Entropy of Hilbert metrics and length spectrum of Hitchin representations in PSL(3,R)\mathrm{PSL}(3,\mathbb{R})

We prove a sharp inequality between the Blaschke and Hilbert distance on a proper convex domain: for any two points $x$ and $y$, $$d^B(x,y) < d^H(x,y) +1.$$ We obtain two interesting consequences: the first one is the volume entropy rigidity for Hilbert geometries : for any proper convex domain of $\mathbb{R}\mathbf{P}^n$, the volume of a ball of radius $R$ grows at most like $e^{(n-1)R}$. The second consequence is the following fact: for any Hitchin representation of a surface group into $\mathrm{PSL}(3,\mathbb{R})$, there exists a Fuchsian representation $j$ in $\mathrm{PSL}(2,\mathbb{R})$ such that the length spectrum of $j$ is uniformly smaller than the length spectrum of $\rho$

The Volume of complete anti-de Sitter 3-manifolds

Up to a finite cover, closed anti-de Sitter $3$-manifolds are quotients of $\mathrm{SO}_0(2,1)$ by a discrete subgroup of $\mathrm{SO}_0(2,1) \times \mathrm{SO}_0(2,1)$ of the form $$j\times \rho(\Gamma)~,$$ where $\Gamma$ is the fundamental group of a closed oriented surface, $j$ a Fuchsian representation and $\rho$ another representation which is "strictly dominated" by $j$. Here we prove that the volume of such a quotient is proportional to the sum of the Euler classes of $j$ and $\rho$. As a consequence, we obtain that this volume is constant under deformation of the anti-de Sitter structure. Our results extend to (not necessarily compact) quotients of $\mathrm{SO}_0(n,1)$ by a discrete subgroup of $\mathrm{SO}_0(n,1) \times \mathrm{SO}_0(n,1)$

Dominating surface group representations by Fuchsian ones

We prove that a representation from the fundamental group of a closed surface of negative Euler characteristic with values in the isometry group of a Riemannian manifold of sectional curvature bounded by -1 can be dominated by a Fuchsian representation. Moreover, we prove that the domination can be made strict, unless the representation is discrete and faithful in restriction to an invariant totally geodesic 2-plane of curvature -1. When applied to representations into PSL(2,R) of non-extremal Euler class, our result is a step forward in understanding the space of closed anti-de Sitter 3-manifolds.Comment: Added details in lemma 2.3. Corrected a mistake about the link with Toledo's theorem. Removed a superfluous assumption in theorem F and added a last section about "perspectives in higher rank

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

Hausdorff dimension of limit sets for projective Anosov representations

We study the relation between critical exponents and Hausdorff dimensions of limit sets for projective Anosov representations. We prove that the Hausdorff dimension of the symmetric limit set in $\mathbf{P}(\mathbb{R}^{n}) \times \mathbf{P}({\mathbb{R}^{n}}^*)$ is bounded between two critical exponents associated respectively to a highest weight and a simple root

The Influence of extracellular matrix on lens epithelial cell viability

Posterior capsular opacification is the main complication of cataract surgery and results from the proliferation, migration and differentiation of lens epithelial cells remaining in the capsular bag. To better understand this pathological cell behaviour, 1 investigated the interactions between lens epithelial cells and the bovine lens capsule in vitro and their effect on cell viability. As determined by a colorimetric cell proliferation assay, in vitro culture of cells directly on the bovine lens capsule resulted in maintained cell viability in the presence of staurosporine in both lens epithelial cell lines tested, but in neither of the two non-lens cell lines tested. As determined by immunoblotting and reverse-transcriptase polymerase chain reaction (RT-PCR), cell viability on the bovine lens capsule could further be correlated to the presence of both ɑA-crystallin and αB-crystallin expression. A positive correlation of cell viability on the lens capsule with vimentin and HSP27 expression was also found in a smaller set of cell lines. As determined by gelatin zymography and immunoblotting, MMP-2 was expressed by lens epithelial cells, led to the release of FGF-2 and IGF-1 from the lens capsule and correlated with lens epithelial cell viability. Taken together, these results suggest that the lens capsule can act as a store of releasable growth factors available to the lens epithelial cells, with effects on their protein expression and cell viability

Simple Anosov representations of closed surface groups

We introduce and study \emph{simple Anosov representations} of closed hyperbolic surface groups, analogous to Minsky's \emph{primitive stable representations} of free groups. We prove that the set of simple Anosov representations into $\mathrm{SL}(d,\mathbb{C})$ with $d \geqslant 4$ strictly contains the set of Anosov representations. As a consequence, we construct domains of discontinuity for the mapping class group action on character varieties which contain non-discrete representations

Gromov-Thurston manifolds and anti-de Sitter geometry

We consider hyperbolic and anti-de Sitter (AdS) structures on $M\times (0,1)$, where $M$ is a $d$-dimensional Gromov-Thurston manifold. If $M$ has cone angles greater than $2\pi$, we show that there exists a "quasifuchsian" (globally hyperbolic maximal) AdS manifold such that the future boundary of the convex core is isometric to $M$. When $M$ has cone angles less than $2\pi$, there exists a hyperbolic end with boundary a concave pleated surface isometric to $M$. Moreover, in both cases, if $M$ is a Gromov-Thurston manifold with $2k$ pieces (as defined below), the moduli space of quasifuchsian AdS structures (resp. hyperbolic ends) satisfying this condition contains a submanifold of dimension $2k-3$. When $d=3$, the moduli space of quasifuchsian AdS (resp. hyperbolic) manifolds diffeomorphic to $M\times (0,1)$ contains a submanifold of dimension $2k-2$, and extends up to a "Fuchsian" manifold, that is, an AdS (resp. hyperbolic) warped product of a closed hyperbolic manifold by~$\R$. We use this construction of quasifuchsian AdS manifolds to obtain new compact quotients of \O(2d,2)/\U(d,1). The construction uses an explicit correspondence between quasifuchsian $2d+1$-dimensional AdS manifolds and compact quotients of \O(2d,2)/\U(d,1) which we interpret as the space of timelike geodesic Killing fields of \AdS^{2d+1}.Comment: 48 page