone angles of the metric of M_{n} on {x_{i}} × ℝ. Then the sequence M_{n} has converging angles if and only if θ_{n}^{i} converges in (0, π) when n goes to ∞ for any i ≤ n_{0}.

Notice that since λ_{n} converge to λ_{∞}, λ_{n} is eventually non trivial. In particular the manifolds M_{n} are not fuchsian (i.e. their convex core is not a surface) except maybe for finitely many of them. Throughout this section, when we consider a quasifuchsian manifold with particle, we will assume that it is not fuchsian so that its convex core is a 3-dimensional manifold with boundary.

The convex core C(M) of M is homeomorphic to S × I. Thus ∂C(M) is homeomorphic to S ⊔ S and each copy of S in this union has k marked points x_{1}, ⋯, x_{n_{0}} corresponding to the endpoints of the particles. The measured bending lamination on the boundary of C(M) is an element of the space ML_{S,n_{0}} × ML_{S,n_{0}} of measured laminations on two copies of S with n_{0} marked points. The space ML_{S,n_{0}} is endowed with the topology of weak-∗ convergence of measures on compact transversals and ML_{S,n_{0}} × ML_{S,n_{0}} is endowed with the product topology. In simple terms, we can fix a finite but sufficiently large set of curves c_{i} which are either closed curves or segments between two singular points, then two measured laminations are close if and only if their intersection with each of the c_{i} are close.

Although Lemma 3.1 is a generalisation of the "Lemme de fermeture" of [BO04], the proof is very different. The reason is that the two main ingredients of the proof in Bonahon-Otal’s paper are Culler-Morgan-Shalen compactification of the character variety by actions on ℝ-trees and the covering Theorem of Canary. Since both these results hardly extend to manifolds with particles we had to use different arguments. Since our proof also works without particles, we get a new proof of the main result of [BO04].

3.2 A finite cover argument

We work under the assumption that the cone angle around each singularity is less than π. This assumption guarantees that the singularities are never too close to each other, see Lemma 2.8, and that the boundary of the convex core is well defined and is orthogonal to the singularities. On the other hand, cone singularities with cone angles less than π can be viewed has singularities with concentrated positive curvature. But some of the results we will use are easier to prove when the curvature is negative. To overcome this difficulty, we will use a branched cover for which the cone angles are all greater than 2π.

Let M be a quasifuchsian manifold with particles. A branched cover \overline{M} → M branched along the singularities is negatively curved if the cone angles around the singularities of the metric induced on \overline{M} are all greater than 2π. We call \overline{M} a negatively curved branched cover of M.

This name comes from the fact that a singularity with cone angle greater than 2π can be viewed as a set of concentrated negative curvature. More precisely \overline{M} can be approximated by Riemannian manifolds with curvature bounded above by −1 (in the bilipschitz topology). It follows that \overline{M} has properties of negatively curved manifolds, in particular the uniqueness of the geodesic segment joining two given points in a given homotopy class.

We will construct such branched covers for sequences. Consider a sequence of quasifuchsian manifold with particles M_{n} with the same topological type (S, x_{1}, ⋯, x_{n_{0}}) (as defined in the preceding section). We denote by g_{n} the metric of M_{n} and by θ_{n}^{i} the cone angles of g_{n} on κ_{i}. Assume that the sequence M_{n} has converging angles, namely θ_{n}^{i} converge to some θ^{i} ∈ (0, π) for any i ≤ n_{0}.

For each singularity x_{i}, we choose an integer k_{i} such that \frac{2π}{k_{i}} is less than the angle θ^{i} (the limit of θ_{n}^{i}). The surface S with cone angle \frac{2π}{k_{i}} at the point x_{i} is a hyperbolic orbifold. As such it has a manifold cover h : S̄ → S which is a branched cover so that the lifts of the point x_{i} have a branching index equal to k_{i}. The branched cover h : S̄ → S extends naturally to a branched cover h : S̄ × I → S × I.

For a fixed n, we have the metric g_{n} on S × ℝ with cone singularities θ_{n}^{i} along {x_{i}} ∩ ℝ. If we pull back g_{n} using the map h, we get a hyperbolic metric ḡ_{n} with cone singularities on S̄ × ℝ for which the covering transformations are isometries. Let \overline{M}_{n} = (S̄ × ℝ, ḡ_{n}) be the manifold with cone singularities thus obtained. By the choice of {k_{1}, ..., k_{n_{0}}}, for n large enough, we have k_{i}θ_{n}^{i} ≥ 2π, hence the cone angle of ḡ_{n} around each singularity of \overline{M} is at least 2π. Thus for n large enough, \overline{M}_{n} is a negatively curved branched cover of M_{n} and the topological type of \overline{M}_{n} does not depend on n.

3.3 Pleated annuli

A technical device which will be useful later on is a simplicial annulus bounded by two given curves. As was mentioned above, when we consider a quasifuchsian manifold with particle, we assume that it is not fuchsian.

Let us first fix some notations. We consider a quasifuchsian manifold with particles M with topological type (S, x_{1}, ⋯, x_{n_{0}}). We denote by g the complete hyperbolic metric with cone singularities of M and by C(M) the convex core of M. We will use a negatively curved branched cover \overline{M} of M (as defined in the previous section). The construction of such a cover is explained above for a sequence M_{n}, here we take the constant sequence, M_{n} = M for any n, to define \overline{M}. We denote by C(\overline{M}) ⊂ \overline{M} the preimage of the convex core C(M) of M under the covering projection, by λ̄ its bending measured geodesic lamination and by m̄ the induced metric on ∂C(\overline{M}). We will use these notations throughout this section.

Now let us construct our simplicial annulus.

Lemma 3.2. Let M be a quasifuchsian manifold with particles and \overline{M} a negatively curved branched cover of M. Let d̄, d̄′ be homotopic simple closed geodesics, respectively on the upper and on the lower boundary component of C(\overline{M}). There exists an immersed annulus Ā in C(M̄) bounded by d̄∪ d̄′ ⊂ ∂C(M̄) such that the metric induced on Ā by g_{n} is a hyperbolic metric with cone singularities with angles at least 2π. The area of Ā is at most max{lm̄(d̄) + lm̄(d̄′), i(λ̄, d̄) + i(λ̄, d̄′)}.

Proof. Let us specify that lm̄(d̄), resp. lm̄(d̄′), is the length of d̄, resp. d̄′, with respect to the metric m̄ induced by ḡ on ∂C(\overline{M}).

Since d̄ and d̄′ are disjoint homotopic simple closed curves, there is an embedded annulus Ā ⊂ C(\overline{M}) with ∂Ā = d̄∪ d̄′. If the bending laminination of C(\overline{M}) intersects d̄ and d̄′ finitely many times, then d̄ and d̄′ are piecewise geodesics. If not, we approximate them by piecewise geodesic curves and work on the approximates. Consider a triangulation T of Ā whose vertices are all contained in d̄∪ d̄′ and such that any vertex of d̄ and d̄′ (when considered as piecewise geodesics) is a vertex of T. As we have said before, in \overline{M}, there is a unique geodesic segment joining 2 given points in a given homotopy class. It follows that we can change Ā by a homotopy so that each edge of T is a geodesic segment in C(\overline{M}). Next, for each triangle T_{i} of T, we choose a vertex v and we substitute T_{i} by the geodesic cone from v to the edge e_{v} of T_{i} not containing v. This geodesic cone is the union of the geodesic segments joining v to the edge e_{v} of T_{i} (the homotopy class of such segment is defined by the corresponding segment of T_{i}). Again the existence of this cone follows from the uniqueness of geodesic paths. From now on we denote this cone by T_{i}. By construction, it is a locally ruled surface and as such has negative curvature:

Claim 3.3. Let M be a hyperbolic manifold with cone singularities with cone angles bigger than 2π. Given a point v ∈ M and a geodesic segment e_{v} ⊂ M, a geodesic cone T_{i} ⊂ M from v to e_{v} is an union of polygons with curvature −1. Furthermore, the sum of the angles of the polygons meeting at an interior vertex is at least 2π.

Proof. The surface T_{i} meets the singular locus \overline{M}_{s} of \overline{M} along segments and at points. For each component κ of \overline{M}_{s} ∩ T_{i} we consider the two extremal segments joining v to e_{v} and intersecting κ. Doing this for each component of \overline{M}_{s} ∩ T_{i}, we get a family of segments which are geodesic for the metric of (\overline{M}, ḡ) and hence for the induced metric on T_{i}. We add the components of \overline{M}_{s} ∩ T_{i} which are segments to this family and get a new family of geodesic segments. The closure of each complementary region is a polygon, i.e. a disc with piecewise geodesic boundary (see Figure 3.3). By construction each such polygon is a locally ruled surface in ℍ^{3} hence it has curvature −1. Thus we have proved the first sentence of this Claim.

By construction, given an interior vertex v of this decomposition into polygons, there is a geodesic segment (for the metric of (\overline{M}, ḡ)) which passes through v. On each side of this segment, the sum of the angles of the polygons has to be at least π. Thus, we can conclude that the sum of the angles of the polygons around v is at least 2π.

PSfrag replacements \overline{M}_{s} ∩ T_{i}

Figure 1: Decomposition of T_{i} into hyperbolic polygons

We change the annulus Ā so that it is a union of geodesic cones as described in Claim 3.3. Thus the induced metric is hyperbolic with cone singularities with angles greater than 2π. By the Gauss-Bonnet Formula, the area of Ā is at most the bending of ∂Ā, namely Area(Ā) ≤ i(∂Ā, λ̄_{n}) = i(d̄, λ̄_{n}) + i(d̄′, λ̄_{n}).

It remains to prove that Area(Ā) ≤ lm̄(d̄) + lm̄(d̄′). By construction, Ā is a union of triangles T_{i} such that one edge of each T_{i} lies in d̄∪d̄′ and by Claim 3.3, the induced metric on each such triangle is a hyperbolic metric with cone singularities with angles greater than 2π. It follows that the induced metric can be approximated by Riemannian metrics with curvature at most −1. Let T_{i}^{h} be a hyperbolic triangle (i.e. a geodesic triangle in ℍ^{2}) such that the length of the edges of T_{i}^{h} are the same as the length of the edges of T_{i}. Since the induced metric on T_{i} as curvature at most −1, we have Area(T_{i}) ≤ Area(T_{i}^{h}). On the other hand the area of a hyperbolic triangle is less than the length of any of its edges (see [Thu80, Lemma 9.3.2]). It follows that Area(T_{i}) is less than the length of any of its edges, in particular it is less than the length of the edge of T_{i} lying in d̄∪ d̄′. Since this holds for all the triangles composing Ā, we have Area(Ā) ≤ lm̄(d̄) + lm̄(d̄′).

3.4 Long geodesics in M.

In this section, we will show that, under the hypothesis of Lemma 3.1, the induced metrics on ∂C(M_{n}) are bounded. In order to do that we will show that if some geodesic is long in the boundary of C(M), then the boundary of some annulus is almost not bent or the bending lamination tends to have a leaf with a weight greater than or equal to π. Since this would contradict the conditions on λ_{∞}, it will follow that any given simple closed curve on ∂C(M_{n}) has bounded length. As earlier, when we consider a quasifuchsian manifold with particle, we assume that it is not fuchsian.

Throughout this section we use the following notations. We have a sequence of quasifuchsian manifolds with particles M_{n} with the same topological type (S, x_{1}, ⋯, x_{n_{0}}). We denote by (g_{n})_{n∈ℕ} the metric of M_{n}. We assume that the sequence M_{n} has converging angles, namely θ_{n}^{i} converge in (0, π) for any i ≤ n_{0}. Since M_{n} is quasifuchsian (with particles), C(M_{n}) is homeomorphic to S ×I. We denote by S and S′ the two components of ∂C(M_{n}), S′ is homeomorphic to S. Let m_{n} be the metric defined on S ⊔ S′ by the identification with ∂C(M_{n}) endowed with the metric induce by the g_{n}-length of paths. This metric m_{n} is a hyperbolic metric with cone singularities of angles θ_{n}^{i} at the points {x_{i}} ∈ S and {x′_{i}} ∈ S′. We denote by λ_{n} ∈ ML_{S,n_{0}} × ML_{S,n_{0}} the bending measured geodesic lamination of ∂C(M_{n}).

We will make use of the branched covers \overline{M}_{n} defined in Section 3.2. Recall that, for n large enough, \overline{M}_{n} is a negatively curved branched cover of M_{n} and that the topological type of \overline{M}_{n} does not depend on n. We denote by C(\overline{M}_{n}), m̄_{n} and λ̄_{n} the preimages of C(M_{n}), m_{n} and λ_{n} under the covering map \overline{M_{n}} → M_{n}.

We will use the next lemma to prove that, under the right hypothesis on λ_{∞}, the induced metric on ∂C(M_{n}) is bounded.

Lemma 3.4. Let M_{n} be a sequence of quasifuchsian manifolds with particles with the same topological type and converging angles. Assume that (λ_{n}) converges to λ_{∞} (without any hypothesis on λ_{∞}). Consider a simple closed curve d ⊂ S. Let d_{n} ⊂ ∂C(M_{n}) be the closed m_{n}-geodesic freely homotopic to d. If l_{m_{n}}(d_{n}) ⟶ ∞, then either λ_{∞} contains a leaf with a weight greater than or equal to π, or there is a sequence of essential annuli E_{n} such that i(λ_{n}, ∂E_{n}) ⟶ 0.

Proof. Let S′ be the other boundary component of C(M) (i.e. not S), and let d′_{n} be the closed m_{n}-geodesic freely homotopic to d lying in S′. Let \overline{M}_{n} be a negatively curved branched cover of M_{n} so that for n large enough the topological type of \overline{M}_{n} does not depend on n. Let d̄_{n} and d̄′_{n} ⊂ ∂C(\overline{M}_{n}) be homotopic lifts of d_{n} and d′_{n} respectively under the covering projection \overline{M} → M. The preimage λ̄_{n} ∈ ML(∂\overline{M}) of λ_{n} is the bending measured lamination of C(\overline{M}_{n}). Furthermore λ̄_{n} converges to the preimage λ̄_{∞} of λ_{∞}.

First we will show that if d̄_{n} is long compared to the area of an annulus \overline{A}_{n} bounded by d̄_{n} ∪ d̄′_{n} then there are shortcuts in \overline{A}_{n}. Namely d̄_{n} ∪ d̄′_{n} contains points which are close to each other in \overline{M}_{n} but far in d̄_{n} ∪ d̄′_{n}. This can happen for instance if d_{n} and d′_{n} are close to each other in C(M_{n}).

Claim 3.5. Let M_{n} be a sequence of quasifuchsian manifolds with particles and let \overline{M}_{n} be a negatively curved branched cover of M_{n} such that the topological type of \overline{M}_{n} does not depend on n. Assume that (λ̄_{n})_{n∈ℕ} converges to λ̄_{∞}. Let d̄ ⊂ S̄ be a simple closed curve and denote by d̄_{n} ⊂ S̄ ⊂ ∂C(\overline{M}_{n}) the simple closed m̄_{n}-geodesic in the homotopy class of d̄.

If lm̄_{n}(d_{n}) ⟶ ∞, then there is a m̄_{n}-geodesic arc k̄_{n} ⊂ C(\overline{M}_{n}) such that ℓm̄_{n}(k̄_{n}) ⟶ 0 and that either k̄_{n} joins the two components of ∂C(\overline{M}_{n}) or the m̄_{n}-geodesic arc κ̄_{n} ⊂ d̄_{n} in the homotopy class of k̄_{n} relative to its boundary satisfies ℓm̄_{n}(κ̄_{n}) ⟶ ∞.

