Rigidity of compact Fuchsian manifolds with convex boundary

Abstract

A compact Fuchsian manifold with boundary is a hyperbolic 3-manifold homeomorphic to $S_g\times [0;1]$ such that the boundary component $S_g\times \{0\}$ is geodesic. We prove that a compact Fuchsian manifold with convex boundary is uniquely determined by the induced path metric on $S_g\times \{1\}$. We do not put further restrictions on the boundary except convexity.