Weighted volume comparison and monotonicity for $L^p$-bound of Bakry-\'{E}mery Ricci curvature

TL;DR

This paper establishes a volume comparison theorem related to Bakry-Émery Ricci curvature bounds and applies it to analyze the Kähler-Ricci flow, providing new insights into geometric analysis.

Contribution

It proves a relative volume comparison theorem for $L^P$ bounds of Bakry-Émery Ricci curvature and gradient of potential functions, extending previous results.

Findings

Proved a relative volume comparison theorem for $L^P$ bounds.

Provided a modified proof for volume comparison and monotonicity in Kähler-Ricci flow.

Extended Petersen-Wei volume comparison to Bakry-Émery Ricci curvature context.

Abstract

We prove a relative volume comparison theorem of Petersen-Wei for both $L^P$-bound of Bakry-\'{E}mery Ricci curvature and gradient of potential function. As an application, we give a modified proof for a volume comparison and monotonicity of K\"{a}hler-Ricci flow established in a recent work of Tian-Zhang-Zhang-Zhu-Zhu.