quasi-Fuchsian Manifolds Research Articles

Quasi-Fuchsian manifolds close to the Fuchsian locus are foliated by constant mean curvature surfaces

Quasi-Fuchsian manifolds close to the Fuchsian locus are foliated by constant mean curvature surfaces

Fenchel–Nielsen coordinates on the augmented moduli space of anti-de Sitter structures

In this paper we combine our recent work on regular globally hyperbolic maximal anti-de Sitter structures with the classical theory of globally hyperbolic maximal Cauchy-compact anti-de Sitter manifolds in order to define an augmented moduli space. Moreover, we introduce a coordinate system in this space that resembles the complex Fenchel–Nielsen coordinates for hyperbolic quasi-Fuchsian manifolds.

Volumes of quasifuchsian manifolds

Quasifuchsian hyperbolic manifolds, or more generally convex co-compact hyperbolic manifolds, have infinite volume, but they have a well-defined ``renormalized'' volume. We outline some relations between this renormalized volume and the volume, or more precisely the ``dual volume'', of the convex core. On one hand, there are striking similarities between them, for instance in their variational formulas. On the other, object related to them tend to be within bounded distance. Those analogies and proximities lead to several questions. Both the renormalized volume and the dual volume can be used for instance to bound the volume of the convex core in terms of the Weil-Petersson distance between the conformal metrics at infinity.

Asymptotically Poincaré surfaces in quasi-Fuchsian manifolds

We introduce the notion of an asymptotically Poincaré family of surfaces in an end of a quasi-Fuchsian manifold. We show that any such family gives a foliation of an end by asymptotically parallel convex surfaces, and that the asymptotic behavior of the first and second fundamental forms determines the projective structure at infinity. As an application, we establish a conjecture of Labourie from [J. London Math. Soc. 45 (1992), pp. 549–565] regarding constant Gaussian curvature surfaces. We also derive consequences for constant mean curvature surfaces.

Complex length of short curves and Minimal Fibrations of hyperbolic three‐Manifolds fibering over the circle

We investigate the maximal solid tubes around short simple geodesics in hyperbolic three-manifolds and how complex length of curves relate to closed, incompressible, least area minimal surfaces. As applications, we prove, there are some closed hyperbolic three-manifolds fibering over the circle which are not foliated by closed incompressible minimal surfaces diffeomorphic to the fiber. We also show, the existence of quasi-Fuchsian manifolds containing arbitrarily many embedded closed incompressible minimal surfaces. Our strategy is to prove main theorems under natural geometric conditions on the complex length of closed curves on a fibered hyperbolic three-manifold, then we find explicit examples where these conditions are satisfied via computer programs.

Entropy of embedded surfaces in quasi-Fuchsian manifolds

We compare critical exponent for quasi-Fuchsian groups acting on the hyperbolic 3-space, $\mathbb{H}^3$, and on invariant disks embedded in $\mathbb{H}^3$. We give a rigidity theorem for all embedded surfaces when the action is Fuchsian and a rigidity theorem for negatively curved surfaces when the action is quasi-Fuchsian.

Compact domains with prescribed convex boundary metrics in quasi-Fuchsian manifolds

We show the existence of a convex compact domain in a quasi-Fuchsian manifold such that the induced metric on its boundary coincides with a prescribed surface metric of curvature $K\geq-1$ in the sense of A. D. Alexandrov.

Counting closed geodesics in globally hyperbolic maximal compact AdS 3-manifolds

We propose a definition for the length of closed geodesics in a globally hyperbolic maximal compact (GHMC) Anti-De Sitter manifold. We then prove that the number of closed geodesics of length less than R grows exponentially fast with R and the exponential growth rate is related to the critical exponent associated to the two hyperbolic surfaces coming from Mess parametrization. We get an equivalent of three results for quasi-Fuchsian manifolds in the GHMC setting: Bowen’s rigidity theorem of critical exponent, Sanders’ isolation theorem and McMullen’s examples lightening the behaviour of this exponent when the surfaces range over Teichmuller space.

