Un Lemme de Morse pour les surfaces convexes

Abstract

We describe the "hyperbolic" properties of a riemann surface lamination M canonically associated to every compact three manifolds of curvature less than 1. More precisely, if the geodesic flow is the phase space attached to an ordinary differential equation, our space M is the "phase space" attached to a certain elliptic equation, which geometrically describes surfaces of constant Gauss-Kronecker curvature k (k-surfaces), 0<k<1. These hyperbolic properties are : a generic leaf is dense, the union of compact leaves is dense, and stability. They follow from an analogous of the Morse lemma for geodesics, which we call Morse lemma for convex surfaces, and which play the role of a shadowing lemma. We also prove theorems on existence and unicity of k-surfaces.