New Volume Comparison Results and Volume Growth Rigidity of Gradient Ricci Almost Solitons

TL;DR

This paper develops new volume comparison theorems for manifolds with Bakry-Emery Ricci curvature bounds, leading to volume rigidity results for gradient Ricci almost solitons and extending previous work to shrinking cases.

Contribution

It introduces novel volume comparison results for manifolds with Bakry-Emery Ricci curvature bounds and extends rigidity theorems to shrinking gradient Ricci almost solitons.

Findings

New volume comparison theorem established

Volume rigidity results for gradient Ricci almost solitons obtained

Extension of Cao and Zhou's results to shrinking cases

Abstract

In this paper, we establish a new volume comparison theorem for a complete manifold with a function $\rho(x)$ as the lower bound of the Bakry-Emery Ricci curvature. As applications, we obtain a new volume rigidity result of the gradient Ricci almost solitons. Furthermore, we extend the results of Cao and Zhou \cite{CZ} to shrinking gradient Ricci almost solitons and get the rigidity result with respect to the maximal volume growth.