Convexity and rigidity of hypersurfaces in Cartan-Hadamard manifolds

Author:

Mohammad Ghomi

Keyword:

Mathematics, Differential Geometry, Differential Geometry (math.DG), Analysis of PDEs (math.AP), Metric Geometry (math.MG)

journal:

date:

2023-08-28 16:00:00

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 characterization of 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 M. 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 A. Petrunin.

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