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.