Surface projective convexe de volume fini

TL;DR

This paper characterizes finite volume convex projective surfaces, showing that such surfaces are strictly convex with smooth boundaries unless the domain is a triangle, and relates the finite volume property to dual surfaces.

Contribution

It provides new characterizations of finite volume convex projective surfaces and establishes a duality property for their volume finiteness.

Findings

Convex projective surfaces of finite volume are strictly convex with C^1 boundary unless the domain is a triangle.

Finite volume property is equivalent for a surface and its dual.

If the domain is not a triangle, it must be strictly convex with smooth boundary.

Abstract

A convex projective surface is the quotient of a properly convex open $\Omega$ of $\mathbb{P}(\R)$ by a discret subgroup $\Gamma$ 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 $\Omega$ 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.