# The Heintze-Karcher inequality for metric measure spaces

## TL;DR

This paper extends the Heintze-Karcher inequality to non-branching metric measure spaces with Ricci curvature bounds, using needle decomposition, and characterizes equality cases in positively curved Riemannian settings.

## Contribution

It generalizes the Heintze-Karcher inequality to a broad class of metric measure spaces with curvature bounds and characterizes equality cases in Riemannian curvature-dimension spaces.

## Key findings

- Heintze-Karcher inequality proven for non-branching metric measure spaces
- Equality case characterized in positively curved Riemannian spaces
- Utilizes needle decomposition technique for the proof

## Abstract

In this note we prove the Heintze-Karcher inequality in the context of essentially non-branching metric measure spaces satisfying a lower Ricci curvature bound in the sense of Lott-Sturm-Villani. The proof is based on the the needle decomposition technique for metric measure spaces introduced by Cavalletti-Mondino. Moreover, in the class of spaces satisfying a Riemannian curvature-dimension condition with positive curvature the equality case is characterized.