Geodesic planes in hyperbolic 3-manifolds

This paper initiates the study of rigidity for immersed, totally geodesic planes in hyperbolic 3-manifolds M of infinite volume. In the case of an acylindrical 3-manifold whose convex core has totally geodesic boundary, we show that the closure of any immersed geodesic plane is a properly immersed submanifold of M. On the other hand, we show that rigidity fails for quasifuchsian manifolds.

Minimal discs in hyperbolic space bounded by a quasicircle at infinity

We prove that the supremum of principal curvatures of a minimal embedded disc in hyperbolic three-space spanning a quasicircle in the boundary at infinity is estimated in a sublinear way by the norm of the quasicircle in the sense of universal Teichmüller space, if the quasicircle is sufficiently close to being the boundary of a totally geodesic plane. As a by-product we prove that there is a universal constant C independent of the genus such that if the Teichmüller distance between the ends of a quasi-Fuchsian manifold M is at most C, then M is almost-Fuchsian. The main ingredients of the proofs are estimates on the convex hull of a minimal surface and Schauder-type estimates to control principal curvatures.

Counting minimal surfaces in quasi-Fuchsian three-manifolds

It is well known that every quasi-Fuchsian manifold admits at least one closed incompressible minimal surface, and at most finitely many stable ones. In this paper, for any prescribed integer $N > 0$, we construct a quasi-Fuchsian manifold which contains at least $2^N$ such minimal surfaces. As a consequence, there exists some simple closed Jordan curve on $S_{\infty }^2$ such that there are at least $2^N$ disk-type complete minimal surfaces in $\mathbb {H}^3$ sharing this Jordan curve as the asymptotic boundary.

The convex core of quasifuchsian manifolds with particles

We consider quasifuchsian manifolds with "particles", i.e., cone singularities of fixed angle less than $\pi$ 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 Teichm\"uller space of a surface with $n$ marked points. This is analoguous to the Bers theorem in the non-singular case. 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 [BO] in the non-singular case.

Polyhedral metrics on the boundaries of convex compact quasi-Fuchsian manifolds

We show the existence of a convex compact subset in a quasi-Fuchsian manifold such that the induced metric on the boundary of the subset coincides with a prescribed hyperbolic polyhedral metric.

Cusp geometry of fibered 3-manifolds

Let $F$ be a surface and suppose that $\varphi\colon F \to F$ is a pseudo-Anosov homeomorphism, fixing a puncture $p$ of $F$. The mapping torus $M = M_\varphi$ is hyperbolic and contains a maximal cusp $C$ about the puncture $p$. We show that the area (and height) of the cusp torus $\partial C$ is equal to the stable translation distance of $\varphi$ acting on the arc complex $\scr{A}(F,p)$, up to an explicitly bounded multiplicative error. Our proof relies on elementary facts about the hyperbolic geometry of pleated surfaces. In particular, the proof of this theorem does not use any deep results from Teichm\"uller theory, Kleinian group theory, or the coarse geometry of $\scr{A}(F,p)$. A similar result holds for quasi-Fuchsian manifolds $N\cong F\times\Bbb{R}$. In that setting, we find a combinatorial estimate for the area (and height) of the cusp annulus in the convex core of $N$, up to explicitly bounded multiplicative and additive error. As an application, we show that covers of punctured surfaces induce quasi-isometric embeddings of arc complexes.

On almost-Fuchsian manifolds

An almost-Fuchsian manifold is a class of complete hyperbolic three-manifolds. Such a three-manifold is a quasi-Fuchsian manifold which contains a closed incompressible minimal surface with principal curvatures everywhere in the range of ( − 1 , 1 ) (-1,1) . In such a manifold, the minimal surface is unique and embedded, hence one can parametrize these hyperbolic three-manifolds by their minimal surfaces. In this paper we obtain estimates on several geometric and analytical quantities of an almost-Fuchsian manifold M M in terms of the data on the minimal surface. In particular, we obtain an upper bound for the hyperbolic volume of the convex core of M M and an upper bound on the Hausdorff dimension of the limit set associated to M M . We also constructed a quasi-Fuchsian manifold which admits more than one minimal surface, and it does not admit a foliation of closed surfaces of constant mean curvature.

