Smooth metric measure spaces and quasi-Einstein metrics

TL;DR

This paper unifies various approaches to smooth metric measure spaces and quasi-Einstein metrics through a conformal geometric perspective, providing new results, interpretations, and examples.

Contribution

It introduces a natural conformal geometric framework that unites previous perspectives on smooth metric measure spaces and quasi-Einstein metrics.

Findings

A variational characterization of quasi-Einstein metrics

New families of quasi-Einstein metric examples

A unified framework connecting Bakry-Émery and Chang-Gursky-Yang approaches

Abstract

Smooth metric measure spaces have been studied from the two different perspectives of Bakry-\'Emery and Chang-Gursky-Yang, both of which are closely related to work of Perelman on the Ricci flow. These perspectives include a generalization of the Ricci curvature and the associated quasi-Einstein metrics, which include Einstein metrics, conformally Einstein metrics, gradient Ricci solitons, and static metrics. In this article, we describe a natural perspective on smooth metric measure spaces from the point of view of conformal geometry and show how it unites these earlier perspectives within a unified framework. We offer many results and interpretations which illustrate the unifying nature of this perspective, including a natural variational characterization of quasi-Einstein metrics as well as some interesting families of examples of such metrics.