Spectral comparison results for the $N$-Bakry-Emery Ricci tensor

TL;DR

This paper proves diameter and volume comparison theorems for manifolds with a lower bound on the $N$-Bakry-Emery Ricci tensor in the spectrum sense, extending geometric analysis tools to weighted manifolds.

Contribution

It introduces new comparison results under spectral lower bounds for the $N$-Bakry-Emery Ricci tensor, broadening the scope of geometric analysis in weighted manifolds.

Findings

Diameter comparison under spectral lower bounds

Global weighted volume comparison results

Extension of classical comparison theorems to weighted settings

Abstract

We establish the diameter and global weighted volume comparison when the $N$-Bakry-Emery Ricci tensor has a positive lower bound in the spectrum sense.