Weighted volume comparison and monotonicity for $L^p$-bound of Bakry-Émery Ricci curvature

Abstract

We establish a Petersen-Wei type relative volume comparison theorem for weighted Riemannian manifolds under both $L^p$-bounds of the Bakry-Émery Ricci curvature and the gradient of potential function. As an application, we give a modified proof for a volume comparison and monotonicity of Kähler-Ricci flow established in a recent work of Tian-Zhang-Zhang-Zhu-Zhu with improved estimate for error term.

We removed the volume growth condition in the main theorem in the first version of paper