Rigidity phenomena on lower N-weighted Ricci curvature bounds with ε-range for nonsymmetric Laplacian

Abstract

Lu, Minguzzi, and Ohta introduced the notion of a lower N-weighted Ricci curvature bound with ε-range and derived several comparison geometric estimates from a Laplacian comparison theorem for a weighted Laplacian. The aim of this paper is to investigate various rigidity phenomena for the equality case of their comparison geometric results. We will obtain rigidity results concerning the Laplacian comparison theorem, diameter comparisons, and volume comparisons. We also generalize their works for a nonsymmetric Laplacian induced from vector field potential.