Proof. Denote by d̄′_{n} the closed m̄_{n}-geodesic lying in S̄′ ⊂ ∂C(\overline{M}_{n}) that is homotopic to d̄_{n} in C(\overline{M}_{n}). Consider the annulus Ā_{n} with ∂Ā_{n} = d̄_{n} ∪d̄′_{n} that was constructed in Lemma 3.2. Since lm̄_{n}(d̄_{n}) ⟶ ∞, there is ε_{n} ⟶ 0 and a segment s̄_{n} ⊂ d̄_{n} such that lm̄_{n}(s̄_{n}) ⟶ ∞ and i(s̄_{n}, λ̄_{n}) ≤ ε_{n}. Let t̄_{n} ⊂ C(\overline{M}_{n}) be the ḡ_{n}-geodesic segment homotopic to s̄_{n} relative to its endpoints. Since s̄_{n} is almost not bent, its length is very close to the length of t̄_{n} (see [Lec06, Lemme A2]). In particular, lḡ_{n}(t̄_{n}) ⟶ ∞. Furthermore, for the same reason, any point in s̄_{n} is close to t̄_{n}. Namely there is η_{n} = η(ε_{n}) ⟶ 0 such that for any point z̄_{n} ⊂ s̄_{n}, there is x̄_{n} ⊂ t̄_{n} with dḡ_{n}(x̄_{n}, z̄_{n}) ≤ η_{n} (see [Lec06, Affirmation A3]).

Since (λ̄_{n})_{n∈ℕ} converges to λ̄_{∞}, then the bending i(λ̄_{n}, ∂Ā_{n}) of ∂Ā_{n} converges. By Lemma 3.2 the area of Ā_{n} is bounded. Now, in d̄_{n}, we replace s̄_{n} by t̄_{n}. By the previous paragraph, we can still consider the annulus Ā_{n} and its area is bounded. For any point in t̄_{n} that is at distance at least \frac{1}{3}lm̄_{n}(t̄_{n}) from ∂t̄_{n}, we consider in Ā_{n} an arc orthogonal to t̄_{n} that either hits ∂Ā_{n} at distance less than η_{n} from its basepoint or has length η_{n} (η_{n} will be specified later on). Let Z̄_{n} ⊂ Ā_{n} be the union of those arcs that have length η_{n} and let z̄_{n} be the union of their starting points (i.e. their intersection with t̄_{n}). The set Z̄_{n} is embedded and its area is the same as the area of a strip of length ℓm̄_{n}(z̄_{n}) and width η_{n}. Notice that since the singularities of Ā_{n} have cone angles at least 2π, the area of this strip at least the area of a hyperbolic strip with the same length and width, i.e. it is at least ℓm̄_{n}(z̄_{n}) sinh(η_{n}). Let K be a number larger than the area of Ā_{n}. Taking η_{n} such that sinh(η_{n}) > \frac{3K}{ℓm̄n (t̄_{n})}, we get

K ≥ Area(Z̄_{n}) ≥ ℓm̄_{n}(z̄_{n}) sinh(η_{n}) > K \frac{3ℓm̄_{n}(z̄_{n})}{ℓm̄_{n}(t̄_{n})} .

Hence ℓm̄_{n}(z̄_{n}) < \frac{1}{3}ℓm̄_{n}(t̄_{n}). It follows that there exists an arc with length less than η_{n} orthogonal to t̄_{n} whose starting point x̄′_{n} ⊂ t̄_{n} is at a distance at least \frac{1}{3}lm̄_{n}(t̄_{n}) from ∂t̄_{n} (distance measured on t̄_{n}) and which hits ∂Ā_{n} on a point ȳ_{n} ⊂ ∂Ā_{n}.

As we have seen in the previous paragraph there is a point x̄_{n} ⊂ k̄_{n} very close to x̄′_{n}. It follows that x̄_{n} ⊂ d̄_{n} and ȳ_{n} ⊂ d̄_{n} ∪ d̄′_{n} are joined in C(\overline{M}_{n}) by an arc k̄_{n} satisfying ℓ_{g_{n}}(k̄_{n}) ⟶ 0.

If ȳ_{n} ∈ d̄′_{n} then we are done. Otherwise x̄_{n} and ȳ_{n} both lie in d̄_{n}. By construction k̄_{n} lies in an annulus connecting d̄_{n} to d̄′_{n}. It follows that there is a m̄_{n}-geodesic arc κ̄_{n} ⊂ d̄_{n} that is homotopic to k̄_{n} relative to \{x̄_{n}\} ∪ \{ȳ_{n}\}. Since x̄_{n} is at distance at least \frac{lm̄n (t̄_{n})}{3} from the points in ∂s̄_{n}, we have

ℓm̄_{n}(κ̄_{n}) ≥ \frac{lm̄_{n}(t̄_{n})}{3} ⟶ ∞ .

Consider the points x̄_{n} and ȳ_{n} constructed in Claim 3.5 and extract a subsequence such that either ȳ_{n} ∈ d̄_{n} for any n or ȳ_{n} ∈ d̄′_{n} for any n. We will show below that if ȳ_{n} lies in d̄_{n} then λ_{∞} has a leaf with a weight greater than or equal to π and that if ȳ_{n} lies in d̄′_{n} then there is a sequence of essential annuli E_{n} ⊂ M such that i(λ_{n}, ∂E_{n}) ⟶ 0.

In the next step we are going to construct m̄_{n}-geodesic loops based at x̄_{n} and ȳ_{n} that are almost not bent.

Claim 3.6. Let M_{n} be a sequence of quasifuchsian manifolds with particles and let \overline{M}_{n} be a negatively curved branched cover of M_{n} such that the topological type of \overline{M}_{n} does not depend on n. Consider 2 points x̄_{n}, ȳ_{n} ∈ ∂C(\overline{M}_{n}) away from the singularities and a ḡ_{n}-geodesic arc k̄_{n} ⊂ C(\overline{M}_{n}) joining x̄_{n} to ȳ_{n} such that ℓḡ_{n}(k̄_{n}) ⟶ 0 and that either x̄_{n} and ȳ_{n} lie on different components of ∂C(M_{n}) or there is a m̄_{n}-geodesic arc κ̄_{n} ∈ ∂C(\overline{M}_{n}) that is homotopic to k̄_{n} relative to its boundary {x̄_{n}} ∪ {ȳ_{n}} and that satisfies ℓm̄_{n}(κ̄_{n}) ⟶ ∞.

Consider a loop l̄_{n} on ∂C(\overline{M}_{n}) based at x̄_{n}, which is geodesic for m̄_{n} (except at x̄_{n}). Let f̄_{n} ⊂ ∂C(\overline{M}_{n}) be the m̄_{n}-geodesic loop based at ȳ_{n} that is homotopic to l̄_{n}. Assume that ℓm̄_{n}(l̄_{n}) is bounded. Then the bending of l̄_{n} and f̄_{n} tends to 0, namely i(l̄_{n}, λ̄_{n}) ⟶ 0 and i(f̄_{n}, λ̄_{n}) ⟶ 0.

Proof. When saying that x̄_{n} and ȳ_{n} are away from the singularities we mean that there is a uniform upper bound on their distance to the singular locus of \overline{M}_{n}.

Let M̃_{n} be the universal cover of \overline{M}_{n}, it is a simply connected hyperbolic 3-manifold with cone singularities. Let C(M̃_{n}) be the lift of C(\overline{M}_{n}) to M̃_{n}. Let l̃_{n}, k̃_{n}, x̃_{n}, ỹ_{n} be lifts of l̄_{n}, k̄_{n}, x̄_{n} and ȳ_{n} with x̃_{n} ∈ l̃_{n} and x̃_{n} ∪ ỹ_{n} = ∂k̃_{n}. The point x̄′_{n} = ∂l̃_{n} \ x̃_{n} is the image of x̄_{n} under a covering transformation. Consider the m̃_{n}-geodesic arc f̃_{n} ⊂ ∂C(M̃_{n}) joining ȳ_{n} to its image ȳ′_{n} under this covering transformation.

Let us first assume that there are no singularities in \overline{M}_{n}. Then M̃_{n} is isometric to ℍ^{3} and we choose the isometry so that x̃_{n} is identified with a fixed point of ℍ^{3} (independantly of n). Let Π(x̃_{n}) be a support plane for C(M̃_{n}) at x̃_{n}, namely a totally geodesic plane that intersects C(M̃_{n}) only along ∂C(M̃_{n}) and contains x̃_{n}. Up to moving x̄_{n} slightly, we may assume that it is disjoint from λ̄_{n} so that there is only one support plane at x̃_{n}. The convex set C(M̃_{n}) lies in a half-space E(x̃_{n}) bounded by Π(x̃_{n}). Similarly let Π(ỹ_{n}) be a support plane at ỹ_{n} and let E(ỹ_{n}) be the half-space bounded by Π(ỹ_{n}) that contains C(M̃_{n}).

If x̄_{n} and ȳ_{n} lie on different components of ∂C(M_{n}) then Π(x̃_{n}) and Π(ỹ_{n}) are disjoint. Otherwise, since ℓm̄_{n}(κ̄_{n}) ⟶ ∞, either Π(x̃_{n}) and Π(ỹ_{n}) are disjoint or their intersection goes to ∞ with n (namely the sequence Π(x̃_{n})∩Π(ỹ_{n}) lies outside larger and larger compact sets in ℍ^{3}). Since ℓḡ_{n}(k̄_{n}) ⟶ 0, ỹ_{n} converges to x̃_{n} (viewed as a fixed point in ℍ^{3}) and, up to extracting a subsequence, Π(x̃_{n}) and Π(ỹ_{n}) converges to the same plane Π_{∞} in ℍ^{3}. Furthermore E(x̃_{n}) converges to a half-space E(x̃_{∞}) bounded by Π_{∞} and E(ỹ_{n}) converges to the other half-space E(ỹ_{∞}) bounded by Π_{∞}.

Let Π(x̃′_{n}) be a support plane at x̃′_{n}. Since ℓm̄_{n}(f̄_{n}) is bounded, up to extracting a subsequence, x̃′_{n} converges in ℍ^{3}. Again since either x̄′_{n} and ȳ_{n} lie on different components of ∂C(M_{n}) or dm̃_{n}(x̃′_{n}, ỹ_{n}) ⟶ ∞, Π(ỹ_{n})∩Π(x̃′_{n}) either is empty or goes to infinity. It follows that Π(x̃′_{n}) also converges to Π_{∞} and that E(x̃′_{n}) converges to E(x̃_{∞}). The external dihedral angle between Π(x̃_{n}) and Π(x̃′_{n}) is an upper bound for i(l̄_{n}, λ̄_{n}), hence i(l̄_{n}, λ̄_{n}) ⟶ 0.

It remains to show that i(f̄_{n}, λ̄_{n}) ⟶ 0.

By construction, dg̃_{n}(x̃′_{n}, ỹ′_{n}) = dg̃_{n}(x̃′_{n}, ỹ′_{n}) = ℓḡ_{n}(k̄_{n}) ⟶ 0, hence Π(x̃′_{n}) and Π(ỹ′_{n}) converge to the same plane Π_{∞} in ℍ^{3}. If x̄_{n} and ȳ_{n} lie on different components of ∂C(M_{n}) then Π(x̃′_{n}) and Π(ỹ′_{n}) are disjoint. It follows that E(ỹ′_{n}) converges to E(ỹ_{∞}) while E(x̃′_{n}) converges to E(x̃_{∞}) which implies i(f̄_{n}, λ̄_{n}) ⟶ 0 as above. If x̄_{n} and ȳ_{n} lie on the same component of ∂C(M_{n}) then dm̃_{n}(x̃′_{n}, ỹ′_{n}) = dm̃_{n}(x̃_{n}, ỹ_{n}) ⟶ ∞. It follows that E(ỹ′_{n}) converge to E(ỹ_{∞}) which again implies i(f̄_{n}, λ̄_{n}) ⟶ 0.

When \overline{M}_{n} have singularities, we cannot define support planes, but we can define local support planes at points which are disjoint from the singularities. Thus we can locally use the same arguments as in the non singular case, leading to the same conclusion.

Let us choose for l̄_{n} a shortest m̄_{n}-geodesic loop based at x̄_{n}. Since the area of (S̄ ⊔ S̄′, m̄_{n}) is bounded, there is a constant Q > 0 such that ℓm̄_{n}(l̄_{n}) ≤ Q. By Claim 3.6, we have i(l̄_{n}, λ̄_{n}) ⟶ 0. Let f̄_{n} be the m̄_{n}-geodesic loop based at ȳ_{n} that is homotopic to l̄_{n}. By Claim 3.6, we have i(f̄_{n}, λ̄_{n}) ⟶ 0. Since l̄_{n} and f̄_{n} are freely homotopic in C(\overline{M}_{n}), there is an annulus Ē_{n} bounded by l̄_{n} and f̄_{n}.

If ȳ_{n} lies in d̄′_{n}, then l̄_{n} and f̄_{n} lie in different components of ∂\overline{M}. In particular Ē_{n} is an essential annulus for any n. Furthermore, we have i(λ̄_{n}, ∂Ē_{n}) ≤ i(l̄_{n}, λ̄_{n}) + i(f̄_{n}, λ̄_{n}) ⟶ 0.

Consider the projection E_{n} of Ē_{n} to C(M_{n}). Although E_{n} may not be embedded, it follows from the Annulus Theorem [Wal67] that any neighbourhood of E_{n} contains an embedded annulus which we still denote by E_{n}. We have then i(λ_{n}, ∂E_{n}) ⟶ 0.

Thus we have proved:

Claim 3.7. Let M_{n} be a converging sequence of quasifuchsian manifolds with particles with the same topological type and converging angles. Consider a geodesic arc k_{n} ⊂ C(M_{n}) joining the two components of ∂C(M_{n}) such that ℓ_{m_{n}}(k_{n}) ⟶ 0. Then there is a sequence of essential annuli E_{n} such that, up to extracting a subsequence, i(λ_{n}, ∂E_{n}) ⟶ 0. □

If ȳ_{n} lies in d̄_{n} then l̄_{n} and f̄_{n} are homotopic on ∂C(\overline{M}_{n}). We are going to show that in this case λ_{n} tends to have a leaf with a weight greater than or equal to π.

Claim 3.8. Let M_{n} be a sequence of quasifuchsian manifolds with converging angles and let \overline{M}_{n} be a negatively curved branched cover of M_{n} such that the topological type of \overline{M}_{n} does not depend on n. Consider a ḡ_{n}-geodesic arc k̄_{n} ⊂ C(\overline{M}_{n}) such that ℓḡ_{n}(k̄_{n}) ⟶ 0 and that there is a m̄_{n}-geodesic arc κ̄_{n} ∈ ∂C(\overline{M}_{n}) that is homotopic to k̄_{n} relative to its boundary and that satisfies ℓm̄_{n}(κ̄_{n}) ⟶ ∞. Then lim inf i(κ̄_{n}, λ̄_{n}) ≥ π.

Proof. The curve k̄_{n} ∪ κ̄_{n} is a skew polygon (up to approximating κ̄_{n} by piecewise geodesic segments) and bounds a disc in C(\overline{M}_{n}). Consider the geodesic cone D̄_{n} from x̄_{n} to k̄_{n} ∪ κ̄_{n}. As in the proof of Lemma 3.2, the induced metric on D̄_{n} is a hyperbolic metric with cone singularities with cone angles of at least 2π. Since k̄_{n} is short, the local support planes at the endpoints x̄_{n} and ȳ_{n} of k̄_{n} are close to each other (compare with the proof of Claim 3.6). It follows that the sum of the internal angles of D̄_{n} at x̄_{n} and ȳ_{n} is close to being greater than π, namely there is ε_{n} ⟶ 0 such that the sum of these 2 angles is greater than π − ε_{n}. Now the Gauss-Bonnet formula shows that lim inf i(κ̄_{n}, λ̄_{n}) ≥ π.

