Maximal surfaces in anti-de Sitter 3-manifolds with particles

We prove the existence of a unique maximal surface in each anti-de Sitter (AdS) convex Globally Hyperbolic Maximal (GHM) manifold with particles (that is, with conical singularities along time-like lines) for cone angles less than $\pi$. We interpret this result in terms of Teichm\"uller theory, and prove the existence of a unique minimal Lagrangian diffeomorphism isotopic to the identity between two hyperbolic surfaces with cone singularities when the cone angles are the same for both surfaces and are less than $\pi$.Comment: Accepted for publication at "Annales de l'institut Fourier

The geometry of maximal representations of surface groups into SO(2,n)

In this paper, we study the geometric and dynamical properties of maximal representations of surface groups into Hermitian Lie groups of rank 2. Combining tools from Higgs bundle theory, the theory of Anosov representations, and pseudo-Riemannian geometry, we obtain various results of interest. We prove that these representations are holonomies of certain geometric structures, recovering results of Guichard and Wienhard. We also prove that their length spectrum is uniformly bigger than that of a suitably chosen Fuchsian representation, extending a previous work of the second author. Finally, we show that these representations preserve a unique minimal surface in the symmetric space, extending a theorem of Labourie for Hitchin representations in rank 2.Comment: 56 pgs, section 3 has been reorganized , former sections 4.2 and 4.3 have been merged into section 4.2 and rewritten to avoid reference to maximal surfaces and Higgs bundles, appendix added on strong version of Ahlfors-Schwarz-Pick lemma. To appear in Duke Math Journa

Minimal Lagrangian diffeomorphisms between hyperbolic cone surfaces and Anti-de Sitter geometry

We study minimal diffeomorphisms between hyperbolic cone-surfaces (that is diffeomor- phisms whose graph are minimal submanifolds). We prove that, given two hyperbolic metrics with the same number of conical singularities of angles less than π, there always exists a minimal diffeomorphism isotopic to the identity. When the cone-angles of one metric are strictly smaller than the ones of the other, we prove that this diffeomorphism is unique. When the angles are the same, we prove that this diffeomorphism is unique and area- preserving (so is minimal Lagrangian). The last result is equivalent to the existence of a unique maximal space-like surface in some Globally Hyperbolic Maximal (GHM) anti-de Sitter (AdS) 3-manifold with particles

Plateau Problems for Maximal Surfaces in Pseudo-Hyperbolic Spaces

We define and prove the existence of unique solutions of an asymptotic Plateau problem for spacelike maximal surfaces in the pseudo-hyperbolic space of signature (2, n): the boundary data is given by loops on the boundary at infinity of the pseudo-hyperbolic space which are limits of positive curves. We also discuss a compact Plateau problem. The required compactness arguments rely on an analysis of the pseudo-holomorphic curves defined by the Gauss lifts of the maximal surfaces.Comment: 85 pages, 3 figures, in the version the statement of the compactness theorem 6.1 has been made more explicit for further use in some other articl

On complete maximal submanifolds in pseudo-hyperbolic space

We provide a full classification of complete maximal $p$-dimensional spacelike submanifolds in the pseudo-hyperbolic space $\mathbf{H}^{p,q}$, and we study its applications to Teichm\"uller theory and to the theory of Anosov representations of hyperbolic groups in $\mathsf{PO}(p,q+1)$.Comment: 60 page

Irreducible decomposition for local representations of quantum Teichmüller space

We give an irreducible decomposition of the so-called local representations \cite{math/0407086} of the quantum Teichmüller space $\mathcal{T}_q(\Sigma)$ where $\Sigma$ is a punctured surface of genus $g>0$ and $q$ is a $N$-th root of unity with $N$ odd

Maximal Surface in AdS convex GHM 3-manifold with particles

We prove the existence of a unique maximal surface in an anti-de Sitter (AdS) convex Globally Hyperbolic Maximal (GHM) manifold with particles (i.e. with conical singularities along timelike lines) for cone-angles less than $\pi$. We reinterpret this result in terms of Teichm\"uller theory, and prove the existence of a unique minimal Lagrangian diffeomorphism isotopic to the identity between two hyperbolic structures with conical singularities of the same angles on a closed surface with marked points.

Minimal diffeomorphism between hyperbolic surfaces with cone singularities

We prove the existence of a minimal diffeomorphism isotopic to the identity between two hyperbolic cone surfaces (Σ,g1) and (Σ,g2) when the cone angles of g1 and g2 are different and smaller than π. When the cone angles of g1 are strictly smaller than the ones of g2, this minimal diffeomorphism is unique

COMPACT CONNECTED COMPONENTS IN RELATIVE CHARACTER VARIETIES OF PUNCTURED SPHERES

International audienceWe prove that some relative character varieties of the fundamental group of a punctured sphere into the Hermitian Lie groups SU(p, q) admit compact connected components. The representations in these components have several counter-intuitive properties. For instance, the image of any simple closed curve is an elliptic element. These results extend a recent work of Deroin and the first author, which treated the case of PU(1, 1) = PSL(2, R). Our proof relies on the non-Abelian Hodge correspondance between relative character varieties and parabolic Higgs bundles. The examples we construct admit a rather explicit description as projective varieties obtained via Geometric Invariant Theory