Convexity and rigidity of hypersurfaces in Cartan-Hadamard manifolds

TL;DR

This paper proves that in Cartan-Hadamard manifolds, certain convex hypersurfaces bound convex flat regions under curvature conditions, extending classical Euclidean results and contributing to rigidity theory in non-positive curvature spaces.

Contribution

It generalizes Euclidean convex hypersurface characterizations to Cartan-Hadamard manifolds and establishes new rigidity results using Alexandrov geometry techniques.

Findings

Convex hypersurfaces bound convex flat regions under curvature conditions.

Extension of classical convexity characterizations to non-positive curvature spaces.

Application to bounds on total absolute curvature of surfaces in Cartan-Hadamard manifolds.

Abstract

We show that in Cartan-Hadamard manifolds $M^n$, $n\geq 3$, closed infinitesimally convex hypersurfaces $\Gamma$ bound convex flat regions, if curvature of $M^n$ vanishes on tangent planes of $\Gamma$. This encompasses Chern-Lashof-Sacksteder characterization of compact convex hypersurfaces in Euclidean space, and some results of Greene-Wu-Gromov on rigidity of Cartan-Hadamard manifolds. It follows that closed simply connected surfaces in $M^3$ with minimal total absolute curvature bound Euclidean convex bodies, as stated by Gromov in 1985. The proofs employ the Gauss-Codazzi equations, a generalization of Schur comparison theorem to CAT($k$) spaces, and other techniques from Alexandrov geometry outlined by Petrunin.