Using this Claim we will now show that under the right hypothesis λ_{n} tends to have a leaf with a weight greater than or equal to π.

Claim 3.9. Let M_{n} be a sequence of quasifuchsian manifolds with particles with the same topological type S × I and converging angles. Let d ⊂ S be a simple closed curve and consider its geodesic representative d_{n} on one component of ∂C(M_{n}). Consider an arc κ_{n} ⊂ d_{n} and denote by k_{n} the geodesic arc in M_{n} in the homotopy class of κ_{n} relative to its boundary. If ℓ_{g_{n}}(k_{n}) ⟶ 0, ℓ_{m_{n}}(κ_{n}) ⟶ ∞ and λ_{n} converges, its limit λ_{∞} has a leaf with a weight greater than or equal to π.

Proof. Let \overline{M}_{n} be a negatively curved branched cover of M_{n} whose topological does not depend on n. Let κ̄_{n} and k̄_{n} be lifts of κ_{n} and k_{n} respectively. Let x̄_{n} and ȳ_{n} be the endpoints of κ̄_{n} and let l̄_{n} ⊂ ∂C(\overline{M}_{n}) be a shortest geodesic loop based at x̄_{n}. Let f̄_{n} ⊂ ∂C(M_{n}) be the geodesic loop based at ȳ_{n} that is homotopic to l̄_{n} on ∂C(\overline{M}_{n}). Let S̃_{n} be the universal cover of the connected component of ∂C(\overline{M}_{n}) containing x̄_{n} endowed with the induced metric. Pick a connected component l̃_{n} ⊂ S̃_{n} of the preimage of l̄_{n} under the covering projection. This broken geodesic l̃_{n} is invariant under a primitive covering transformation γ_{n} and we denote by f̃_{n} the component of the preimage of f̄_{n} that is also invariant under γ_{n}. The line l̃_{n} and f̃_{n} are disjoint and bound an infinite band B̃_{n}, they are connected by a lift κ̃_{n} of κ̄_{n} and by its translates by γ_{n}^{k}, k ∈ ℤ. Pick a simple closed geodesic ē_{n} ⊂ ∂C(\overline{M}_{n}) and let ẽ_{n} be a lift of ē_{n} to S̃_{n}. It is easy to check that a component ã_{n} of ẽ_{n} ∩ B̃_{n} which is an arc connecting l̃_{n} to f̃_{n} satisfies i(ã_{n}, λ̃_{n}) ≥ i(λ̃_{n}, κ̃_{n}) − ♯{ã_{n} ∩ ⋃_{k} g_{n}^{k}κ̃_{n}}(i(l̄_{n}, λ̄_{n}) + i(f̄_{n}, λ̄_{n})) where λ̃_{n} is the preimage of λ̄_{n} under the covering projection. By Claim 3.6, i(l̄_{n}, λ̄_{n}) ⟶ 0 and i(f̄_{n}, λ̄_{n}) ⟶ 0 and by Claim 3.8, lim inf i(κ̄_{n}, λ̄_{n}) ≥ π. Notice that ♯{ã_{n} ∩ ⋃_{k} g_{n}^{k}κ̃_{n}} is bounded except if ē_{n} spirals more and more toward c̄_{n}. If, for example, we assume that ē_{n} and c̄_{n} converge to intersecting geodesic laminations, ē_{n} does not spiral toward c̄_{n} and we find that lim inf i(ē_{n}, λ̄_{n}) ≥ i(ē_{n}, c̄_{n})π.

We will now use this inequality to conclude that, when λ̄_{n} converge, its limit λ̄_{∞} has a leaf with a weight greater than or equal to π.

Let us notice that, up to extracting a subsequence, the homotopy class of c̄_{n} does not depend on n. Otherwise there is a simple closed curve ē ⊂ S̄ such that i(ē, c̄_{n}) ⟶ ∞. To see that, extract a subsequence such that c̄_{n} converges in the Hausdorff topology, pick a simple closed curve ē that intersects this limit transversally and apply the inequality above to {ē_{n}} = {ē}. But i(ē, c̄_{n}) ⟶ ∞ would contradict the assumption that λ̄_{n} converge.

Let c̄ ⊂ S be a simple closed curve in the homotopy class defined by c̄_{n}. By the inequality above, we have lim inf i(ē, λ̄_{n}) ≥ i(ē, c̄)π for any simple closed curve ē. It follows easily that c̄ is a leaf of λ̄_{n} with a weight greater than or equal to π. Taking the quotient, we conclude that λ_{∞} has a a leaf with a weight greater than or equal to π.

It is now easy to conclude the proof of Lemma 3.4. Under the assumptions of Lemma 3.4, namely when there is a simple closed curve d such that ℓ_{m_{n}}(d_{n}) ⟶ ∞, it follows from Claims 3.5, 3.7 and 3.9 that either there is a sequence of essential annuli E_{n} ⊂ C(M_{n}) such that i(λ_{n}, ∂A_{n}) ⟶ 0 or λ_{∞} contains a leaf with a weight equal to at least π.

We can now deduce from Lemma 3.4 that under the assumptions of Lemma 3.1, the sequence of induced metrics (m_{n})_{n∋ℕ} on ∂C(M_{n}) is bounded.

Lemma 3.10. Let M_{n} be a sequence of quasifuchsian manifolds with particles with the same topological type and converging angles. Let λ_{n} be the measured bending laminations on the boundary of the convex core of M_{n} and suppose that λ_{n} → λ_{∞}. Let λ^{±} be the respective restrictions of λ_{∞} to the two components of the boundary ∂C(M_{n}) of the convex core of M_{n}. Suppose that

• λ_{−} and λ_{+} fill S,

• each closed curve in the support of λ_{−} (resp. λ_{+}) has weight less than π.

Then the sequence of induced metrics (m_{n})_{n∈ℕ} on ∂C(M_{n}) is bounded.

Proof. If (m_{n}) is unbounded, then there is a simple closed curve d ⊂ S with geodesic representative d_{n} ⊂ ∂C(M_{n}) such that l_{m_{n}}(d_{n}) is unbounded. By Lemma 3.4 and the assumptions on λ, there is a sequence E_{n} of essential annuli such that i(λ_{n}, ∂E_{n}) ⟶ 0. Such a sequence of annuli contradicts the assumption that λ_{−} and λ_{+} fill S.

3.5 Convergence of convex cores

The last step in the proof of Lemma 3.1 is to show that, under the assumption that the sequence of metrics on the boundary are bounded, a subsequence of convex cores converges for the bilipschitz topology. Before starting the proof, we will discuss the Margulis Lemma for quasifuchsian manifolds with particles. Let us first review the Margulis Lemma for manifolds with variable curvature.

Theorem 3.11 (Margulis Lemma). Given n ∈ ℕ there are constant μ = μ(n) > 0 and I(n) ∈ ℕ with the following property. Let X be an n-dimensional Hadamard manifold which satisfies the curvature condition −1 ≤ K ≤ 0 and let Γ be a discrete group of isometries acting on X. For x ∈ X let Γ_{μ}(x) = {γ ∈ Γ|d_{γ}(x) ≤ μ} be the subgroup generated by the elements γ with d_{γ}(x) ≤ μ. Then Γ_{μ}(x) is almost nilpotent, thus it contains a nilpotent subgroup of finite index. The index is bounded in I(n).

This statement is taken from [BGS85, §8.3]. Since we are considering a manifold M homeorphic to S × I, an almost nilpotent subgroup of π_{1}(M) is cyclic.

Theorem 3.11 does not hold for hyperbolic manifolds with cone singularities since the curvature is not defined at the singularities. On the other it is not hard to replace the metric in a neighborhood of the singular locus with a Riemannian metric. Furthermore, if the cone angle is at least 2π, one can choose the Riemannian metric so that it has negative curvature with a lower bound depending on the cone angles and the choice of the neighborhood of the singularities. Now we consider a quasifuchsian manifold with particle M and a negatively curved branched cover \overline{M} of M. By Lemma 2.8, there are R, ε > 0 such that any closed curve with length at most ε is at distance at least R from the singularities. We replace the R-neighborhood of the singularities with a smooth Riemannian metric and apply Lemma 3.11 to the resulting manifold. Notice that the lower bound on the curvature of the Riemannian metric thus obtained will depend on R and the cone angle. Thus we get ε depending on R and the cone angles so that for a given point x in the universal cover M̃ of \overline{M} the subgroup of π_{1}(\overline{M}) generated by the set \{γ ∈ π_{1}(\overline{M})|d(x, γx) ≤ ε\} is cyclic. Since \overline{M} is a finite branched cover, we have a similar statement for M, replacing ε with ε/p where p is the index of the cover which depends on the cone angles of the singularities of M. It follows that we have a Margulis decomposition for quasifuchsian manifolds with particles:

Lemma 3.12. Let M_{n} be a sequence of quasifuchsian manifolds with particles with the same topological type and converging angles. There is ε depending on the limit angles such that, for n large enough, each component of the ε-thin part of M_{n} is a neighborhood of a closed geodesic.

Notice that the Margulis tubes we obtain here are disjoint from the singularities so they are isometric to regular neighborhoods of geodesics in hyperbolic 3-manifolds. We can now discuss the convergence of quasifuchsian manifolds with particles.

Lemma 3.13. Let M_{n} be a sequence of quasifuchsian manifolds with particles with the same topological type and converging angles. Suppose that the sequence (m_{n})_{n∈ℕ} (the induced metrics on the boundary of the convex cores) converges. Then, after taking a subsequence, (M_{n})_{n∈ℕ} converges to a quasifuchsian manifold with particles with the same topological type as M_{n}.

Proof. First notice that since the cone angles are less than π, by Lemma 2.8 there is a positive lower bound for the distance between two components of the singularity locus. Consider a point x_{n} ∈ C(M_{n}), extract a subsequence such that the sequence (x_{n}, M_{n}) converges in the Gromov-Hausdorff topology (such a subsequence always exists). By [BP01, Proposition 3.2.6], the limit (x_{∞}, M_{∞}) is a hyperbolic manifold with cone singularities. By [BP01, Proposition 3.3.1], the sequence (x_{n}, M_{n}) converges to (x_{∞}, M_{∞}) in the bilipschitz topology.

It remains to show that M_{∞} has the same topological type as M_{n} and that its metric is convex co-compact. To do that we will show that the diameter of C(M_{n}) is uniformly bounded. It will follow that C(M_{n}) converges to a convex set with the same topological type.

Lemma 3.14. Let (M_{n})_{n∈ℕ} be a sequence of quasifuchsian manifolds with particles with the same topological type and converging angles. Suppose that the sequence of induced metrics (m_{n})_{n∈ℕ} on the boundary of the convex cores converges. Then the diameter of C(M_{n}) is uniformely bounded.

Proof. Consider a negatively curved ramified cover \overline{M}_{n} of M_{n} whose topological type does not eventually depend on n. It follows from the Margulis Lemma that a very short geodesic in \overline{M}_{n} lies in a very deep embedded tube. Using this observation we will show that there is a uniform lower bound on the length of any fixed curve in C(\overline{M}_{n}).

Claim 3.15. Let M_{n} be a sequence of quasifuchsian manifolds with particles and let \overline{M}_{n} be a negatively curved ramified cover of M_{n} whose topological type does not depend on n. Suppose that the sequence (m̄_{n})_{n∈ℕ} converges. Let c̄ ⊂ S̄ be a simple closed curve. Then there is Q > 0 such that if c̄_{n} ⊂ C(\overline{M}_{n}) denotes the geodesic representative of c̄, lḡ_{n}(c̄_{n}) ≥ Q for any n ∈ ℕ.

Proof. Assume the contrary, that is (after extracting a subsequence), lim lḡ_{n}(c̄_{n}) = 0. Then c̄_{n} is the core of a deeper and deep Margulis tube \overline{T}_{n}. Notice that since (m̄_{n}^{+}, m̄_{n}^{−})_{n∈ℕ} converges, there is no short curve in ∂C(\overline{M}_{n}), more precisely, there is a uniform lower bound on the length of any simple closed geodesic in ∂C(\overline{M}_{n}). Since the induced metric on ∂C(\overline{M}_{n}) is negatively curved, it can have a large diameter only if it contains a short curve. Thus the uniform lower bound on the length of any simple closed geodesic in ∂C(\overline{M}_{n}) provides us with a bound on the diameter of each component of ∂C(\overline{M}_{n}). It follows that ∂C(\overline{M}_{n}) does not go too deep in a Margulis tube (compare with [Min99, Lemma 6.3]):

Let ε_{0} be a Margulis constant for the sequence g_{n} as provided by Lemma 3.12, namely the ε_{0}-thin part M_{n}^{<ε_{0}} of M_{n} is a union of Margulis tubes for n large enough. Pick a number ε_{1} < ε_{0}. By [Mey87] and [?] (see also [Min99, Lemma 6.1]) if ε_{1} is small enough, the diameter of M_{n}^{<ε_{0}} − M_{n}^{<ε_{1}} is large. In particular, if a component of ∂C(\overline{M}_{n}) intersects M_{n}^{<ε_{1}}, it has a large diameter. Hence the bound on the diameter of each component of ∂C(\overline{M}_{n}) provides us with a lower bound ε so that ∂C(\overline{M}_{n}) is disjoint frolm the ε-thin part of M_{n}.

Thus taking \overline{T}_{n} to be an ε-Margulis tube, we get that \overline{T}_{n} lies entirely in the interior of C(\overline{M}_{n}). Consider a simple closed curve d̄⊂ S̄ that intersects c̄ essentially. By Lemma 3.2, there is an essential annulus \overline{A}_{n} ⊂ C(\overline{M}_{n}) which is in the homotopy class defined by d̄× I such that the area of \overline{A}_{n} is at most ℓ_{}_{m̄_{n}^{+}}(d̄_{+}) + ℓ_{}_{m̄_{n}^{−}}(d̄_{−}). In particular, since the sequence (m̄_{n})_{n∈ℕ} converges, the area of \overline{A}_{n} is bounded. On the other hand, since d̄ intersects c̄ essentially, \overline{A}_{n} intersects c̄_{n} essentially. In particular, \overline{A}_{n} intersects \overline{T}_{n} along a disc \overline{D}_{n}. When the length of c̄_{n} tends to 0, d(c̄_{n}, ∂\overline{T}_{n}) ⟶ ∞ (see [Min99, Lemma 6.1]). It follows that the diameter of \overline{D}_{n}, and hence its area, tends to ∞ when the length of c̄_{n} tends to 0. Thus an upper bound for the area of \overline{A}_{n} ⊃ \overline{D}_{n} yields a lower bound for the length of c̄_{n}. This concludes the proof of Claim 3.15.

Consider now two simple closed curves c̄, d̄⊂ S̄ such that the components of S̄ \ (c̄∪ d̄) are discs. Two such curves are said to fill the surface S̄. Consider essential annuli \overline{A}_{n} and \overline{B}_{n} in C(\overline{M}_{n}) in the homotopy classes defined by c̄ and d̄, constructed as in Lemma 3.2. In particular \overline{A}_{n} and \overline{B}_{n} have bounded area. Since \overline{A}_{n} and \overline{B}_{n} have bounded areas and negative curvature, the only way for them to have a large diameter is to have a very short core curve. But this would contradict Claim 3.15. Thus we can conclude that \overline{A}_{n} and \overline{B}_{n} have uniformly bounded diameters.

