Rigidity of nonpositively curved manifolds with convex boundary

Abstract

We show that a compact Riemannian 3 3 -manifold M M with strictly convex simply connected boundary and sectional curvature K ≤ a ≤ 0 K\leq a\leq 0 is isometric to a convex domain in a complete simply connected space of constant curvature a a , provided that K ≡ a K\equiv a on planes tangent to the boundary of M M . This yields a characterization of strictly convex surfaces with minimal total curvature in Cartan-Hadamard 3 3 -manifolds, and extends some rigidity results of Greene-Wu, Gromov, and Schroeder-Strake. Our proof is based on a recent comparison formula for total curvature of Riemannian hypersurfaces, which also yields some dual results for K ≥ a ≥ 0 K\geq a\geq 0 .