Total absolute curvature and rigidity of surfaces in Cartan-Hadamard manifolds

TL;DR

This paper proves that closed surfaces with minimal total absolute curvature in Cartan-Hadamard 3-manifolds bound flat convex bodies, extending classical theorems and solving a long-standing problem.

Contribution

It generalizes Chern-Lashof's theorem to Cartan-Hadamard manifolds and introduces a new embedding construction using holonomy.

Findings

Closed surfaces with minimal total absolute curvature bound flat convex bodies.

Established a regularity result for convex hulls in this setting.

Proved a Schur-type comparison theorem for curves in Cartan-Hadamard manifolds.

Abstract

We show that closed surfaces with minimal total absolute curvature in Cartan-Hadamard 3-manifolds bound flat convex bodies. This generalizes Chern-Lashof's theorem for surfaces in Euclidean space and solves a problem posed by Gromov in 1985. Our proof is based on an isometric embedding construction via holonomy, and uses Pogorelov's theory of surfaces with bounded extrinsic curvature. Along the way, we obtain a regularity result for convex hulls and a Schur-type comparison theorem for curves in Cartan-Hadamard manifolds.