Ricci tensor on smooth metric measure space with boundary

TL;DR

This paper investigates the measure-valued Ricci tensor on smooth metric measure spaces with boundary, extending previous concepts to new settings and providing a novel approach to curvature-dimension conditions.

Contribution

It introduces a generalized Ricci tensor for spaces with boundary and applies it to analyze curvature-dimension conditions in this context.

Findings

Generalization of Ricci tensor to measure-valued form on spaces with boundary

New method for studying curvature-dimension conditions

Extension of Bakry-Emery Ricci tensor concepts

Abstract

The aim of this note is to study the measure-valued Ricci tensor on smooth metric measure space with boundary, which is a generalization of Bakry-Emery's modified Ricci tensor on weighted Riemannian manifold. As an application, we offer a new approach to study curvature-dimension condition of smooth metric measure space with boundary.