arXiv subjects

Francois Labourie

Publications and source records attributed to Francois Labourie.

8 recordsLinked to original sources

arXiv2013

The pressure metric for Anosov representations

Using the thermodynamics formalism, we introduce a notion of intersection for projective Anosov representations, show analyticity results for the intersection and the entropy, and rigidity results for the intersection. We use the renormalized intersection to produce a $Out(\Gamma)$-invariant Riemannian metric on the smooth points of the deformation space of irreducible, projective Anosov representations of a word hyperbolic group $\Gamma$ into $SL(m,R)$ whose Zariski closure contains a generic element. In particular, we produce mapping class group invariant Riemannian metrics on Hitchin components which restrict to the Weil--Petersson metric on the Fuchsian loci. Moreover, we produce $Out(\Gamma)$-invariant metrics on deformation spaces of convex cocompact representations into $PSL(2,C)$ and show that the Hausdorff dimension of the limit set varies analytically over analytic families of convex cocompact representations into any rank 1 semi-simple Lie group.

math.DG↗

arXiv2011

Geodesics on Margulis spacetimes

Let M be a Margulis spacetime whose associated complete hyperbolic surface S has compact convex core. Generalizing the correspondence between closed geodesics on M and closed geodesics on S, we establish an orbit equivalence between recurrent spacelike geodesics on M and recurrent geodesics on S. In contrast, no timelike geodesic recurs in either forward or backwards time.

math.DS↗

arXiv2005

Maximal Representations of Surface Groups: Symplectic Anosov Structures

Let G be a connected semisimple Lie group such that the associated symmetric space X is Hermitian and let Gamma be the fundamental group of a compact orientable surface of genus at least 2. We survey the study of maximal representations, that is the subset of Hom(Gamma,G) which is a union of components characterized by the maximality of the Toledo invariant. Then we concentrate on the particular case G=SP(2n,R), and we show that the image of Gamma under any maximal representation is a discrete faithful realization of Gamma as a Kleinian group of complex motions in X with an associated Anosov system, and whose limit set in an appropriate compactification of X is a rectifiable circle.

math.DG↗

arXiv2005

Anosov Flows, Surface Groups and Curves in Projective Space

In 1990, Hitchin's proved a component of the space of representations of a surface group in SL(n,R) is homeomorphic to a ball. For n=2,3 this component has been identified with the holonomies of geometric structures (hyperbolic for n=2, or real projective for n=3). In the preprint "Anosov flows, Surface groups and Curves in Projective Space", we extend this interpretation to higher dimension and show every representation in Hitchin's component is attached to a (special) curve in projective space, thus giving a geometric interpretation of these representations. We also prove these representations are faithful, discrete and purely loxodromic (or hyperbolic)

math.DG↗

arXiv2004

Proper affine actions and geodesic flows of hyperbolic surfaces

We give necessary and sufficient conditions for an affine deformation of a Schottky subgroup of O(2,1) to act properly on affine space. There exists a real-valued biaffine map between the cohomology of the Schottky group and the space of geodesic currents on the corresponding hyperbolic surface S. For a fixed cohomology class, this map is uniformly positive or uniformly negative on the space of geodesic currents if and only if the corresponding affine deformation is proper. As a corollary, the deformation space of proper affine deformations is an open convex cone.

math.DG↗

arXiv2000

Random k-Surfaces

The main result is the construction of ergodic transversal measures of full support on the space of all k-surfaces of a compact hyperbolic 3-manifold. This space is a laminated space, each of its leaf being identified with a "complete" k-surface, i.e. a surface of constant (extrinsic) curvature k, where k belongs to ]0,1[. Elsewhere, this space has been shown to have chaotic properties mimicking those of the geodesic flow. This construction is achieved through a coding of the space.

math.DG↗

arXiv2000

Fuchsian affine actions of surface groups

We define a fuchsian affine action of a surface group to be such that the linear part factors through a representation of $SL(2,{\mathbb R})$. We prove a fuchsian affine action of a surface group is never proper.

math.DG↗

arXiv1999

Un Lemme de Morse pour les surfaces convexes

We describe the "hyperbolic" properties of a riemann surface lamination M canonically associated to every compact three manifolds of curvature less than 1. More precisely, if the geodesic flow is the phase space attached to an ordinary differential equation, our space M is the "phase space" attached to a certain elliptic equation, which geometrically describes surfaces of constant Gauss-Kronecker curvature k (k-surfaces), 0<k<1. These hyperbolic properties are : a generic leaf is dense, the union of compact leaves is dense, and stability. They follow from an analogous of the Morse lemma for geodesics, which we call Morse lemma for convex surfaces, and which play the role of a shadowing lemma. We also prove theorems on existence and unicity of k-surfaces.

math.DG↗