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Continuity of total curvatures of Riemannian hypersurfaces

Mohammad Ghomi

TL;DR

The paper addresses the stability of total curvatures M_r for hypersurfaces with positive reach in Riemannian manifolds and for convex bodies in Cartan–Hadamard spaces under Hausdorff convergence. It develops a self-contained approach using universal differential forms Φ_r to express M_r as pullbacks and applies Stokes' theorem to obtain continuity in the general Riemannian setting; a parallel-hypersurface estimate and coarea arguments are employed to handle convex, Cartan–Hadamard cases. Key contributions include a geometric construction of universal forms avoiding full exterior algebra, a direct continuity proof without smooth valuation machinery, and a precise bound for variations via parallel hypersurfaces that informs isoperimetric-type questions in nonpositive curvature spaces. The results provide a robust stability tool for total curvature functionals, with implications for the Cartan–Hadamard conjecture and related geometric-analytic problems.

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.

Continuity of total curvatures of Riemannian hypersurfaces

TL;DR

The paper addresses the stability of total curvatures M_r for hypersurfaces with positive reach in Riemannian manifolds and for convex bodies in Cartan–Hadamard spaces under Hausdorff convergence. It develops a self-contained approach using universal differential forms Φ_r to express M_r as pullbacks and applies Stokes' theorem to obtain continuity in the general Riemannian setting; a parallel-hypersurface estimate and coarea arguments are employed to handle convex, Cartan–Hadamard cases. Key contributions include a geometric construction of universal forms avoiding full exterior algebra, a direct continuity proof without smooth valuation machinery, and a precise bound for variations via parallel hypersurfaces that informs isoperimetric-type questions in nonpositive curvature spaces. The results provide a robust stability tool for total curvature functionals, with implications for the Cartan–Hadamard conjecture and related geometric-analytic problems.

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.

Paper Structure

This paper contains 4 sections, 6 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Universal Differential Forms
  3. Proof of Theorem \ref{['thm:main']}
  4. Proof of Theorem \ref{['thm:main2']}

Key Result

Theorem 1.1

Let $\Gamma$ be a closed hypersurface with positive reach embedded in a Riemannian manifold $M$. Suppose there exists a sequence of closed embedded hypersurfaces $\Gamma_i\subset M$ with uniformly positive reach such that $\Gamma_i\to\Gamma$ with respect to Hausdorff distance. Then $\mathcal{M}_r(\G

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1: Bernig-Bröcker bernig-brocker2003
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 4.1
  • proof