A generalization of Reilly's formula and its applications to a new Heintze-Karcher type inequality

TL;DR

This paper generalizes Reilly's formula and applies it to derive new geometric inequalities and proofs for classical theorems in Riemannian geometry, including Alexandrov's Theorem and the Heintze-Karcher inequality.

Contribution

It introduces a broad generalization of Reilly's formula and demonstrates its utility in establishing new inequalities and alternative proofs in geometric analysis.

Findings

New generalized Reilly's formula proved.

Alternative proofs of Alexandrov's Theorem and Heintze-Karcher inequality provided.

Established a new Heintze-Karcher inequality for manifolds with curvature bounds.

Abstract

In this paper, we prove a generalization of Reilly's formula in \cite{Reilly}. We apply such general Reilly's formula to give alternative proofs of the Alexandrov's Theorem and the Heintze-Karcher inequality in the hemisphere and in the hyperbolic space. Moreover, we use the general Reilly's formula to prove a new Heintze-Karcher inequality for Riemannian manifolds with boundary and sectional curvature bounded below.