We investigate properties of the pseudo-Riemannian volume, entropy, and diameter for convex cocompact representations \(ρ: Γ\to \mathrm S \mathrm O(p,q+1)\)of closed \(p\)-manifold groups. In particular: We provide a uniform lower bound of the product entropy times volume that depends only on the geometry of the abstract group \(Γ\). We prove that the entropy is bounded from above by \(p-1\)with equality if and only if \(ρ\)is conjugate to a representation inside \(\mathrm S(\mathrm O(p,1)\times\mathrm O(q))\), which answers affirmatively to a question of Glorieux and Monclair. Lastly, we prove finiteness and compactness results for groups admitting convex cocompact representations with bounded diameter.

Geom. Topol.

The infimum of the dual volume of convex co-compact hyperbolic \(3\)-manifolds

We show that the infimum of the dual volume of the convex core of a convex co-compact hyperbolic \(3\)-manifold with incompressible boundary coincides with the infimum of the Riemannian volume of its convex core, as we vary the geometry by quasi-isometric deformations. We deduce a linear lower bound of the volume of the convex core of a quasi-Fuchsian manifold in terms of the length of its bending measured lamination, with optimal multiplicative constant.

Forum Math. \(Σ\)

Length functions in Teichmüller and anti-de Sitter geometry

We establish a link between the behavior of length functions on Teichmüller space and the geometry of certain anti de Sitter \(3\)-manifolds. As an application, we give new purely anti de Sitter proofs of results of Teichmüller theory such as (strict) convexity of length functions along shear paths and geometric bounds on their second variation along earthquakes. Along the way, we provide shear-bend coordinates for Mess’ anti de Sitter \(3\)-manifolds.

Math. Ann.

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

Even though it is known that there exist quasi-Fuchsian hyperbolic three-manifolds that do not admit any monotone foliation by constant mean curvature (CMC) surfaces, a conjecture due to Thurston asserts the existence of CMC foliations for all almost-Fuchsian manifolds, namely those quasi-Fuchsian manifolds that contain a closed minimal surface with principal curvatures in \((-1,1)\). In this paper we prove that there exists a (unique) monotone CMC foliation for all quasi-Fuchsian manifolds that lie in a sufficiently small neighborhood of the Fuchsian locus.

ArXiv preprint

\(d\)-pleated surfaces and their shear-bend coordinates

In this article, we single out representations of surface groups into \(\mathsf P \mathsf S \mathsf L_d(\mathbb C)\,\,\)which generalize the well-studied family of pleated surfaces into \(\mathsf P \mathsf S \mathsf L_2(\mathbb C)\). Our representations arise as sufficiently generic \(λ\)-Borel Anosov representations, which are representations that are Borel Anosov with respect to a maximal geodesic lamination \(λ\). For fixed \(λ\,\,\)and \(d \), we provide a holomorphic parametrization of the space \(\mathcal R(λ,d)\, \)of \((λ,d)\)-pleated surfaces which extends both work of Bonahon for pleated surfaces and Bonahon and Dreyer for Hitchin representations.

2022

ArXiv preprint

\(\mathrm S \mathrm O_0(2,n+1)\)-maximal representations and hyperbolic surfaces

We study maximal representations of surface groups \(ρ: \pi_1(Σ)\to\mathrm S\mathrm O(2,n+1)\,\,\)via the introduction of \(ρ\)-invariant pleated surfaces inside the pseudo-Riemannian space \(\mathbb H^2,n\,\,\)associated to maximal geodesic laminations of \(Σ\). We prove that \(ρ\)-invariant pleated surfaces are always embedded, acausal, and possess an intrinsic pseudo-metric and a hyperbolic structure. We describe the latter by constructing a shear cocycle from the cross ratio naturally associated to \(ρ\). The process developed to this purpose applies to a wide class of cross ratios, including examples arising from Hitchin and \(Θ\)-positive representations in \(\mathrm S\mathrm O(p,q)\). We also show that the length spectrum of \(ρ\,\,\)dominates the ones of \(ρ\)-invariant pleated surfaces, with strict inequality exactly on curves that intersect the bending locus. We observe that the canonical decomposition of a \(ρ\)-invariant pleated surface into leaves and plaques corresponds to a decomposition of the Guichard-Wienhard domain of discontinuity of \(ρ\,\,\)into standard fibered blocks, namely triangles and lines of photons. Conversely, we give a concrete construction of photon manifolds fibering over hyperbolic surfaces by gluing together triangles of photons. The tools we develop allow to recover various results by Collier, Tholozan, and Toulisse on the (pseudo-Riemannian) geometry of \(ρ\,\,\)and on the correspondence between maximal representations and fibered photon manifolds through a constructive and geometric approach, bypassing the use of Higgs bundles.

Trans. AMS

The dual volume of quasi-Fuchsian manifolds and the Weil-Petersson distance

Making use of the dual Bonahon-Schläfli formula, we prove that the dual volume of the convex core of a quasi-Fuchsian manifold \(M\,\,\)is bounded by an explicit constant, depending only on the topology of \(M\), times the Weil-Petersson distance between the hyperbolic structures on the upper and lower boundary components of the convex core of \(M\).

