Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Integral Inequalities and Rigidity for $V$-Static-Type Equations on Manifolds with Boundary

Maria Andrade

Abstract

In this work, we study compact Riemannian manifolds with boundary satisfying V-static-type equations. By combining a generalized Reilly formula with Steklov-type boundary value problems, we derive integral inequalities for geometric quantities associated with the boundary. These inequalities lead to rigidity results, including characterizations of geodesic balls in space forms. In particular, our results offer new insights into several known rigidity theorems in the literature.

Integral Inequalities and Rigidity for $V$-Static-Type Equations on Manifolds with Boundary

Abstract

In this work, we study compact Riemannian manifolds with boundary satisfying V-static-type equations. By combining a generalized Reilly formula with Steklov-type boundary value problems, we derive integral inequalities for geometric quantities associated with the boundary. These inequalities lead to rigidity results, including characterizations of geodesic balls in space forms. In particular, our results offer new insights into several known rigidity theorems in the literature.
Paper Structure (10 sections, 20 theorems, 102 equations)

This paper contains 10 sections, 20 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Examples of $V$-static-type equation
  4. Proof of main results
  5. Proof of Theorem \ref{['QXRic']}
  6. Proof of Corollary \ref{['cordne']}
  7. Proof of Corollary \ref{['CQXRic']}
  8. Proof of Theorem \ref{['QXR']}
  9. Proof of Corollary \ref{['cordne2']}
  10. Proof of Theorem \ref{['generalizedAmb']}

Key Result

Theorem 1.1

Let $(M^n,g)$ be a compact $n$-dimensional Riemannian manifold with smooth boundary $\partial M$ and nonnegative Ricci curvature. Let $H$ be the mean curvature of $\partial M$. If $H > 0$ everywhere, then with equality in eq:ros-inequality if and only if $M$ is isometric to a Euclidean ball.

Theorems & Definitions (38)

  • Theorem 1.1: mulero1987compact
  • Theorem 1.2: brendle2013constant
  • Proposition 1.3: qiu2015generalization
  • Theorem 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Remark 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Proposition 1.10: ambrozio2017static, Proposition 6
  • ...and 28 more