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

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.