Rigidity of nonpositively curved manifolds with convex boundary

TL;DR

This paper proves a rigidity theorem for compact 3-manifolds with convex boundary and nonpositive curvature, characterizing when they are isometric to convex domains in space forms, extending previous results in geometric rigidity.

Contribution

It establishes a new rigidity result for manifolds with convex boundary and curvature bounds, generalizing earlier theorems by Greene-Wu, Gromov, and Schroeder-Strake.

Findings

Manifolds with boundary and curvature bounds are isometric to space forms under certain conditions.

The proof uses a comparison formula for total curvature of hypersurfaces.

Results also apply to manifolds with nonnegative curvature bounds.

Abstract

We show that a compact Riemannian $3$-manifold $M$ with strictly convex simply connected boundary and sectional curvature $K\leq a\leq 0$ 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 $K\geq a\geq 0$.