Boundary Effect of Ricci Curvature

TL;DR

This paper investigates how Ricci curvature influences boundary geometry in compact Riemannian manifolds, establishing inequalities and rigidity results that connect interior curvature with boundary properties.

Contribution

It introduces new integral inequalities and rigidity theorems linking Ricci curvature to boundary geometry in manifolds with boundary.

Findings

Derived integral inequalities for boundary functions

Established geometric inequalities involving total mean curvature

Proved rigidity results for Ricci curvature with boundary conditions

Abstract

On a compact Riemannian manifold with boundary, we study how Ricci curvature of the interior affects the geometry of the boundary. First we establish integral inequalities for functions defined solely on the boundary and apply them to obtain geometric inequalities involving the total mean curvature. Then we discuss related rigidity questions and prove Ricci curvature rigidity results for manifolds with boundary.