Rigidity of nonpositively curved manifolds with convex boundary

Abstract

We show that a compact Riemannian $3$-manifold $M$ with strictly convex simply connected boundary and sectional curvature $K\le a\le 0$ is isometric to a convex domain in a complete simply connected space of constant curvature $a$, provided that $K\equiv a$ on planes tangent to the boundary of $M$. This yields a characterization of strictly convex surfaces with minimal total curvature in Cartan-Hadamard $3$-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 $K\ge a\ge 0$.

arXiv comment6 pages; Minor revisions; Accepted for publications in Proceedings of AMS