Continuity of total curvatures of Riemannian hypersurfaces

TL;DR

This paper proves the continuity of total generalized mean curvatures for certain hypersurfaces and convex bodies within Riemannian and Cartan-Hadamard spaces, with respect to Hausdorff distance.

Contribution

It establishes the continuity of total generalized mean curvatures for hypersurfaces with positive reach and convex bodies in specific geometric spaces.

Findings

Total generalized mean curvatures are continuous under Hausdorff convergence.

Continuity holds for hypersurfaces with positive reach in Riemannian manifolds.

Convex bodies in Cartan-Hadamard spaces also exhibit this continuity.

Abstract

We show that total generalized mean curvatures of hypersurfaces with positive reach in Riemannian manifolds, and convex bodies in Cartan-Hadamard spaces, are continuous with respect to Hausdorff distance.