Let B_{1≤k≤p} be the closure of the components of S̄ × I \ (c̄× I ∪ d̄× I). Our manifold N̄ = S̄ × I is the union of the B_{k} and the B_{k} are all balls. Define a surjective map f_{n} : S̄ × I → C(\overline{M}_{n}) that maps c̄ × I and d̄× I to \overline{A}_{n} and \overline{B}_{n} respectively and such that the restriction of f_{n} to each B_{k} is an immersion. For each k, the image of ∂B_{k} lies in \overline{A}_{n} ∪ \overline{B}_{n} ∪ ∂C(M_{n}). Since \overline{A}_{n} and \overline{B}_{n} have bounded diameters and since the induced metric on ∂C(\overline{M}_{n}) is bounded, the diameter of f_{n}(∂B_{k}) is bounded for any k. It follows that f_{n}(B_{k}) has a bounded diameter for any k. Since f_{n} is surjective, this implies that C(\overline{M}_{n}) has a uniformly bounded diameter. Since the index of the cover \overline{M}_{n} → M_{n} does not depend on n, C(M_{n}) has a bounded diameter.

It remains to show that the convex core of M_{∞} is compact and homeomorphic to S × I (when M_{n} is homeomorphic to S × ℝ). Once again we will use the negatively curved ramified cover \overline{M}_{n}. Since (x̄_{n}, \overline{M}_{n}) converge to (x̄_{∞}, \overline{M}_{∞}), there is R_{n} ⟶ ∞ and a sequence of bilipschitz map ϕ_{n} : B(x̄_{n}, R_{n}) → B(x̄_{∞}, R_{n}) such that the bilipschitz constants tend to 1. By Lemma 3.14, for n large enough, C(\overline{M}_{n}) ⊂ B(x_{n}, R_{n}). Given a geodesic segment γ_{n} ⊂ C(\overline{M}_{n}), ϕ_{n}(γ_{n}) almost realizes the distance between its endpoints. Since \overline{M}_{n} is a hyperbolic manifold with cone singularities, ϕ_{n}(γ_{n}) is very close to the geodesic segment joining its endpoints. It follows that, for n large enough, the convex hull of ϕ_{n}(C(\overline{M}_{n})) lies in a small neighborhood V_{n}(ϕ_{n}(C(\overline{M}_{n}))) of ϕ_{n}(C(\overline{M}_{n})). This convex hull has to contain C(\overline{M}_{∞}) since it is the smallest convex set. Thus we have C(\overline{M}_{∞}) ⊂ V_{n}(ϕ_{n}(C(\overline{M}_{n}))). It follows that C(\overline{M}_{∞}) is compact. Furthermore, since the induced metric on ∂C(\overline{M}_{n}) is bounded, V_{n}(ϕ_{n}(C(\overline{M}_{n}))) is homeomorphic to S̄ × I for n large enough. It follows that C(\overline{M}_{∞}) is homeomorphic to S̄ × I. Thus we have proved that M_{∞} is a quasifuchsian manifold with cone singularities with the same topological type as M_{n}.

In contrast to the other results of this section, we do not need to assume that the quasifuchsian manifolds under consideration in Lemma 3.13 are not fuchsian.

3.6 The bending lamination of the convex core

To finish the proof of Lemma 3.1 we only have to check that the induced bending lamination on the boundary of the convex core of the limit manifold is the limit of the bending laminations. We can state the result as follows.

Lemma 3.16. Let N = S × ℝ, let x_{1}, ⋯, x_{n_{0}} be distincts points on S, and let κ_{i} = {x_{i}} × ℝ, 1 ≤ i ≤ k. Let (g_{n})_{n∈ℕ} be a sequence of quasifuchsian metrics on N with particles of angles θ_{n}^{i} along κ_{i}, 1 ≤ i ≤ k. Let λ_{n}, resp. m_{n}, be the measured bending laminations, resp. the induced metric, on the boundary of the convex core of (N, g_{n}). Suppose that (g_{n}) converges in bilipschitz topology towards a quasifuchsian metric with particles g on N, with cone angles θ^{i} ∈ (0, π) along κ_{i}. Then (m_{n})_{n∈ℕ} converges to the induced metric m on the boundary of the convex core of (N, g), while (λ_{n})_{n∈ℕ} converges to the measured bending lamination λ of the boundary of the convex core of (N, g).

Proof. Set M = (N, g) and M_{n} = (N, g_{n}) and denote by C(M) the convex core of M and by C(M_{n}) the convex core of M_{n}. We consider as above the finite cover N̄ of N ramified along the cone singularities, chosen so that all cone angles in N̄ have angle larger than 2π. This is useful below since we will use negative curvature arguments, in particular the existence of a geodesic segment in a homotopy class with fixed endpoints. Clearly it is sufficient to prove the lemma for N̄, where the “convex core” considered is C(\overline{M}_{n}), the lift to N̄ of C(M_{n}), since once the result is obtained in N̄, we can take the quotient by the group of deck transformations to obtain the result on N.

Let (γ_{n})_{n∈ℕ} be a sequence of segments in N̄, with γ_{n} geodesic for ḡ_{n} for all n ∈ ℕ. Suppose that (γ_{n})_{n∈ℕ} converges to a segment γ. We know that ḡ_{n} → ḡ in the bilipschitz topology and, in hyperbolic geometry, any segment which is close to realizing the distance between its endpoints is close to a geodesic segment. So γ is geodesic for ḡ. Conversely, any geodesic segment for ḡ is a Hausdorff limit of geodesic segments for the g_{n}. The same holds for closed geodesics.

Let (Ω_{n})_{n∈ℕ} be a sequence of compact subsets of N̄ such that, for all n ∈ ℕ, Ω_{n} is convex for ḡ_{n}. Suppose that Ω_{n} → Ω in the Hausdorff topology. The definition of a convex subset and the previous paragraph show that Ω is convex, since any geodesic segment γ in (N̄, ḡ) with endpoints in the interior of Ω is the limit of a sequence of geodesic segments γ_{n}, with γ_{n} geodesic for ḡ_{n}. Since γ_{n} has endpoints in Ω_{n} (for n large enough) and Ω_{n} is convex for ḡ_{n}, γ_{n} ⊂ Ω_{n}, and therefore γ̄ ⊂ Ω, and Ω is convex for ḡ. Conversely, a similar argument shows that any compact convex subset for ḡ is the Hausdorff limit of a sequence of compact convex subsets of the metrics ḡ_{n}.

For all n, C(\overline{M}_{n}) contains all closed geodesics in (N̄, ḡ_{n}). Given a non-trivial homotopy class α in N̄_{r} (the complement of the singular curves in N̄), it is realized for each n ∈ ℕ by a (unique) closed geodesic γ_{n} in (N̄, ḡ_{n}), and the sequence (γ_{n})_{n∈ℕ} converges to the closed geodesic γ which realizes α in (N̄, ḡ). For each n ∈ ℕ, γ_{n} ⊂ C(\overline{M}_{n}). Moreover we have seen that the diameter of the C(\overline{M}_{n}) is bounded. It follows that (C(\overline{M}_{n}))_{n∈ℕ} converges – after extracting a subsequence – to a limit subset C′ which contains all closed geodesics in (N̄, ḡ).

Since C′ is the limit of a sequence of convex subset of the (N̄, ḡ_{n}), it is convex. Moreover if Ω ⊂ C′ is convex, then it is the limit of a sequence of convex subsets Ω_{n} ⊂ C(\overline{M}_{n}). But then (Ω_{n} ∩ C(\overline{M}_{n}))_{n∈ℕ} is a sequence of convex subsets converging to Ω. Because the C(\overline{M}_{n}) are minimal convex subsets, Ω_{n} ∩ C(\overline{M}_{n}) = C(\overline{M}_{n}) for all n, so that Ω = C′. So C′ = C(\overline{M}). This shows that C(\overline{M}) is the Hausdorff limit of the C(\overline{M}_{n}).

Note that it is not clear at this point whether ∂C(\overline{M}_{n}) → ∂C(\overline{M}) in the C^{1} topology. However, a general fact is that, if ϕ : S → H^{3} is a smooth embedding of a surface, and if (ϕ_{n})_{n∈ℕ} is a sequence of Lipschitz embeddings of S in H^{3} which converges to ϕ in the C^{0} topology, then the distance d_{n} induced on S by the ϕ_{n} are larger, in the limit, than the distance d induced by ϕ:

∀x, y ∈ S, lim sup_{n→∞} d_{n}(x, y) ≥ d(x, y) .  (1)

The same holds if ϕ is Lipschitz with locally convex image rather than smooth, see [AZ67]. Moreover, in case of equality in Equation (1) and if the ϕ_{n} also have locally convex images, then the convergence of (ϕ_{n}) to ϕ is stronger, in the sense that the tangent plane to ϕ_{n}(S) almost everywhere converges to the tangent plane to ϕ(S).

Coming back to ∂C(\overline{M}_{n}), the C^{0} convergence towards ∂C(\overline{M}) (together with the bilipschitz convergence of ḡ_{n} to ḡ) is sufficient to insure that the metric m̄_{n} on ∂C(M_{n}) is larger in the limit than the metric m̄ induced by ḡ on ∂C(\overline{M}). In other terms, if x, y ∈ ∂C(\overline{M}) and x_{n}, y_{n} ∈ ∂C(\overline{M}_{n}) are such that lim x_{n} = x, lim y_{n} = y, then there exists for each ϵ > 0 some N_{0} ∈ ℕ such that, for all n ≥ N_{0},

dm̄_{n}(x_{n}, y_{n}) ≥ (1 − ϵ)dm̄(x, y) .

It follows that the lengths of the closed geodesics in S for m̄_{n} are bounded from below by (1 − ϵ) times their lengths for m̄.

Since the metrics m̄_{n} and m̄ are hyperbolic metrics with cone singularities of fixed angles, this shows, using standard arguments based for instance on pants decompositions, that m̄_{n} → m̄ (see Section A.2, or [DP07]).

It then follows that as n → ∞, m̄_{n} is bounded from above by (1 + ϵ)m̄: under the same hypothesis as above,

dm̄_{n}(x_{n}, y_{n}) ≤ (1 + ϵ)dm̄(x, y) .  (2)

Since m̄_{n} → m̄ in this sense of Equation (2), and moreover ∂C(\overline{M}_{n}) → ∂C(\overline{M}) in the C^{0} topology, it follows that the convergence is actually stronger, and the tangent plane to ∂C(\overline{M}_{n}) converges almost everywhere to the tangent plane to ∂C(\overline{M}) (both exist almost everywhere by convexity). This implies that the measured laminations λ_{n} of ∂C_{n} converge to λ.

4 Prescribing the measured bending lamination on the boundary of the convex core

The goal of this section is to prove Theorem 1.12 and then Theorem 1.13. The proof of Theorem 1.12 is largely based on a well-known doubling argument already used for non-singular manifolds, which reduces the infinitesimal rigidity with respect to the measured lamination (when the support of the lamination is along closed curves) to a rigidity statement proved by Hodgson and Kerckhoff [HK98] for hyperbolic cone-manifolds.

Theorem 1.13 is then a consequence, using the compactness statement proved in section 3.

4.1 A doubling argument

Let M be convex co-compact manifold with particles, and let C(M) be its convex core. Suppose that the support of the measured bending lamination of C(M) is a disjoint union of closed curves.

Definition 4.1. The doubled convex core of M is the 3-dimensional hyperbolic manifold with cone singularities DC(M) obtained by gluing two copies of C(M) isometrically using the identification of their boundaries.

We have seen that the singular locus of M does not intersect the support of the bending lamination on the boundary of the convex core — actually it even remains at a distance which is bounded from below by a constant depending only on the cone angles. So the “particles” intersect the boundary of the convex core inside faces, and moreover it does so orthogonally. It follows that the singular locus of DC(M) is a disjoint union of closed curves, which are of two types:

• each “particle” p of M corresponds to a cone singularity along a closed curve in DC(M), of length equal to twice the length of the intersection of p with C(M),

• each closed curve in the support of the measured bending lamination of the boundary of C(M) corresponds to a closed curve (of the same length) in DC(M).

Still by definition, DC(M) admits an isometric involution — exchanging the two copies of C(M) which are glued to obtain DC(M) — and the set of fixed points of this involution is a (non connected)) closed surface S, which corresponds to the boundaries of both copies of C(M). This surface is orthogonal to the singularities of the first kind, and contains the singularities of the second kind.

4.2 Local deformations

The doubling trick explained above leads directly to a rigidity statement. We consider again a convex cocompact manifold M with particles, for which the measured bending lamination of the convex core is along closed curves γ_{1}, ⋯, γ_{N}, for which the bending angles are equal to α_{1}, ⋯, α_{N} ∈ (0, π). As in the introduction, we call θ_{1}, ⋯, θ_{n_{0}} the cone angles at the “particles”, and let θ = (θ_{1}, ⋯, θ_{n_{0}}).

Lemma 4.2. There exists a neighborhood U of (α_{1}, ⋯, α_{N}) in (0, π)^{N} and a neighborhood V of the hyperbolic metric g on M in QF_{S,n_{0},θ} such that, if (α′_{1}, ⋯, α′_{N}) ∈ U, there is a unique g′ ∈ V for which the support of the measured bending lamination on C(M) is γ_{1} ∪ ⋯ ∪ γ_{N} and the bending angle on γ_{i} is α′_{i}, 1 ≤ i ≤ N.

Proof. Hodgson and Kerckhoff [HK98] proved a local deformation result for hyperbolic cone-manifolds. It follows from their result that there exists a unique cone-manifold close to DC(M) with the same topology as DC(M) (including the singular locus), the same angles at the cone singularities corresponding to the particles in M, and angles 2α′_{1}, ⋯, 2α′_{N} instead of 2α_{1}, ⋯, 2α_{N} at the cone singularities corresponding to the pleating lines of C(M).

The uniqueness of D′ shows that it has the same symmetry as DC(M), that is, it admits an isometric involution fixing a surface S′ isotopic to the surface S fixed by the isometric involution on DC(M). By an easy symmetry argument, the cone singularities in D′ corresponding to the particles in M still have to be orthogonal to S′, while those corresponding to the pleating lines of ∂C(M) have to be contained in S′ (see [BO04, Section 8] for details on the uniqueness part of this argument; the same argument can basically be used when particles are present).

Therefore, D′ is the double of a hyperbolic manifold with convex boundary (obtained as the metric completion of one half of the complement of S′ in D′) with cone singularities orthogonal to the boundary. The boundary of this manifold is convex with no extremal point, so that it is the convex core of a quasifuchsian manifold with particles M′, with the same cone angle as M at the particles and such that ∂C(M′) is pleated along the same lines as ∂C(M), but with pleating angles α′_{1}, ⋯, α′_{N} instead of α_{1}, ⋯, α_{N}.

The uniqueness of such a manifold, in the neighborhood of M, follows from the uniqueness of D′ in the neighborhood of DC(M).

4.3 Proof of Theorem 1.12

Let γ_{1}, ⋯, γ_{N} be the curves in the support of λ, considered as curves in ∂N. Following the doubling construction above, we define a closed manifold D(N) by gluing two copies of N along their boundary. D(N) contains two families of curves, which we still call c_{1}, ⋯, c_{n_{0}} (corresponding to the particles in N) and γ_{1}, ⋯, γ_{N} (corresponding to the pleating lines on the boundary of the convex core).

Let θ′_{1}, ⋯, θ′_{n_{0}} ∈ (0, π) and α′_{1}, ⋯, α′_{N} ∈ (0, π) be chosen such that:

• for all i ∈ {1, ⋯, n_{0}}, 0 ≤ θ′_{i} ≤ θ_{i}, and θ′_{i} = π/k_{i} for some k_{i} ∈ ℕ,

• for all j ∈ {1, ⋯, N}, 0 ≤ α′_{j} ≤ α_{j}/2, and α′_{j} = π/2l_{j} for some l_{j} ∈ ℕ.

