Convexity and rigidity of hypersurfaces in Cartan-Hadamard manifolds

Geometry Topology Seminar
Monday, September 11, 2023 - 2:00pm for 1 hour (actually 50 minutes)
Skiles 006
Mohammad Ghomi – Georgia Tech
John Etnyre

We show that in Cartan-Hadamard manifolds M^n, n≥ 3, closed infinitesimally convex hypersurfaces S bound convex flat regions, if curvature of M^n vanishes on tangent planes of S. 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(0) spaces, and other techniques from Alexandrov geometry outlined by A. Petrunin, including Reshetnyak’s majorization theorem, and Kirszbraun’s extension theorem.