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

Let $\Delta_V=\Delta-\langle V, \nabla\cdot\rangle$ be a non-symmetric diffusion operator on a complete smooth Riemannian manifold $(M,g)$ with its volume element $\mathfrak{m}=\mathrm{vol}_g$, and $V$ a $C^1$-vector field. In this paper, we prove a Laplacian comparison theorem on $(M,g,V)$ with a lower bound for modified $m$-Bakry-Émery Ricci tensor for $m\leq 1$ in terms of $V$. As consequences, we give the optimal conditions on modified $m$-Bakry-Émery Ricci tensor for $m\leq1$ 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 on complete Riemannian manifolds under mild conditions. Some of these results were well-studied for $m$-Bakry-Émery Ricci curvature for $m\geq n$ by Qian, Lott, Xiangdong Li, Wei–Wylie, or $m=1$ by Wylie and Wylie–Yeroshkin for $V=\nabla\phi$ with some $\phi\in C^2(M)$. When $m<1$, our results are new in the literature. This is a joint work with my master course student Toshiki Shukuri.