Abstract:
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.
