Surface projective convexe de volume fini

Abstract

A convex projective surface is the quotient of a properly convex open $\Omega$ of $\mathbb{P}(\mathbb{R})$ by a discret subgroup $\Gamma$ of ${\mathrm{SL}}_3(\mathbb{R})$. 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 $\Omega$ is not a triangle then $\Omega$ is strictly convex, with $\Cc^1$ boundary and that a convex projective surface $S$ is of finite volume if and only if the dual surface is of finite volume.

arXiv commentTo be publish in: Annales de l'Institut Fourier