2021

ArXiv preprint

Para-hyperKähler geometry of the deformation space of maximal globally hyperbolic anti-de Sitter three-manifolds

In this paper we study the para-hyperKähler geometry of the deformation space of MGHC anti-de Sitter structures on \(Σ\times\mathbb R\), for \(Σ\,\,\)a closed oriented surface. We show that a neutral pseudo-Riemannian metric and three symplectic structures coexist with an integrable complex structure and two para-complex structures, satisfying the relations of para-quaternionic numbers. We show that these structures are directly related to the geometry of MGHC manifolds, via the Mess homeomorphism, the parameterization of Krasnov-Schlenker by the induced metric on \(K\)-surfaces, the identification with the cotangent bundle \(T^*\mathcal T(Σ) \), and the circle action that arises from this identification. Finally, we study the relation to the natural para-complex geometry that the space inherits from being a component of the \(\mathrm P \mathrm S \mathrm L(2,\mathbb B)\)-character variety, where \(\mathbb B\,\,\)is the algebra of para-complex numbers, and the symplectic geometry deriving from Goldman symplectic form.

Given a differentiable deformation of geometrically finite hyperbolic \(3\)–manifolds \((M_t)_t\), the Bonahon–Schläfli formula (J. Differential Geom. 50 (1998) 25–58) expresses the derivative of the volume of the convex cores \((C M_t)_t\,\,\)in terms of the variation of the geometry of their boundaries, as the classical Schläfli formula (Q. J. Pure Appl. Math. 168 (1858) 269–301) does for the volume of hyperbolic polyhedra. Here we study the analogous problem for the dual volume, a notion that arises from the polarity relation between the hyperbolic space \(\mathbb H^3\,\,\)and the de Sitter space \(\mathrm d\mathbb S^3\). The corresponding dual Bonahon–Schläfli formula has been originally deduced from Bonahon’s work by Krasnov and Schlenker (Duke Math. J. 150 (2009) 331–356). Applying the differential Schläfli formula (Electron. Res. Announc. Amer. Math. Soc. 5 (1999) 18–23) and the properties of the dual volume, we give a (almost) self-contained proof of the dual Bonahon–Schläfli formula, without making use of Bonahon’s results.

2019

ArXiv preprint

Constant Gaussian curvature foliations and Schläfli formulas of hyperbolic \(3\)-manifolds

Filippo Mazzoli

to appear in Ann. Sc. norm. super. Pisa - Cl. sci., Oct 2019

We study the geometry of the foliation by constant Gaussian curvature surfaces \((\Sigma_k)_k\,\,\)of a hyperbolic end, and how it relates to the structures of its boundary at infinity and of its pleated boundary. First, we show that the Thurston and the Schwarzian parametrizations are the limits of two families of parametrizations of the space of hyperbolic ends, defined by Labourie in 1992 in terms of the geometry of the leaves \(\Sigma_k\). We give a new description of the renormalized volume using the constant curvature foliation. We prove a generalization of McMullen’s Kleinian reciprocity theorem, which replaces the role of the Schwarzian parametrization with Labourie’s parametrizations. Finally, we describe the constant curvature foliation of a hyperbolic end as the integral curve of a time-dependent Hamiltonian vector field on the cotangent space to Teichmüller space, in analogy to the Moncrief flow for constant mean curvature foliations in Lorenzian space-times.

2016

ArXiv preprint

Intertwining operators of the quantum Teichmüller space

In arXiv:0707.2151 the authors introduced the theory of local representations of the quantum Teichmüller space \(\mathcal T^q_S \,\) (\(q\,\,\)being a fixed primitive \(N\)-th root of \( (-1)^N + 1 \)) and they studied the behaviour of the intertwining operators in this theory. One of the main results [Theorem 20, arXiv:0707.2151] was the possibility to select one distinguished operator (up to scalar multiplication) for every choice of a surface \(S\), ideal triangulations \(λ, λ’\,\,\)and isomorphic local representations \(ρ, ρ’\), requiring that the whole family of operators verifies certain Fusion and Composition properties. By analyzing the constructions of arXiv:0707.2151, we found a difficulty that we eventually fix by a slightly weaker (but actually optimal) selection procedure. In fact, for every choice of a surface \(S\), ideal triangulations \(λ, λ’\,\,\)and isomorphic local representations \(ρ, ρ’\), we select a finite set of intertwining operators, naturally endowed with a structure of affine space over \(H_1(S;\mathbb Z_N) \,\) (\(\mathbb Z_N \,\,\)is the cyclic group of order \(N\)), in such a way that the whole family of operators verifies augmented Fusion and Composition properties, which incorporate the explicit behavior of the \(\mathbb Z_N\)-actions with respect to such properties. Moreover, this family is minimal among the collections of operators verifying the weak Fusion and Composition rules (in practice the ones considered in arXiv:0707.2151). In addition, we adapt the derivation of the invariants for pseudo-Anosov diffeomorphisms and their hyperbolic mapping tori made in arXiv:0707.2151 and arXiv:math/0407086 by using our distinguished family of intertwining operators.