arXiv research

A convex projective surface is the quotient of a properly convex open $Ω$ of $\mathbb{P}(\R)$ by a discret subgroup $Γ$ of $\mathrm{SL}_3(\R)$. We give some caracterisations of the fact that a convex projective surface is of finite volume for the Busemann's measure. We deduce of this that if $Ω$ is not a triangle then …

Entropy study of geodesic flow on convex projective surfaces.

Cooper and Long generalised Epstein and Penner's Euclidean cell decomposition of cusped hyperbolic manifolds of finite volume to non-compact strictly convex projective manifolds of finite volume. We show that Weeks' algorithm to compute this decomposition for a hyperbolic surface generalises to strictly convex projecti…

In this paper we study the degeneration of convex real projective structures on bordered surfaces.

The paper studies the correlation of Hilbert lengths for convex projective surfaces.

We define convex projective structures on 2D surfaces with holes and investigate their moduli space. We prove that this moduli space is canonically identified with the higher Teichmuller space for the group PSL_3 defined in our paper math/0311149. We define the quantum version of the moduli space of convex projective s…

Affine deformations of convex cones on projective surfaces.

We prove that the space of convex real projective structures on a surface of genus $g\ge 2$ admits a mapping class group invariant Kähler metric where Teichmüller space with Weil-Petersson metric is a totally geodesic complex submanifold.

We show that the volume entropy of the Hilbert metric on a closed convex projective surface tends to zero as the corresponding Pick differential tends to infinity. The proof is based on the theorem, due to Benoist and Hulin, that the Hilbert metric and Blaschke metric are comparable.

We initiate the study of holomorphically convex groups: groups that can be realized as fundamental groups of smooth complex projective varieties with holomorphically convex universal covers. If $G$ is a holomorphically convex group of cohomological dimension two, we show that $G$ is isomorphic to the fundamental group …

The area of a convex projective surface of genus $g\ge 2$ is at least $(g-1)π^2/2+\|τ\|^2/8$ where $τ=(\log t_i)$ is the vector of triangle invariants of Bonahon-Dreyer and $t_i$ are the Fock-Goncharov triangle coordinates.

There is a canonical identification, due to the author, of a convex real projective structure on an orientable surface of genus g and a pair consisting of a conformal structure together with a holomorphic cubic differential on the surface. The Deligne-Mumford compactification of the moduli space of curves then suggests…

Study surface subgroups acting on projective space, finding bending laminations and spheres.

This article follow the article {http://hal.archives-ouvertes.fr/hal-00361030/fr/} in which the author characterize the fact of being of finite volume for a convex projective surface. We show here that the moduli space $β_f(Σ_{g,p})$ of the convex projective structure on the surface $Σ_{g,p}$ of genius $g$ with $p$ pun…

We determine that the deformation space of convex real projective structures, that is, projectively flat torsion-free connections with the geodesic convexity property on a compact 2-orbifold of negative Euler characteristic is homeomorphic to a cell of certain dimension. The basic techniques are from Thurston's lecture…

Extending the Labourie-Loftin correspondence, we establish, on any punctured oriented surface of finite type, a one-to-one correspondence between convex projective structures with specific types of ends and punctured Riemann surface structures endowed with meromorphic cubic differentials whose poles are at the puncture…

The study proves non-orientable surfaces can map to a torus.

Using results by Donaldson and Auroux on pseudo-holomorphic curves as well as Duval's rational convexity construction, the paper investigates the existence of smooth Lagrangian surfaces representing 2-dimensional homology classes in complex projective surfaces. We prove that if the projective surface X is minimal, of g…

New Finsler metrics describe trace function growth rates in convex projective surfaces.

In this paper we show that many projective Anosov representations act convex cocompactly on some properly convex domain in real projective space. In particular, if a non-elementary word hyperbolic group is not commensurable to a non-trivial free product or the fundamental group of a closed hyperbolic surface, then any …

This paper deals with non-Archimedean representations of punctured surface groups in PGL(3), associated actions on Euclidean buildings (of type A2), and degenerations of real convex projective structures on surfaces. The main result is that, under good conditions on Fock-Goncharov generalized shear parameters, non-Arch…

Study on projective structures and their foliations on surfaces.