The Orbifold Hyperbolization Theorem for cyclic orbifolds (initially stated by Thurston, and proved in [BP01, CHK00]) can be applied to show that there is a unique hyperbolic orbifold structure on D(N) with singularities of angles θ′_{i} on the c_{i} and 2α′_{j} on the γ_{j}.

Since the θ_{i} are in (0, π), the result of Kojima [Koj98] shows that this orbifold structure can be deformed to a unique cone-manifold structure, with cone angles θ_{i} on the curves c_{i} and 2α′_{j} on the curves γ_{j}.

Let (α_{t})_{t∈[0,1]} = (α_{1,t}, ⋯, α_{N,t})_{t∈[0,1]} be the 1-parameter family defined by

α_{j,t} = (1 − t)α′_{j} + tα_{j}, 1 ≤ j ≤ N .

Then for all j ∈ {1, ⋯, N}, α_{j,0} = α′_{j}, α_{j,1} = α_{j}. Let I ∈ [0, 1] be the maximal interval containing 0 such that, for all t ∈ I:

• there exists a hyperbolic structure on D(N) with cone singularities of angle θ_{i} on c_{i}, 1 ≤ i ≤ n_{0}, and a cone singularity of angle 2α_{j,t} on γ_{j}, 1 ≤ j ≤ N, • this hyperbolic structure has an isometric involution exchanging the two copies of N glued to obtain

D(N).

By construction, I ≠ ∅. Lemma 4.2 shows that I is open, while Lemma 3.1 shows that I is closed. So I = [0, 1], this proves the existence part of the statement because D(N) with the hyperbolic cone-structure for t = 1 is obtained by doubling the convex core of a convex co-compact hyperbolic manifold with particles of angles θ_{i} and pleating angles α_{i} on the boundary, as needed.

For the uniqueness, the same deformation argument can be used to start from a cone-manifold structure on D(N) and decrease the angles along the curves γ_{j}, 1 ≤ j ≤ N, from 2α_{j} to 2α′_{j}. Lemma 4.2 shows that the corresponding deformation of the hyperbolic cone-manifold structure exists and is unique. Since the endpoint of the deformation is unique (by the Orbifold Hyperbolization Theorem) there can be only one cone-manifold structure on D(N) with angles θ_{i} on the curves c_{i}, 1 ≤ i ≤ n_{0}, angle 2α_{j} on the curve γ_{j}, 1 ≤ j ≤ N, and the necessary symmetry property.

4.4 Proof of Theorem 1.13.

Given λ_{−}, λ_{+} ∈ ML_{S,x} satisfying the hypothesis of Theorem 1.13, both are limits of a sequence of measured laminations (λ_{−,n})_{n∈ℕ}, (λ_{+,n})_{n∈ℕ} with support along a union of closed curves, which satisfy the hypothesis of Theorem 1.12.

For all n, Theorem 1.12 shows that λ_{−,n} and λ_{+,n} are the upper and lower measured bending laminations of the boundary of the convex core for a unique quasifuchsian hyperbolic structure with particles g_{n} on S × ℝ. Lemma 3.1, applied to this sequence of hyperbolic structures, shows that it has a subsequence which converges to a quasifuchsian hyperbolic structure with particles, for which the lower and upper measured bending laminations of the boundary of the convex core are λ_{−} and λ_{+}, respectively.

4.5 The conditions are necessary

Finally we check here that the hypothesis in Theorem 1.13 are necessary. It obviously follows that the hypothesis in Theorem 1.12 are also necessary.

Lemma 4.3. Let M be a non Fuchsian quasifuchsian manifold with particles, let λ be the measured bending lamination on the boundary of its convex core. Then λ satisfies the hypothesis of Theorem 1.13.

Proof. The hypothesis that the weight of each closed curve in the support of λ_{−} and λ_{+} is less than π is clearly a consequence of the fact that C(M) is convex and compact.

Suppose by contradiction that λ_{−} and λ_{+} do not fill S. There exists then a sequence (c_{n})_{n∈ℕ} of simple closed curve in S such that

i(λ_{−}, c_{n}) + i(λ_{+}, c_{n}) → 0 .

Let c_{n}^{−} and c_{n}^{+} be the geodesic representatives of c_{n} in the lower and upper boundary components of C(M), respectively.

Let c̄_{n}^{−} and c̄_{n}^{+} be lifts of c_{n}^{−} and c_{n}^{+}, respectively, to M̄, corresponding to the same lift of c. Lemma 3.2 shows that there exists an annulus A_{n} ⊂ M̄ bounded by c̄_{n}^{−} and c̄_{n}^{+} on which the induced metric is hyperbolic with cone points of negative singular curvature (cone angle larger than 2π). Moreover, the boundary of A_{n} is convex (for the induced metric) and its total curvature goes to 0 as n → ∞. The Gauss-Bonnet formula then implies that the area of A_{n} goes to 0 as n → ∞. Since the lengths of the c̄_{n}^{−} and c̄_{n}^{+} are bounded from below, this means that the distance between c̄_{n}^{−} and c̄_{n}^{+} in A_{n} goes to 0 as n → ∞. Therefore, the distance between the upper and lower boundary of C(M) is zero, a contradiction because we have supposed that M is not Fuchsian.

5 Earthquakes estimates

In this section we consider a convex co-compact manifold with particles M. The arguments in this more general case are the same as in the specific situation of quasifuchsian manifolds with particles. Its boundary ∂M has a number of marked points x_{1}, ⋯, x_{2n_{0}} which are the endpoints of the n_{0} “particles”, and to each is attached an angle θ_{k} ∈ (0, π), 1 ≤ k ≤ 2n_{0}, which is the angle at the corresponding particle.

We identify ∂M with the boundary of its convex core (see Lemma 2.5). We will use the following notations.

• λ is the measured bending lamination of the boundary of the convex core.

• m is its induced metric.

• t is the (unique) hyperbolic metric in the conformal class at infinity τ, with cone singularities of prescribed angle θ_{k} at the marked point x_{k}.

• G_{λ}(m) is the metric obtained by grafting the hyperbolic metric m along the measured lamination λ, so that G_{λ}(m) has curvature in [−1, 0]. If for instance λ is rational, then G_{λ}(m) is obtained by inserting a flat annulus in (∂M, m) for each closed curve in the support of λ, see e.g. [Dum08].

This section contains a basic estimates relating t to m. It will be useful in proving the compactness of a certain map and Theorem 1.7. Its statement is based on the following extension to hyperbolic surfaces with cone singularities of Thurston’s Earthquake Theorem (as found in [Thu86b, Ker83]).

Theorem 5.1 ([BS09]). For any h, h′ ∈ T_{Σ,x,θ}, there is a unique measured lamination ν ∈ ML_{Σ,x} such that the right earthquake along ν sends h to h′.

The main estimate proved in this section, and the main tool for the proof of Theorems 1.7 and 1.14, is the following.

Proposition 5.2. There exists a constant C > 0 (depending only on the topology of M) such that, if ν ∈ ML_{∂M,x} is the measured lamination such that t = E_{r}(ν)(m), then the length L_{m}(ν) is at most C.

It is proved in Section 5.4, after some preliminary considerations. It is used below in Section 6.

5.1 The average curvature of geodesics

In this part we prove a technical statement which is useful at several points below. It is an extension to convex co-compact manifolds with particles of a result proved earlier by Bridgeman [Bri98] for the convex core of non-singular convex co-compact manifolds, or more generally of pleated surfaces in H^{3}. However the argument used here is inspired by Bonahon and Otal [BO04].

We consider a quasifuchsian manifold with particles, M, and call θ_{1}, ⋯, θ_{n_{0}} the cone angles at the particles. By definition, θ_{1}, ⋯, θ_{n_{0}} ∈ (0, π). Here S is one of the connected components of ∂C(M).

Proposition 5.3. There exists a constant C_{0} > 0 such that, if γ is a geodesic segment on S transverse to λ, i(γ, λ) ≤ C_{0}(l_{m}(γ) + 1).

Note that C_{0} depends on the θ_{i} (at least the argument we use here does depend on the maximum of the θ_{i}) but not otherwise on M.

Proposition 5.3 will follow from the following lemma. We use here the constant ϵ_{0} coming from Lemma 2.7 and Lemma 2.8.

Lemma 5.4. There exists λ_{1} > 0 such that if γ is at distance at least ϵ_{0}/2 from the intersection of S with the singular set of M and if l_{m}(γ) ≤ ϵ_{0}/4, then i(λ, γ) ≤ λ_{1}.

The proof of this lemma is based on some intermediate steps. The first is a consequence of Lemma 2.7 and Lemma 2.8.

Claim 5.5. There exists ρ_{0} > 0 such that any point in S at distance at least ϵ_{0} in S from the singular points of S is also at distance at least ρ_{0} in M from the singular set of M.

Proof. Let x ∈ S which is at distance at least ϵ_{0} from the singular points of S, suppose that it is at distance strictly less than ϵ_{0} from a particle p. Let y be a point in p which is closest from x, and let D be the totally geodesic disk of radius ϵ_{0} orthogonal to p at y. This disk does not encounter any other particle by Lemma 2.8. Moreover x ∈ D because y is at minimal distance from x among the points of p. We can therefore apply the second point in Lemma 2.7 to D, with Ω equal to the intersection of D with the convex core of M. The result follows.

Corollary 5.6. Let x ∈ S be contained in the support of the bending lamination λ, and let D′ be the totally geodesic disk of radius ρ_{0} in M orthogonal to λ at x. Then D′ does not intersect the singular set of M.

After taking ρ_{0} smaller if necessary, we have another simple statement which will be necessary below.

Claim 5.7. Let y ∈ λ be a point in the connected component of x in the intersection of S with D′, and let g_{y} be the geodesic segment in the support of λ centered at y and of length 2ϵ_{0}. Then the angle between g_{y} and D′ at y is at least π/4.

Proof. We call g_{x} the geodesic segment contained in the support of λ centered at x and of length 2ϵ_{0}. g_{x} is disjoint from g_{y} on S while x is at distance at most ρ_{0} from y, it follows that there exists c > 0 (depending on ϵ_{0} and ρ_{0}, and going to 0 as ρ_{0} → 0 for fixed ϵ_{0}) such that the distance to g_{x} in S of any point of g_{y}([−ϵ_{0}+ρ_{0}, ϵ_{0}−ρ_{0}]) is at most cϵ_{0}. The same estimate holds in M̃, the universal cover of M. If ρ_{0} is small enough — relative to ϵ_{0} — the result follows.

Remark 5.8. There exists k_{0} > 0, depending only on ρ_{0}, such that, if Ω is a convex subset in the disk of radius ρ_{0} in H^{2}, the total curvature of the boundary of Ω is at most k_{0}.

Proof. This follows from the Gauss-Bonnet Theorem applied to Ω, with k_{0} equal to 2π plus the area of the hyperbolic disk of radius ρ_{0}.

Proof of Lemma 5.4. If γ does not intersect the support of λ the statement obviously holds, so we suppose that some point x ∈ γ is in the support of λ. Let D be the totally geodesic disk of radius ϵ_{0}/2 centered at x and orthogonal, at x, to the support of λ. By construction D is disjoint from M_{s}.

Remark 5.8 shows that the total curvature of the connected component c of D ∩S containing x is at most k_{0}. By Claim 5.7, each geodesic in the support of λ which intersects c makes with D an angle at least π/4. It follows that i(c, λ) ≤ 2k_{0}. It also follows, since the length of γ is less than ϵ_{0}/4, that γ can be deformed transversally to λ to a segment of c, so that i(γ, λ) ≤ i(c, λ). Therefore i(γ, λ) ≤ 2k_{0}, and this proves the lemma.

Proof of Proposition 5.3. Notice first that Lemma 5.4, although stated only for geodesic segments γ that are at distance at least ϵ_{0}/2 from the cone singularities, actually applies without this hypothesis. This is because, by Lemma 2.7, the support of λ cannot enter the ϵ_{0}-neighborhood of the singular points, so that any part of γ at distance less than ϵ_{0} from the singular set of S has zero intersection with λ.

Let n ∈ ℕ be the unique integer such that nϵ_{0}/4 ≤ l_{m}(γ) < (n + 1)ϵ_{0}/4. Then γ can be cut into a sequence of segments γ_{1}, ⋯, γ_{n} of length ϵ_{0}/4 and one last segment γ_{n+1} of length smaller than ϵ_{0}/4. Lemma 5.4 can be applied to each of those segments, it yields that i(λ, γ_{i}) ≤ λ_{1}, 1 ≤ i ≤ n + 1, so that

i(λ, γ) ≤ (n + 1)λ_{1} ≤ (\frac{4l_{m}(γ)}{ϵ_{0}} + 1) λ_{1} ,

this proves the proposition.

5.2 The grafted metric and the hyperbolic metric at infinity.

We consider here the relation between the grafted metric G_{λ}(m) and the hyperbolic metric at infinity t. We first recall the definition of the grafting map

G : ML × T → T
(l, m) ↦ G_{l}(m)

on a closed surface S.

The definition of G_{l}(m) is simpler when l is a weighted multicurve, that is, its support is a disjoint union of closed curves c_{1}, ⋯, c_{N}. The transverse measure is then described by a positive weight w_{i} on c_{i}, for 1 ≤ i ≤ N. Then G_{l}(m) is obtained by realizing each curve c_{i} as a closed geodesics in (S, m), cutting S open along each c_{i}, and gluing in a flat strip of width w_{i}. Thurston showed that this map extends by continuity from weighted multicurves to measured laminations, see [KP94].

We now return to the setting where m and λ are the induced metric and measured bending lamination of the convex core of a quasifuchsian manifold M. The grafted metric G_{λ}(m) is then isometric to the induced metric on the unit normal bundle of C(M) in the unit tangent bundle of M. (The unit normal bundle of C(M) is the space of unit vectors at points of ∂C(M) which are the oriented normals of a support plane of C(M).)

One of the key properties of the grafted metric G_{λ}(m) (see e.g. [KP94]) is that it is in the conformal class τ at infinity — more precisely, there is a natural “Gauss map” defined from the unit normal bundle of ∂C(M) with its “grafted metric” to the boundary at infinity of M, which is conformal. This means that G_{λ}(m) is conformal to t. Moreover, since the angles θ_{i} are in (0, π), the intersection of the boundary of the convex core with the particles is at non-zero distance (for m) from the support of λ, so that the cone angles of the grafted metric at the intersections with the particles of the boundary of the convex core is equal to the cone angle of the corresponding singularities.

The fact that t is conformal to G_{λ}(m) translates as

t = e^{2u}G_{λ}(m) ,

where u : ∂M → ℝ is a function.

Lemma 5.9. The function u is non-positive on ∂M.

Proof. Consider two metrics g and g′ with g′ = e^{2u}g, and let K and K′ be their curvatures. Then (see e.g. chapter 1 of [Bes87])

K′ = e^{−2u}(Δu + K) .

We can apply this formula here with g = G_{λ}(m) and g′ = t, so that K′ = −1 while K ∈ [−1, 0]. It takes the form:

Δu = −K − e^{2u} = |K| − e^{2u} ,

with K ∈ [−1, 0] (this equation is understood in a distributional sense).

Since the cone angles are the same for t and for G_{λ}(m), u is continuous and bounded at the singular points (see [Tro91]). Let x_{M} ∈ S be a point where u achieves its maximum. Suppose first that x_{M} is not a singular point, then u is C^{2} at x_{M} by elliptic regularity (see [Tro91]). Moreover Δu ≥ 0 at x_{M} since x_{M} is a maximum of u. It follows that e^{2u} ≤ |K| ≤ 1, so that u ≤ 0. To complete the proof it is sufficient to prove that u cannot achieve a positive maximum at a singular point of S. So we consider a singular point x_{0} of S, and suppose that u > 0 at x_{0}. We will show that u cannot have a maximum at x_{0}.

