Rigidity of Quasi-Einstein Metrics

TL;DR

This paper investigates the properties and rigidity of quasi-Einstein metrics, a generalization of Einstein metrics, providing new results including splitting theorems for certain Kähler quasi-Einstein metrics.

Contribution

It establishes several rigidity results and a splitting theorem for Kähler quasi-Einstein metrics, advancing understanding of their geometric structure.

Findings

Rigidity results for quasi-Einstein metrics

Splitting theorem for Kähler quasi-Einstein metrics

Connections to warped product Einstein metrics

Abstract

We call a metric quasi-Einstein if the $m$-Bakry-Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of Einstein metrics, which contains gradient Ricci solitons and is also closely related to the construction of the warped product Einstein metrics. We study properties of quasi-Einstein metrics and prove several rigidity results. We also give a splitting theorem for some K\"ahler quasi-Einstein metrics.