We review some basic concepts related to convex real projective structures from the differential geometry point of view. We start by recalling a Riemannian metric which originates in the study of affine spheres using the Blaschke connection (work of Calabi and of Cheng-Yau) mentioning its relation with the Hilbert metr…

Let P(S) be the space of convex projective structures on a surface S with negative Euler characteristic. Goldman and Bonahon-Dreyer constructed two different sets of global coordinates for P(S), both associated to a pair of pants decomposition of the surface S. The article explicitly describes the coordinate change bet…

In this article we give a geometric interpretation of the Hitchin component for PSL(4,R) in the representation variety of a closed oriented surface of higher genus. We show that representations in the Hitchin component are precisely the holonomy representations of properly convex foliated projective structures on the u…

Generalising a seminal result of Epstein and Penner for cusped hyperbolic manifolds, Cooper and Long showed that each decorated strictly convex projective cusped manifold has a canonical cell decomposition. Penner used the former result to describe a natural cell decomposition of decorated Teichmüller space of puncture…

In this survey, we study representations of finitely generated groups into Lie groups, focusing on the deformation spaces of convex real projective structures on closed manifolds and orbifolds, with an excursion on projective structures on surfaces. We survey the basics of the theory of character varieties, geometric s…

In this paper we study the deformation of strictly convex real projective structures on a closed surface. Specially we study the deformation in terms of the entropy on bulging deformations. As a byproduct we construct a sequence of divergent structures whose topological entropy converges to a designated number between …

Let M be a compact surface of negative Euler characteristic and let C(M) be the deformation space of convex real projective structures on M. For every choice of pants decomposition for M, there is a well known parameterization of C(M) known as the Goldman parameterization. In this paper, we study how some geometric pro…

We give a short and simple proof of Cauchy's surface area formula, which states that the average area of a projection of a convex body is equal to its surface area up to a multiplicative constant in the dimension.

Extends illumination bodies to non-Euclidean spaces and proves their volume derivative defines surface area.

Study foliations in PSL(4,R)-Teichmüller theory, proving two invariant foliations.

We construct and study a natural homeomorphism between the moduli space of polynomial cubic differentials of degree d on the complex plane and the space of projective equivalence classes of oriented convex polygons with d+3 vertices. This map arises from the construction of a complete hyperbolic affine sphere with pres…

In this note we introduce a natural Finsler structure on convex surfaces, referred to as the projective Finsler structure, which is dual in a sense to the obvious inclusion of a convex surface in a normed space. It has an associated projective girth, which is similar to the notion of girth defined by Schäffer. We prove…

We study primitive stable representations of free groups into higher rank semisimple Lie groups and their properties. Let $Σ$ be a compact, connected, orientable surface (possibly with boundary) of negative Euler characteristic. We first verify the $σ_{mod}$-regularity for convex projective structures and positive repr…

Generalized Cauchy's surface area formula to arbitrary submanifolds in R^n.

Let S be an orientable, finite type surface with negative Euler characteristic. The augmented moduli space of convex real projective structures on S was first defined and topologized by the first author. In this article, we give an explicit description of this topology using explicit coordinates. More precisely, given …

We show that a properly convex projective structure $\mathfrak{p}$ on a closed oriented surface of negative Euler characteristic arises from a Weyl connection if and only if $\mathfrak{p}$ is hyperbolic. We phrase the problem as a non-linear PDE for a Beltrami differential by using that $\mathfrak{p}$ admits a compatib…

Study on volumes of random inscribed polytopes in projective geometries.

New algebra invariant distinguishes Legendrian knots in convex surfaces.

These notes grew out of our learning and applying the methods of Fock and Goncharov concerning moduli spaces of real projective structures on surfaces with ideal triangulations. We give a self-contained treatment of Fock and Goncharov's description of the moduli space of framed marked properly convex projective structu…

The Funk metric connects billiards, projective geometry, and convex geometry.

We introduce the notion of a minimal Lagrangian connection on the tangent bundle of a manifold and classify all such connections in the case where the manifold is a compact oriented surface of non-vanishing Euler characteristic. Combining our classification with results of Labourie and Loftin, we conclude that every pr…

Investigate pseudoconvexity of locally trivial holomorphic ball bundles over compact Riemann surfaces.

We introduce a new functional $\mathcal{E}_{\mathfrak{p}}$ on the space of conformal structures on an oriented projective manifold $(M,\mathfrak{p})$. The nonnegative quantity $\mathcal{E}_{\mathfrak{p}}([g])$ measures how much $\mathfrak{p}$ deviates from being defined by a $[g]$-conformal connection. In the case of a…

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 struc…

Develops new methods for Epstein surfaces and W-volume.

Let S be a closed, connected, orientable surface of genus at least 2, and let C(S) denote the deformation space of convex real projective structures S. In this article, we introduce two new flows on C(S), which we call the internal bulging flow and the eruption flow. These are geometrically defined flows associated to …