Let D be the geodesic disk of radius r centered at x_{0} in (S, G_{λ}(m)). Since λ does not enter a small neighborhood of x_{0}, D is hyperbolic, with only one cone singularity at x_{0}, if r is small enough. Let i_{0} be the isometric map between D, with the metric G_{λ}(m), and the hyperbolic disk H_{α}^{2} with one cone singularity of angle α, where α is the cone angle of S at x_{0}. Let i_{1} : D → H_{α}^{2} be the isometric embedding of (D, t) in H_{α}^{2}. Call v_{0} the vertex of H_{α}^{2}, i.e., its singular point. Since u > 0 at x_{0}, if r is small enough, then

∀x ∈ D \ {x_{0}}, d(i_{1}(x), v_{0}) > d(i_{0}(x), v_{0}) .

There is a natural complex map ϕ : H_{α}^{2} → H^{2}, given in holomorphic coordinates centered at the singular point by z → z^{2π/α}. It is conformal and multiplies the metric by a factor (2π/α)^{2}d(x, v_{0})^{2(2π/α−1)}. Consider the composition

Φ := ϕ ∘ i_{1} ∘ i_{0}^{−1} ∘ ϕ^{−1} : (ϕ ∘ i_{0})(D) → (ϕ ∘ i_{1})^{−1}(D) .

It is a conformal map, with conformal factor equal to

(2π/α)^{2}d(i_{1}(x), v_{0})^{2(2π/α−1)}e^{2v}(2π/α)^{−2}d(i_{0}(x), v_{0})^{−2(2π/α−1)} ,

with v = u ∘ (ϕ ∘ i_{0})^{−1}. This can be written as

(\frac{d(i_{1}(x), v_{0})}{d(i_{0}(x), v_{0})})^{2(2π/α−1)} e^{2v} ,

and is bigger than 1 since u > 0 and d(i_{1}(x), v_{0}) > d(i_{0}(x), v_{0}).

Since Φ is a conformal map between two hyperbolic domains, its conformal factor cannot have a local maximum bigger than 1 at an interior point by the argument given at the beginning of this proof. Therefore, u cannot have a positive maximum at x_{0}.

The following notion will be useful in this section and the next.

Definition 5.10. A c-curve on ∂M is either a closed curve or a segment with endpoints at cone singularities, which does not contain any singular point (except at its endpoints if it’s not a closed curve).

We will sometimes implicitly consider c-curves up to homotopy in the complement of the singular points in ∂M. Each homotopy class (with fixed endpoints) contains a unique geodesic for any non-positively curved metric on ∂M (in particular for m, t and G_{λ}(m)). Given a c-curve γ, we will denote by L_{m}(γ) (resp. L_{t}(γ), L_{G_{λ}(m)}(γ)) the length of that geodesic for the corresponding metric.

Corollary 5.11. Let γ be a c-curve in ∂M, then L_{t}(γ) ≤ L_{G_{λ}(m)}(γ).

This follows directly from Lemma 5.9, since any minimizing c-curve in (S, G_{λ}(m)) has shorter length for t, and the minimizing curve in (S, t) in the same homotopy class is even shorter. Note also that for any c-curve γ, i(λ, γ) ≤ CL_{m}(γ). This follows from Proposition 5.3, and by the fact that the lengths of the c-curves which are segments between two singular points of S is bounded from below.

5.3 An upper bound on the lengths of the curves at infinity

The second step in the proof of Proposition 5.2 is a comparison between the lengths of c-curves in the metrics t and m.

Proposition 5.12. There exists a constant C > 0 (independent of M) such that:

1. for each c-curve γ in ∂M, L_{t}(γ) ≤ CL_{m}(γ),

2. for each long tube T in the thin part of (∂M, m), T might also be a long tube for t, but its length for t is at most its length for m plus C.

The proof uses some simple statements on the geometry of long hyperbolic tubes in (S, t). Recall (see [DP07]) that the Margulis Lemma applies to hyperbolic surfaces with cone singularities of angle at most θ, when θ ∈ (0, π): there exists a constant c_{M}, depending on θ only, such that the set of points where the injectivity radius is less than c_{M} is a disjoint union of cusps, disks centered at a cone singularity, and tubes with core of length less than 2c_{M}.

We consider in this subsection a hyperbolic tube T, which can be described as isometric to the set of points at distance at most L (for some L > 0) from the unique simple closed geodesic in the quotient of the hyperbolic plane H^{2} by a hyperbolic translation of length l. Moreover l is supposed to be small and L large, so that the lengths of the boundary components of T — which are both equal to l cosh(L) — are equal to c_{M}. We call σ the core of T, in other terms the unique simple closed geodesic contained in T, and we denote by σ_{M} the c_{M}-neighborhood of σ — the set of points at distance at most c_{M} from σ in T.

Lemma 5.13. There exists a constant C > 0 such that i(λ, ∂σ_{M}) ≤ Ce^{−L}: the intersection of λ with the boundary of σ_{M} is at most Ce^{−L}.

Proof. Any maximal embedded geodesic segment in T intersects exactly once σ, but also each of the two connected components of ∂σ_{M}. It follows that the intersection with λ of each of the connected components of ∂σ_{M} is equal to i(λ, σ). But since the length of σ is l = e^{−L}, Proposition 5.3 — applied to long segments that wrap many times around σ — shows that i(λ, σ) ≤ Ce^{−L}.

Lemma 5.14. There exists a constant C > 0 such that, if g is an embedded maximal geodesic segment in T, then the length of the orthogonal projection on σ_{M} of g ∩ (T \ σ_{M}) is at most C.

Proof. If g ⊂ σ, then g ∩ (T \ σ_{M}) = ∅ and the result applies. We suppose from here on that g is not contained in σ. If g is contained in one connected component of the complement of σ in T then, since g is embedded, its orthogonal projection on σ is injective, so that the length of its orthogonal projection is bounded by the length of σ. Otherwise, it follows from standard hyperbolic geometry arguments that g intersects σ exactly once.

Consider the universal cover T̃ of T, it is isometric to the set of points at distance at most L from a geodesic σ̃ ⊂ H^{2} which is the lift of σ. Choose one of the connected components, say g̃, of the lift of g to T̃. It intersects the lift of ∂σ_{M} with an angle which is bounded from below — otherwise g could not intersect σ. It follows from this, and from elementary geometric properties of the hyperbolic plane, that the length of the orthogonal projection on σ̃ of each of the segment of g̃ outside the lift of σ_{M} is bounded from above by a constant.

Corollary 5.15. There exists a constant C > 0 such that, whenever g_{0} is a maximal geodesic segment in T such that the orthogonal projection of g_{0} on σ is injective, then i(λ_{|T \σ_{M}} , g_{0}) ≤ C.

Proof. Let c be a maximal geodesic segment in the intersection with T of the support of λ, and let c′ be one of the connected components of c∩T \σ_{M}. Since both c and g_{0} are geodesic segments, the union of the orthogonal projections on σ of the segments of g_{0} and of c between two successive intersections between them covers σ.

It follows that the number of intersections between c′ and g_{0} is at most equal to (l_{c′} + l_{g_{0}})/l, where l_{c′} is the length of the orthogonal projection of c′ on σ and l_{g_{0}} is the length of the orthogonal projection of g_{0} on σ (and l is the length of σ).

But the hypothesis on g_{0} shows that l_{g_{0}} ≤ l, while Lemma 5.14 shows that l_{c′} ≤ C. So the number of intersections between c′ and g_{0} is at most Ce^{L}, where C is some positive constant.

Since this inequality applies to all geodesic segments in the support of λ, we find that

i(λ_{T \σ_{M}} , g_{0}) ≤ Ce^{L}i(λ, ∂σ_{M}) ,

and Lemma 5.13 then shows that i(λ_{T \σ_{M}} , g_{0}) is bounded by a positive constant.

Proof of Proposition 5.12. Let γ be a c-curve in ∂M. It follows from Proposition 5.3 that

L_{G_{λ}(m)}(γ) ≤ L_{m}(γ) + i(λ, γ) ≤ C(L_{m}(γ) + 1) ,

where here again C is a constant depending only on the topology of M. Moreover Corollary 5.11 indicates that

L_{t}(γ) ≤ L_{G_{λ}(m)}(γ) ,

and point (1.) follows.

For point (2.) consider a closed geodesic γ contained in the union of T and of the thick part of ∂M, such that

• the intersection of γ with T has two connected components γ_{1} and γ_{2},

• the intersection of γ with the thick part of ∂M (for m) has two connected components γ′_{1} and γ′_{2}, and each has length bounded by C.

If T separates the boundary component of ∂M containing it, γ has to go through T twice, otherwise it is not necessary but it is still possible to choose γ with this property, and both cases can then be treated in a uniform manner.

Once such a curve γ has been found, it is possible to change it by Dehn twists so that, in addition to the conditions above, the segments γ_{1} and γ_{2} “wrap” at most once around T, i.e., their orthogonal projection to the core σ of T is injective. This is achieved by “untwisting” γ as much as is necessary.

Denote as above by σ the core of T, and by σ_{M} the set of points at distance at most c_{M} from σ. Since γ wraps at most once around T, the length of the intersection of γ_{1} and γ_{2} with σ_{M} is at most 3c_{M}. It then follows from Proposition 5.3 that

i(λ_{|σ_{M}} , γ) ≤ C ,

where C is some positive constant. By the same proposition, the intersection of γ with the restriction of λ to the thick part of (∂M, m) is at most C. But Corollary 5.15 shows that

i(λ_{|T \σ_{M}} , γ) ≤ C .

Putting together those estimates we obtain that i(λ, γ) ≤ C, where C is yet another positive constant. The definition of the grafted metric then proves that

L_{G_{λ}(m)}(γ) ≤ L_{m}(γ) + C .

Finally Lemma 5.9 indicates that the length of γ for t is less than that for G_{λ}(m). The result follows.

5.4 A bound on the length of the earthquake lamination

We now switch from 3-dimensional to 2-dimensional geometry to show that an upper bound on the length of curves in (∂M, t) — relative to the length of the same curves in (∂M, m), as stated in Proposition 5.12 — implies a lower bound on the same lengths. Proposition 5.2 will follow.

We consider a closed surface Σ, with some marked points x_{1}, ⋯, x_{n_{0}}, and an angle θ_{i} ∈ (0, π) attached to x_{i}.

Proposition 5.16. For each C > 0 there is a constant C′ > 0 as follows. Let h, h′ ∈ H_{Σ,x,θ} be two hyperbolic metrics such that:

1. for each c-curve γ in Σ, L_{h′}(γ) ≤ CL_{h}(γ),

2. for each long tube T in the thin part of (Σ, h), T might also be a long tube for h′, but its length for h′ is at most its length for h plus C.

Let ν ∈ M_{Σ,x} be the measured lamination such that h′ = E_{r}(ν)(h). Then the length L_{h}(ν) is at most C′.

The proof of Proposition 5.16 will use a basic estimate on the variation of the length of curves under an earthquake, essentially taken from [BS09].

Proposition 5.17. Let m ∈ ML_{Σ,x} be a measured lamination, let g ∈ H_{Σ,x,θ} be a hyperbolic metric with cone singularities, and let g′ := E_{m}^{r}(g). Let γ be a c-curve. Then

|L_{g}(γ) − L_{g′}(γ)| ≤ i(m, γ) ≤ L_{g}(γ) + L_{g′}(γ) .

Proof. The upper bound on i(m, γ) can be found in [BS09] (Lemma 7.1, p. 76); it is stated there for closed curves, but the proof extends directly to segments between two singular points.

For the lower bound on i(m, γ), suppose first that the support of m is a disjoint union of simple closed curves. Consider the geodesic (for g) γ_{0} homotopic to γ in Σ_{x}, and let γ_{1} be its image by the earthquake E_{m}^{r}, along with the union of the segments in the support of m between two points corresponding — after the earthquake — to one intersection of m with γ_{0}. γ_{1} is homotopic to γ_{0} in Σ_{x}. Clearly L_{g′}(γ_{1}) = L_{g}(γ) + i(m, γ), while L_{g′}(γ) ≤ L_{g′}(γ_{1}). It follows that L_{g′}(γ) ≤ L_{g}(γ) + i(m, γ). The same inequality also holds with g and g′ exchanged, and the lower bound on i(m, γ) follows. The result when m is a general lamination — not rational — holds by density of the rational laminations in ML_{Σ,x}.

We now return to the notations used in Proposition 5.16. Note that the support of ν is a geodesic lamination in (Σ, g). It is therefore possible to consider the intersection of ν with the thin (resp. thick) part of Σ for g, which we call ν_{t} (resp. ν_{T}). The same decomposition can be done for g′, leading to ν′_{t} and ν′_{T}.

We first state a basic property of hyperbolic surfaces, which is necessary below.

Lemma 5.18. There exist r > 0, C > 0 and θ_{0} ∈ (0, π), depending only on the supremum θ_{M} of the θ_{i} and on the genus of Σ, such that, for any x ∈ Σ_{T} and any geodesic segment γ_{0} of length 2r centered at x, there exists a closed geodesic in Σ of length at most C intersecting γ_{0} with angle at least θ_{0}.

Proof. Note that any maximal segment in the thick part of a topologically non-trivial hyperbolic surface (with cone singularities of angle less than π) intersects some closed geodesic, of length bounded by a constant C.

The statement therefore follows from a straightforward compactness argument. Indeed, if the constant θ < θ_{M} did not exist, there would be a sequence of thick hyperbolic surfaces with boundary Σ_{T,n} (with cone singularities of angles less than π), for which the optimal value of r would go to infinity, or the optimal value of θ would go to π, as n → ∞. This sequence could be taken of fixed topology, and the diameter of those surfaces would then be bounded, so that r would necessarily be bounded.

We could then choose a converging subsequence, and obtain a thick hyperbolic surface (with cone singularities of angle less than θ_{M}) for which some maximal geodesic segment intersects no closed geodesic of length less than C transversally, a contradiction.

Lemma 5.19. There exists a constant C (depending only on the genus of Σ) so that the length of ν_{T} is at most C.

Proof. Let r_{0} > 0 be smaller than the injectivity radius of (Σ, h) at each point of Σ_{T}. There exists another number r_{1} ∈ (0, r_{0}) with the following property: if γ_{0} and γ_{1} are two disjoints geodesics in H^{2} and x ∈ γ_{0} is at distance at most r_{1} from γ_{1}, then any geodesic intersecting γ_{0} at distance less than r_{1} from x and making an angle bigger than θ_{0} with γ_{0} intersects γ_{1} at distance at most r_{0} from x.

Choose a large constant C_{1} > 0. If the length of ν_{T} were bigger than some large constant, the sum of the weights of the segments of the support of ν intersecting some geodesic disk of radius r_{1} and center x ∈ supp(ν) ∩ Σ_{T} would be bigger than C_{1}. Applying the previous lemma, with γ_{0} equal to a segment containing x in the support of ν, would yield a closed curve c in Σ_{T}, of bounded length, such that i(c, ν) is arbitrarily large.

Proposition 5.17 would then show that the length of c for h′ is much larger than the length of c for h, contradicting point (1) in the hypothesis of Proposition 5.16.

Lemma 5.20. There exists a constant C > 0 as follows. Let γ ⊂ Σ_{T} be a geodesic segment of length at most c_{M}. Then

i(ν, γ) ≤ CL(ν_{T}) .

