Tohoku Mathematical Journal
2022
March
SECOND SERIES VOL. 74, NO. 1

Tohoku Math. J.
74 (2022), 83-107

Title LAPLACIAN COMPARISON THEOREM ON RIEMANNIAN MANIFOLDS WITH MODIFIED $m$-BAKRY-EMERY RICCI LOWER BOUNDS FOR $m\leq1$

Author Kazuhiro Kuwae and Toshiki Shukuri

(Received November 7, 2019, revised October 26, 2020)
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 $m\leq 1$ in terms of vector fields. As consequences, we give the optimal conditions for modified $m$-Bakry-Émery 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-Émery Ricci curvature under $m\geq n$ ([19, 21, 27, 33]) or $m=1$ ([34, 35]) if the vector field is a gradient type. When $m<1$, our results are new in the literature.

Mathematics Subject Classification. Primary 53C20; Secondary 53C21, 53C22, 53C23, 53C24, 58J60.
Key words and phrases. Modified $m$-Bakry-Émery Ricci curvature, Laplacian comparison theorem, weighted Myers' theorem, Bishop-Gromov volume comparison theorem, Ambrose-Myers' theorem, Cheng's maximal diameter theorem, Cheeger-Gromoll splitting theorem.

To the top of this page

Back to the Contents