Rigidity phenomena on lower $N$-weighted Ricci curvature bounds with $\varepsilon$-range for non-symmetric Laplacian

TL;DR

This paper explores rigidity phenomena in lower $N$-weighted Ricci curvature bounds with $ extpsilon$-range, extending comparison theorems to non-symmetric Laplacians and analyzing equality cases for geometric estimates.

Contribution

It introduces rigidity results for the equality cases in Laplacian, diameter, and volume comparison theorems under weighted Ricci curvature bounds, generalizing to non-symmetric Laplacians.

Findings

Rigidity results for Laplacian comparison theorem

Rigidity in diameter and volume comparison

Extension to non-symmetric Laplacian cases

Abstract

Lu-Minguzzi-Ohta have introduced the notion of a lower $N$-weighted Ricci curvature bound with $\varepsilon$-range, and derived several comparison geometric estimates from a Laplacian comparison theorem for 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 non-symmetric Laplacian induced from vector field potential.