Proof. We will consider the case when ν is rational, the general case follows by density of the rational measured laminations.

Let r be the injectivity radius of Σ_{T}. Let ν_{γ} be the union of the intersections with Σ_{T} of all geodesic segments centered at a point x ∈ γ, of length 2r, in the support of ν. Each of those segments has length at least r, since at least one side of x is contained in Σ_{T}.

By definition of r, those segments intersect γ exactly once. Moreover, the length of ν_{T} is larger than the sum over the segments of their length times their weight (this sum is finite since ν is rational). But this sum is at least ri(ν, γ), so that L(ν_{T}) ≥ ri(ν, γ). This proves the lemma.

Lemma 5.21. There exists a constant C > 0 such that, if T is a tube of length 2L in (Σ_{t}, h), with core σ, then i(ν, σ) ≤ Ce^{−L}.

Proof. L_{h}(σ) = c_{1}e^{−L}, where c_{1} is some constant. Point (1) in the hypothesis of Proposition 5.16 shows that the length of σ for h′ is at most c_{2}e^{−L}, where c_{2} is another positive constant. But Proposition 5.17 then yields the result.

Recall that we call σ_{M} the set of points at distance at most c_{M} from σ.

Lemma 5.22. There exists a constant C > 0 such that, if T is a tube of length 2L in (Σ_{t}, h), with core σ, then the length for h of the restriction of ν to σ_{M} is at most C.

Note that there is no reason to believe that this statement is optimal; indeed, it appears quite reasonable to think that the bound could be improved to Ce^{−L}. The bound given here, however, is both sufficient for our needs and easier to obtain.

Proof. We know by Lemma 5.21 that i(ν, σ) ≤ C_{1}e^{−L}, where C_{1} > 0 is some constant. It follows that, if L_{h}(ν_{|σ_{M}}) is larger than some constant C_{2}, then each leaf of ν intersects σ with a very small angle, and for any geodesic segment γ going through T and intersecting σ with angle bigger than π/4, i(γ, ν_{|σ_{M}}) ≥ C_{2}e^{L}.

Let γ_{1} be a closed geodesic in (Σ, h) which has two segments in Σ_{T} and two segments going through T. Furthermore we choose γ_{1} with the smallest length. Since Σ_{T} has bounded diameter, there is C > 0 such that ℓ_{h}(γ_{1}) ≤ C + 2T. Since γ_{1} has minimal length, it intersects σ with angle at most π/4 at each of the two intersections. Then i(ν, γ_{1}) ≥ 2C_{2}e^{L}, so that, by Proposition 5.17, L_{h′}(γ_{1}) is much larger than L_{h}(γ_{1}). This contradicts the hypothesis of Proposition 5.16.

Proof of Proposition 5.16. According to Lemma 5.19, the length of the restriction of ν to the thick part of Σ is bounded by a constant (depending only on the genus of Σ). Σ is the union of Σ_{T}, a finite set of neighborhoods of cusps and cone singularities (which can be disregarded because geodesic laminations do not enter them), and a finite set of long thin tubes, the number of those tubes being at most 3g − 3, where g is the genus of Σ. Let T be one of those tubes, and let σ be its core. Then L(ν_{|σ_{M}}) is bounded by a constant by Lemma 5.22. Moreover, the length of each maximal segment of the support of ν in T \ σ_{M} is at most 2e^{L}, and each is contained in a maximal segment in T (contained in the support of ν) which intersects σ once. Since i(ν, σ) ≤ Ce^{L}, the length of the restriction of ν to T \ σ_{M} is at most 4C. Summing all contributions to the length of ν yields the desired result.

Proof of Proposition 5.2. The statement clearly follows from Proposition 5.12 and from Proposition 5.16.

6 The conformal structure at infinity

This section contains the proof of Theorems 1.7 and 1.14, mostly as a consequence of Lemma 5.2.

6.1 A topological lemma

We first state a simple topological lemma, necessary below to apply Proposition 5.2 as directly as possible. We fix a closed surface S of genus at least 2, a n-tuple of points x = (x_{1}, ⋯, x_{n_{0}}) and a n_{0}-tuple of angles θ = (θ_{1}, ⋯, θ_{n_{0}}) ∈ (0, π)^{n}.

Lemma 6.1. Let c > 0, and let K ⊂ H_{S,x,θ} be a compact subset. The set of all elements of H_{S,x,θ} obtained by a right earthquake along a measured lamination of length at most c on an element of K is relatively compact.

Proof. Let m ∈ K. The set of measured laminations l ∈ ML of length less than c for m is compact in ML_{S,x}. Since the earthquake map is continuous relative to the measured lamination factor, the set

E_{r}({l ∈ ML_{S,x} | L_{m}(l) ≤ c} × {m})

is compact in H_{S,x,θ}.

Again because the earthquake map E_{r} is continuous, it follows that there is a neighborhood U_{m} of m in H_{S,x,θ} such that the image by E_{r} of

{(l, m′) ∈ ML_{S,x} × U_{m} | L_{m′}(l) ≤ C}

is relatively compact.

Since K is compact, it is covered by finitely many such neighborhoods U_{m_{i}}, for m_{i} in K. The result follows.

6.2 Compactness relative to the conformal structure at infinity

The previous considerations lead to a simple proof of Proposition 1.6 from Proposition 5.2 and Lemma 3.13.

Consider a sequence (g_{n})_{n∈ℕ} of quasifuchsian metrics with particles, as in Proposition 1.6. Let (m_{n})_{n∈ℕ} be the sequence of induced metrics on the boundary of the convex core, and let t_{n} be the sequence of hyperbolic metrics in the conformal class at infinity τ_{n}. Since (τ_{n})_{n∈ℕ} converges by the hypothesis of Proposition 1.6, (t_{n})_{n∈ℕ} converges to a limit t, so it remains in a compact subset of H_{S,x,θ} × H_{S,x,θ}.

But m_{n} is obtained from t_{n} by an earthquake along a measured lamination ν_{n} which, by Proposition 5.2, has bounded length. Lemma 6.1 therefore shows that (m_{n})_{n∈ℕ} remains in a compact subset of H_{S,x,θ} × H_{S,x,θ}. We can therefore extract a sub-sequence so that (m_{n})_{n∈ℕ} converges.

Lemma 3.13 then shows that (g_{n})_{n∈ℕ} has a subsequence which converges to a quasifuchsian metric with particles. This proves the proposition.

6.3 Proof of Theorem 1.7

We are now ready to prove Theorem 1.7. It is helpful to introduce additional notations:

• M_{S,x} := ∪_{θ∈(0,π)^{N}} M_{S,x,θ} is the space of quasifuchsian metrics with particles on S ×ℝ with a fixed number of particles but with varying angles,

• H_{S,x} := ∪_{θ∈(0,π)^{N}} H_{S,x,θ} is the space of hyperbolic metrics on S with a fixed number of cone singularities but with varying angles,

• Δ_{S,x} := ∪_{θ∈(0,π)^{N}} H_{S,x,θ} × H_{S,x,θ} is a kind of diagonal with respect to the angle variable in H_{S,x} × H_{S,x}.

Note that, by a result of Troyanov [Tro91] already mentioned above, H_{S,x} can be naturally identified with T_{S,x} × (0, π)^{N}. The notation is nonetheless useful in the argument presented here.

Consider the natural map:

Φ_{S,x} : M_{S,x} → H_{S,x} × H_{S,x}

sending a hyperbolic metric with particles on S×ℝ (with cone angles given by the θ_{i}) to the conformal structures at ±∞. It follows from the definition that the image of Φ_{S,x} is contained in Δ_{S,x}.

Let Φ_{S,x,θ} be the restriction of Φ_{S,x} to M_{S,x,θ}, for a fixed θ ∈ (0, π)^{N}. The main result of [MS09] is that — in a slightly more general context, allowing for more topology and for singularities along a graph — Φ_{S,x,θ} is a local homeomorphism from M_{S,x,θ} to H_{S,x,θ} × H_{S,x,θ}. It follows that Φ_{S,x} is a local homeomorphism from M_{S,x} to Δ_{S,x}. Moreover, Φ_{S,x} is proper by Proposition 1.6, so that it is a covering of T_{S,x,θ} × T_{S,x,θ}.

To prove that Φ_{S,x} is a (global) homeomorphism we need to show that some elements of the target space have exactly one inverse image. Suppose that for all i ∈ {1, ⋯, N}, θ_{i} = 2π/k_{i}, where k_{i} ∈ ℕ, k_{i} ≥ 2. Let τ_{+}, τ_{−} ∈ H_{S,x,θ}. There exists a finite covering π : \overline{S} → S, with ramification of order k_{i} at the x_{i}, such that τ_{+} (resp. τ_{−}) lifts to a non-singular hyperbolic metric \overline{τ}_{+} (resp. \overline{τ}_{−}). By the Bers double uniformization theorem [Ber60] \overline{τ}_{+} and \overline{τ}_{−} are in the conformal class at infinity of a unique quasifuchsian hyperbolic metric, say \overline{g}, on \overline{S} × ℝ. Since it is unique, \overline{g} is invariant under the deck transformations of π, so that \overline{g} is the pull-back to \overline{S} × ℝ of a hyperbolic metric g on S × ℝ, with cone singularities of angle θ_{i} along \{x_{i}\} × ℝ, 1 ≤ i ≤ N. This construction also shows that g is unique — since any other hyperbolic metric with particles of the given angles would lift to a non-singular quasifuchsian metric on \overline{S} × ℝ, which would have to be \overline{g}. This shows that (τ_{+}, τ_{−}) has a unique inverse image by Φ_{S,x}, so that Φ_{S,x} is a homeomorphism from M_{S,x} to Δ_{S,x}.

6.4 Proof of Theorem 1.14

We need another natural map.

Definition 6.2. Let Ψ_{S,x,θ} : H_{S,x,θ} × H_{S,x,θ} → H_{S,x,θ} × H_{S,x,θ} be defined as follows. Given (t_{+}, t_{−}) ∈ H_{S,x,θ} × H_{S,x,θ} and θ = (θ_{1}, ⋯, θ_{n_{0}}) ∈ (0, π)^{n}, there is by Theorem 1.7 a unique quasifuchsian metric with particles g ∈ M_{S,x,θ} such that Φ_{S,x,θ}(g) = ([t_{+}], [t_{−}]). Then

Ψ_{S,x,θ}(t_{+}, t_{−}) = (m_{+}, m_{−}) ,

where m_{+} and m_{−} are the conformal classes of the induced metrics on the upper and lower boundary components of the convex core of (S × ℝ, g), respectively.

According to Proposition 5.2 and Lemma 6.1, if (m_{+}, m_{−}) = Ψ_{S,x,θ}(t_{+}, t_{−}), then m_{+} is C_{q}-quasi-conformal to t_{+}, and m_{−} is C_{q}-quasi-conformal to t_{−}. This shows that Ψ_{S,x,θ} is proper and extends continuously to a map which is the identity on the boundary at infinity of H_{S,x,θ} × H_{S,x,θ}, so that it is onto. This proves Theorem 1.14.

7 Some questions and remarks

7.1 Some questions.

The quasifuchsian cone-manifolds described above are direct extensions of the “usual” quasifuchsian hyperbolic manifolds which have received much attention over the last couple of decades. It is quite natural to wonder whether some properties which have been conjectured in the smooth case can be extended to the singular setting.

Question 7.1. Does uniqueness hold in Theorem 1.14?

Another natural question, which is “dual” to the previous one in a precise sense (see e.g. [Sch03]) concerns the measured bending lamination on the boundary of the convex core.

Question 7.2. Does uniqueness hold in Theorem 1.13?

The same questions can be asked for submanifolds of quasifuchsian cone-manifolds which are convex but have a smooth boundary, which is orthogonal to the singular locus. In the smooth case it is known [Sch06] that one can prescribe the induced metric on the boundary, as well as the its third fundamental form (the smooth analog of the measured bending lamination) and that each is obtained uniquely, it would be interesting to know whether the same is true for quasi-fuchsian cone-manifold. The methods of [Sch06] do not appear to extend directly to the singular case.

Since the Alhfors-Bers theorem extends as Theorem 1.7 to quasifuchsian manifolds with particles, it is quite natural to ask whether the Ending Lamination Conjecture (see [BMM10, BCM12]) can also be extended to hyperbolic manifolds with particles. A natural starting point would be to consider manifolds homeomorphic to S × ℝ, where S is a closed surface of genus at least 2.

Note also that those questions are not necessarily restricted to quasifuchsian cone-manifolds, and could also be asked for “convex co-compact cone-manifolds”, if that term is understood in a proper way.

7.2 AdS manifolds with particles.

Mess [Mes07, ABB^{+}07] discovered a remarkable analogy between quasifuchsian hyperbolic 3-manifolds and globally hyperbolic maximal compact (GHMC) AdS manifolds. In particular he proved an analog of the Bers double uniformization theorem form GHMC AdS manifolds: on a manifold homeomorphic to S × ℝ, where S is a closed surface of genus at least 2, the space of GHMC AdS manifolds is parametrized by the product of two copies of the Teichmüller space of S, through the “left” and “right” parts of the holonomy representation.

GHMC AdS manifolds also have a convex core, whose boundary has a hyperbolic induced metric, as in the quasifuchsian setting, and is pleated along a measured lamination. The analog of Theorem 1.13 holds in that context [BS12]: any two measured laminations that fill S can be obtained as the bending lamination of the boundary of the convex core. But the uniqueness remains elusive, as in the quasifuchsian setting. Moreover, the analog of Theorem 1.14 is also conjectured to be true but no result is known.

It is also possible to consider GHMC AdS manifolds with “particles”, i.e., cone singularities along time-like geodesics, for which the angle is less than π. The analog of the Bers double uniformization theorem (more directly, the analog of Theorem 1.7) holds in this AdS setting [BS09]. The analog of Theorem 1.13 is also true in that setting [BS12]. However no analog of Theorem 1.14 is known.

Still in the AdS setting, new phenomena can occur when the singularity is along a graph (so that the particles are allowed to interact), see [BBS11, BBS12]. One can associate to a GHMC AdS manifold with a graph of interacting particles a sequence of pairs of points in the Teichmüller space of the underlying surface, with each pair corresponding to a “slice” where no interaction occurs. It would be interesting to know whether any analog of this description holds for quasifuchsian hyperbolic manifolds with cone singularities along a graph (perhaps with some conditions on the cone angles, for instance cone angles less tha 2π).

7.3 The renormalized volume.

Theorem 1.7 for quasifuchsian manifolds with particles has possible applications to the Teichmüller theory of hyperbolic surfaces with cone singularities (of fixed angles) on a surface. Indeed it was remarked in [KS08a] that the definition of the renormalized volume of a quasifuchsian 3-manifolds extends to manifolds with particles. Knowing Theorem 1.7, it is possible to remark that the key property of the renormalized volume — to be a Kähler potential for the Weil-Petersson metric on Teichmüller space — extends to the natural Weil-Petersson metric on the Teichmüller space of hyperbolic metrics with cone singularities (of prescribed angle less than π) on a surface; the proof from [KS08a, KS12] directly extends to this setting.

One direct consequence is that this Weil-Petersson metric is Kähler, as was discovered by Schumacher and Trapani [ST08] by other means. This metric, however, seems to depend on the choice of the cone angles.

Another possible application is to some properties of the grafting map considered on hyperbolic surfaces with cone singularities of angle less than π, as considered in [KS08b, KS12]. This is however less directly related to what we are doing here, since it only uses the geometry of 3-dimensional hyperbolic ends — rather than quasifuchsian metrics — with particles.

