Laplacian comparison theorem on Riemannian manifolds with modified m-Bakry-Emery Ricci lower bounds for $m\le 1$

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-Émery Ricci tensor under $m\le 1$ in terms of vector fields. As consequences, we give the optimal conditions for modified $m$-Bakry-Émery Ricci tensor under $m\le 1$ 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-Émery Ricci curvature under $m\ge n$ if the vector field is a gradient type. When $m<1$, our results are new in the literature.

27 pages. arXiv admin note: text overlap with arXiv:2001.00444

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• $W$-entropy formula for the Witten Laplacian on manifolds with time dependent metrics and potentials
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