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

TL;DR

This paper establishes integral inequalities and rigidity results for compact Riemannian manifolds with boundary satisfying V-static-type equations, characterizing geodesic balls and providing new insights into geometric rigidity theorems.

Contribution

It introduces new integral inequalities using a generalized Reilly formula and Steklov problems, leading to novel rigidity characterizations of manifolds with boundary.

Findings

Rigidity characterizations of geodesic balls in space forms

New integral inequalities for boundary geometric quantities

Connections to existing rigidity theorems

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.