A Quasiconformal estimates

This appendix contains the proof of Proposition 1.15. The first step is a simple extension to hyperbolic surfaces with cone singularities of some classical tools concerning pants decompositions.

A.1 Pants decompositions

The content of this subsection is probably well known, see e.g. [DP07] for closely related considerations. We include this material for completeness.

Let S be a closed surface, and let h be a hyperbolic metric on S with cone singularities at some points x_{1}, ⋯, x_{n_{0}}, with cone angles θ_{1}, ⋯, θ_{n_{0}} ∈ (0, π). If h had cusps — or geodesic boundary components — at the x_{i} rather than cone singularities, it would be quite natural to consider pants decompositions of (S, h). With cone singularities of angles less than π, it remains possible.

Definition A.1. A singular pair of pants is a hyperbolic surface with geodesic boundary, possibly containing cone singularities of angle less than π, which is either:

• a hyperbolic pair of pants (with geodesic boundary) containing no cone singularity,

• a hyperbolic annulus with geodesic boundary containing exactly one cone singularity,

• a hyperbolic disk with geodesic boundary containing exactly two cone singularities.

Given a singular hyperbolic pair of pants, its three geodesic boundary components or cone singularities will be called its legs. We hope that the reader will excuse us for this weird and perhaps confusing terminology.

Definition A.2. A pants decomposition of S is a decomposition S = S_{1} ∪ ⋯ ∪ S_{n} of S as the union of finitely many subsurfaces with disjoint interior, each of which is a singular pair of pants.

It is implicit in this definition that the boundary of the S_{i} contains no cone singularities; the cone singularities are each contained in the interior of one of the singular pairs of pants.

Lemma A.3. There exists a constant C_{p} > 0 such that, for any choice of S and h, (S, h) has a pants decomposition with all boundary curves of length less than C_{p}.

Sketch of the proof. A standard recursive argument (see e.g. [BP92]) reduces the proof to showing that, for any hyperbolic surface with cone singularities (of angle less than π) and geodesic boundary, there is a simple closed geodesic of length at most C_{p} which is not homotopic to a singular point or to a boundary component.

This in turn follows from other standard arguments, for instance based on comparing the area of the surface (given by a suitable Gauss-Bonnet formula, see e.g. [Tro91]) to the area of embedded geodesic disks.

Definition A.4. Let P be a singular pant. Its leg invariants are the length of its geodesic boundary components and the angles at its cone singularities.

For instance, the boundary invariants of a (non-singular) hyperbolic pair of pants are the lengths of its boundary components.

Lemma A.5. Each hyperbolic pair of pants is uniquely determined, up to isometry, by its leg invariants and by the type of its “legs” — whether they are boundary components or cone singularities.

The proof follows the classical arguments used for non-singular hyperbolic pairs of pants, it is based on elementary properties of some hyperideal hyperbolic triangles stated below in three propositions (the first two have probably been known since Lobachevsky).

Recall that a hyperideal triangle can be defined, using the projective model of the hyperbolic plane, as a triangle which might have its vertices either in the hyperbolic plane, on its ideal boundary, or outside the closure of the hyperbolic plane (considered as the interior of a disk in the projective plane), but with all edges intersecting the hyperbolic plane. A vertex is then ideal if it is on the ideal boundary, and strictly hyperideal if it is outside the closed disk.

Recall also that given a point v_{0} outside the closure of the projective model of H^{2} (in the projective plane), there is a unique hyperbolic geodesic, v_{0}^{∗}, such that any the intersection with the projective model of H^{2} of any projective line containing v_{0} is orthogonal to v_{0}^{∗}. This geodesic is called the line dual to v_{0}.

We introduce here a slightly restricted notion of hyperideal triangle.

Definition A.6. An extended hyperbolic triangle is a hyperbolic triangle with one or more strictly hyperideal vertices and its other vertices in the “interior” of the hyperbolic plane. A truncated hyperbolic triangle is the intersection of an extended hyperbolic triangle with the hyperbolic half-planes bounded by the lines dual to its strictly hyperideal vertices (and not containing the endpoints of the edges going towards those vertices).

For instance, a right-angle hyperbolic hexagon can be considered — in two ways — as a truncated hyperbolic triangle, with three strictly hyperideal vertices. Given a hyperbolic triangle, its angles are the hyperbolic angles at the non-hyperideal vertices and the length of its intersections with the lines dual to the strictly hyperideal vertices. Notes that those lengths can quite naturally be considered as angles (they are then imaginary numbers) but it is not necessary to enter such considerations here (see e.g. [Sch98, Sch01] for more details).

There is a natural way to define the edge lengths of an extended hyperbolic triangle. The length of the edge joining two vertices v and v′ is:

• the hyperbolic distance between v and v′, if neither v nor v′ is strictly hyperideal,

• the hyperbolic distance between v and the line dual to v′, when v′ is hyperideal but v is not,

• the distance between the lines dual to v and v′, when both are strictly hyperideal.

It is useful to remark that the lengths and angles of an extended hyperbolic triangle satisfy a natural extension of the cosine formula. Moreover it is quite easy to check that an extended hyperbolic triangle, with vertices of given type, is uniquely determined by two lengths and one angle, or by two angles and one length.

Lemma A.7. An extended hyperbolic triangle is uniquely determined by the type of its vertices — whether they are “usual” or strictly hyperideal vertices — and its edge lengths.

Proof. This statement is classical for “usual” hyperbolic triangles, with no strictly hyperideal vertex. It is also well-known for triangles with three strictly hyperideal vertices, see [BP92].

Consider an extended hyperbolic triangle, with exactly one strictly hyperideal vertex, say v_{1}, and two usual vertices, v_{2} and v_{3}. Let l_{i} be the length of the edge between v_{j} and v_{k}, for {i, j, k} = {1, 2, 3}. Consider l_{2}, l_{3} as fixed, then l_{1} is easily seen (using for instance the cosine formula for extended hyperbolic triangles) to be a strictly increasing function of the angle δ at v_{1}. This proves the lemma in this case.

Consider now the situation where v_{1} is a “usual” vertex, while v_{2} and v_{3} are strictly hyperideal. Then given v_{1}, the positions of the lines dual to v_{2} and v_{3} are completely determined by the angle α at v_{1}. Moreover the distance between those lines, which by definition is equal to l_{1}, is a strictly increasing function of α. This shows the result in this last case.

The same arguments can be used to prove the “dual” lemma, concerning the possible angles. Here we need a more precise statement, in addition to the fact that the angles determine the triangle we need to know that a large class of triples of angles can actually be realized.

Lemma A.8. An extended hyperbolic triangle is uniquely determined by the type of its vertices — whether they are “usual” or strictly hyperideal vertices — and its angles. The angles at the “usual” vertices can take any value in (0, π/2), while the “angles” at the strictly hyperideal vertices can be any numbers in (0, ∞).

Proof. Again the case where all vertices are “usual” is classical, while the case where all three vertices are strictly hyperideal is well-known.

Consider a triangle T with one “usual” vertex, say v_{1}, and two strictly hyperideal vertices, v_{2} and v_{3}. Let e_{1}, e_{2}, e_{3} be the edges opposite to v_{1}, v_{2}, v_{3} respectively. The triangle T is completely determined by the length l_{1} of the edge e_{1} and by the “angles” α_{2} and α_{3}, that is, the lengths of the segments of the lines v_{2}^{∗}, v_{3}^{∗} dual to v_{2}, v_{3} between their intersections with e_{1} and with e_{3} and e_{2}, respectively.

Given α_{2} and α_{3}, the possible values of l_{1} vary between a minimal value l_{1,m} and a maximal value l_{1,M}. Suppose for instance that α_{2} ≥ α_{3}, then l_{1,m} corresponds to the situation where e_{3} is reduced to a point. Then, after truncation, T corresponds to a quadrilateral with 3 right angles. The last angle, between v_{2}^{∗} and e_{2}, has to be less than π/2 by the Gauss-Bonnet theorem. This means that for l_{1} slightly larger than l_{1,m}, α_{3} > π/2. On the other hand, α_{3} → 0 as l_{1} → l_{1,M}, and α_{3} is a decreasing function of l_{1} ∈ (l_{1,m}, l_{1,M}). This proves the lemma for triangles with two strictly hyperideal vertices.

Consider now a triangle T′ with one strictly hyperideal vertex, say v_{1}, and two “usual” vertices, v_{2} and v_{3}. Consider α_{2}, α_{3} ∈ (0, π/2) as fixed, T′ is then entirely determined by l_{1}. l_{1} can vary between a minimal value l_{1,m} > 0 and ∞, where l_{1,m} corresponds to the case where v_{1} is an ideal vertex. The angle α_{1} then varies between 0 and ∞, and is a strictly increasing function of l_{1}. The result follows.

Lemma A.9. Each singular pair of pants has a unique decomposition as the union of two copies of a truncated hyperbolic triangle (glued along their common boundary).

Proof. Let v_{1}, v_{2}, v_{3} be the three legs — which could be either singular points or boundary components. There is a unique homotopy class of embedded segment joining v_{i} to v_{j}, for i ≠ j. Those three segments can be uniquely realized as minimizing geodesics, which are then orthogonal to the boundary components. Cutting the pair of pants along those three geodesic segments yields two extended hyperbolic triangles, glued along their edges. Those two extended triangles have the same edge lengths, so that they are isometric by Lemma A.7.

Proof of Lemma A.5. By Lemma A.8, the two extended triangles glued to obtain a hyperbolic pair of pants are uniquely determined by their angles, which can take any value as long as the angles at the “usual” vertices are less than π/2. This shows that hyperbolic pairs of pants are uniquely determined by their leg invariants, and any values are possible as long as the angles at the singular points are less than π.

We now turn to the parameterization of hyperbolic metrics with cone singularities by Fenchel-Nielsen type coordinates. We first state a lemma on the existence and uniqueness of a pants decomposition from topological data, leaving the proof to the reader since it is the same as in the non-singular case.

Lemma A.10. A pants decomposition is uniquely determined by the choice of the boundary curves γ_{1}, ⋯, γ_{N}, considered as simple closed curves in S \ {x_{1}, ⋯, x_{n_{0}}}, under the hypothesis that:

• the γ_{i} can be realized as pairwise disjoint curves,

• each connected component of their complement is either a pair of pants containing none of the x_{i}, or a cylinder containing exactly one of the x_{i}, or a disk containing exactly two of the x_{i}.

Finally we state the main consequence, on the parameterization of hyperbolic metrics with cone singularities of fixed angle by Fenchel-Nielse coordinates, again leaving the proof to the reader. Note that the Dehn twist parameters are defined only in a relative way, however this is exactly the same as in the non-singular case (see e.g. [BP92]).

Corollary A.11. Given a (topological) pants decomposition of S with boundary curves γ_{1}, ⋯, γ_{N}, there is a homeomorphism

T_{S,x,θ} → (ℝ_{>0} × ℝ)^{N}

sending a hyperbolic metric to the length and fractional Dehn twist parameters at the γ_{i}.

The fractional Dehn twist parameters used here are the translation length of one side with respect to the other so that, for a boundary curve of length l, a parameter equal to l corresponds to a “usual” Dehn twist (the other possibility is to use an “angle” parameter, where 2π corresponds to full Dehn twist).

A.2 Proof of Proposition 1.15

It is now possible to use the pants decomposition provided by Lemma A.3 to prove Proposition 1.15: the induced metric on the boundary of the convex core is (uniformly) quasi-conformal to the conformal structure at infinity.

The starting point is that a pants decomposition of (∂M, m) with boundary curves of bounded length defines a pants decomposition of (∂M, τ) with boundary curves of approximately the same length. Recall that the constant C_{p} was introduced in Lemma A.3.

Lemma A.12. There exists a constant C > 0 as follows. Let γ_{1}, ⋯, γ_{N} be simple closed curves on ∂M, defining a pants decomposition, of lengths less than C_{p} for m. Then

∀i ∈ \{1, ⋯, N\}, \frac{L_{m}(γ_{i})}{C} ≤ L_{τ} (γ_{i}) ≤ CL_{m}(γ_{i}) .

Proof. The upper bound is a direct consequence of the first point in Proposition 5.12. If γ_{i} is short for m — i.e., it is the core of a long tube in the thin part of (∂M, m) — then the second point of Proposition 5.12 proves the lower bound for γ_{i}.

Suppose now that γ_{i} is realized in (∂M, m) as a closed geodesic in the thick part of ∂M. Then there exists a closed geodesic γ′ intersecting γ_{i} of length at most C_{p}. If the length of γ_{i} in (∂M, τ) were small, than γ_{i} would be realized in (∂M, τ) as the core of a long tube T in the thin part of (∂M, τ). But then γ′ would have to be long (at least as long as the T). This would contradict the first point in Proposition 5.12, and this proves the lower bound for γ_{i}.

Lemma A.13. There exists a constant C > 0 such that, for each of the γ_{i}, the difference in the Dehn twist parameter corresponding to γ_{i} in m and in τ is at most C(| log(L_{m}(γ_{i}))| + 1).

The precise form of the estimate is important only if γ_{i} is short for m (and therefore for τ), in which case | log(L_{m}(γ_{i}))| is half the length of the tube in the thin part of (∂M, m) containing γ_{i}.

Proof. Suppose first that γ_{i} is not short. Then it is contained in the thick part of (∂M, m), and there exists another curve γ′, intersecting γ_{i}, of uniformly bounded length. A Dehn twist parameter bigger than some constant would extend the length of γ′ by more than is allowed by Proposition 5.12, this proves the lemma in this first case.

The same argument can be used when γ_{i} is short (i.e. when it is the core of a long thin tube), then γ′ can be chosen to have length bounded by a constant time | log(L_{m}(γ_{i}))|, and this defines the maximal Dehn twist parameter along γ_{i}.

Proof of Proposition 1.15. Let γ_{1}, ⋯, γ_{N} be disjoint closed curves, defining a pants decomposition of (∂M, m) with boundary curves of length less than C_{p}, as in Lemma A.3. Let l_{1}, ⋯, l_{N} be the length of the γ_{i} for m, and let d_{1}, ⋯, d_{N} be the Dehn twist parameters for the same curves.

Lemma A.10 shows that the γ_{i} also define a pant decomposition of (∂M, τ), let l′_{i} be the length of the γ_{i} for τ, and let d′_{i} be their Dehn twist parameters. Lemma A.12 indicates that the l′_{i} are within a fixed multiplicative constant from the l_{i}, while, by Lemma A.13,

|d′_{i} − d_{i}| ≤ C(| log(l_{i})| + 1) ,  (3)

where C is some positive constant.

Let m′ be the hyperbolic metric with cone singularities obtained by gluing pairs of pants with boundary lengths equal to the l_{i}, but with Dehn twist parameters equal to the d′_{i}.

Note that m′ is C_{1}-quasi-conformal to m, for some uniform constant C_{1} > 0. To prove this remark that for each i ∈ {1, ⋯, N} the set of points at distance at most C(| log(L_{m}(γ_{i}))| + c_{M}) from γ_{i} is an annulus, and that those annuli are disjoint. One can therefore build a C_{1}-quasi-conformal diffeomorphism between m and m′ which is an isometry in the complement of those annuli around the γ_{i}, and which is “twisted” in those annuli, with a twisting parameter which is an affine function of the distance to the γ_{i}.

The second and last step is that m′ is C_{3}-quasi-conformal τ. Since those two metrics differ only by the lengths of the boundary curves γ_{i}, and in view of (3), this follows again from a simple and explicit construction which we leave to the interested reader.

Note that it might be possible to prove Proposition 1.15 using the same type of arguments as those used by Epstein and Marden [EM86] in the non-singular case. This would have the advantage of providing directly a quasiconformal constant independent on the genus of the boundar.

Acknowledgements

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