The renormalized volume and the volume of the convex core of quasifuchsian manifolds

We show that the renormalized volume of a quasifuchsian hyperbolic 3-manifold is equal, up to an additive constant, to the volume of its convex core. We also provide a precise upper bound on the renormalized volume in terms of the Weil-Petersson distance between the conformal structures at infinity. As a consequence we show that holomorphic disks in Teichmuller space which are large must have enough negative curvature. 1. Results 1.1. Notations. In all the paper we consider a closed surface S of genus g ≥ 2, and we call TS the Teichmuller space of S. Given a complex structure c ∈ TS, we denote by Qc the vector space of holomorphic quadratic differentials on (S,c). We also call GS the space of quasifuchsian metrics on S × R, considered up to isotopy. The Bers Simultaneous Uniformization Theorem provides a homeomorphism between T S∪S and GS. So GS is parameterized by T+ × T−, where T+ and T− are two copies of TS corresponding respectively to the upper and lower boundaries at infinity of S × R. For q ∈ G, we denote by VR(q) the renormalized volume of (S × R,g) (as defined in Section 3 following e.g. (12)), while C(q) is the convex core of (S × R,g) and VC(q) is its volume. 1.2. Comparing the renormalized volume to the volume of the convex core. The first result pre- sented here is a precise comparison between the volume of the convex core and the renormalized volume of a quasifuchsian hyperbolic manifold.

The Surface Area Preserving Mean Curvature Flow in Quasi-Fuchsian Manifolds

In this paper, we consider the surface area preserving mean curvature flow in quasi-Fuchsian 3-manifolds. We show that the flow exists for all times and converges exponentially to a smooth surface of constant mean curvature with the same surface area as the initial surface.

Minimal Lagrangian surfaces in $${\mathbb {CH}^2}$$ and representations of surface groups into SU(2, 1)

We use an elliptic differential equation of Ţiţeica (or Toda) type to construct a minimal Lagrangian surface in $${\mathbb {CH}^2}$$ from the data of a compact hyperbolic Riemann surface and a cubic holomorphic differential. The minimal Lagrangian surface is equivariant for an SU(2, 1) representation of the fundamental group. We use this data to construct a diffeomorphism between a neighbourhood of the zero section in a holomorphic vector bundle over Teichmuller space (whose fibres parameterise cubic holomorphic differentials) and a neighborhood of the $${\mathbb {R}}$$ -Fuchsian representations in the SU(2, 1) representation space. We show that all the representations in this neighbourhood are complex-hyperbolic quasi-Fuchsian by constructing for each a fundamental domain using an SU(2, 1) frame for the minimal Lagrangian immersion: the Maurer–Cartan equation for this frame is the Ţiţeica-type equation. A very similar equation to ours governs minimal surfaces in hyperbolic 3-space, and our paper can be interpreted as an analog of the theory of minimal surfaces in quasi-Fuchsian manifolds, as first studied by Uhlenbeck.

Quasi-Fuchsian 3-Manifolds and Metrics on Teichmüller Space

An almost Fuchsian 3-manifold is a quasi-Fuchsian manifold which contains an incompressible closed minimal surface with principal curvatures in the range of (−1, 1). By the work of Uhlenbeck, such a 3-manifold M admits a foliation of parallel surfaces, whose locus in Teichmuller space is represented as a path γ, we show that γ joins the conformal structures of the two components of the conformal boundary of M . Moreover, we obtain an upper bound for the Teichmuller distance between any two points on γ, in particular, the Teichmuller distance between the two components of the conformal boundary of M , in terms of the principal curvatures of the minimal surface in M . We also establish a new potential for the Weil-Petersson metric on Teichmuller space.

Quasi-Fuchsian manifolds with particles

We consider 3-dimensional hyperbolic cone-manifolds which are “convex co-compact” in a natural sense, with cone singularities along infinite lines. Such singularities are sometimes used by physicists as models for massive spinless point particles. We prove an infinitesimal rigidity statement when the angles around the singular lines are less than $\pi$: any infinitesimal deformation changes either these angles, or the conformal structure at infinity with marked points corresponding to the endpoints of the singular lines. Moreover, any small variation of the conformal structure at infinity and of the singular angles can be achieved by a unique small deformation of the cone-manifold structure. These results hold also when the singularities are along a graph, i.e., for “interacting particles”.