Laplacian comparison theorem on Riemannian manifolds with modified m-Bakry-Emery Ricci lower bounds for $m\leq1$

TL;DR

This paper establishes a Laplacian comparison theorem for Riemannian manifolds with modified m-Bakry-Emery Ricci bounds for m ≤ 1, extending classical geometric results to new non-gradient vector field cases.

Contribution

It introduces novel Laplacian comparison results for non-symmetric diffusion operators with m ≤ 1, broadening the scope of geometric theorems under these conditions.

Findings

Derived optimal conditions for modified m-Bakry-Emery Ricci bounds

Extended Myers' and Bishop-Gromov theorems to new settings

Established splitting and diameter theorems for m<1 cases

Abstract

In this paper, we prove a Laplacian comparison theorem for non-symmetric diffusion operator on complete smooth $n$-dimensional Riemannian manifold having a lower bound of modified $m$-Bakry-\'Emery Ricci tensor under $m\leq 1$ in terms of vector fields. As consequences, we give the optimal conditions for modified $m$-Bakry-\'Emery Ricci tensor under $m\leq1$ such that the (weighted) Myers' theorem, Bishop-Gromov volume comparison theorem, Ambrose-Myers' theorem, Cheng's maximal diameter theorem, and the Cheeger-Gromoll type splitting theorem hold. Some of these results were well-studied for $m$-Bakry-\'Emery Ricci curvature under $m\geq n$ if the vector field is a gradient type. When $m<1$, our results are new in the literature.