2021/11/30 by Kazuhiro Kuwae, Kuwae, Kazuhiro, Toshiki Shukuri +1
Mathematics · #53C20 #53C22 #53C23 #53C24 #58J60 #Comparison theorem #Curvature #Differential Geometry (math.DG) #FOS: Mathematics #Geology #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Laplace operator #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Partial Differential Equations #Pure mathematics #Ricci curvature #Riemannian manifold #Secondary 53C21 #Type (biology) #Vector field #math.DG #msc:53C20 #msc:53C21 #msc:53C22 #msc:53C23 #msc:53C24 #msc:58J60
paper · pdf · doi:10.48550/arxiv.2111.15508
published in arXiv (Cornell University) (Cornell University) · 27 pages. arXiv admin note: text overlap with arXiv:2001.00444
arxiv created 2021/11/30 · openalex publication_date 2021/11/30 · arxiv updated 2021/12/01 · openalex created_date 2021/12/06 · openalex updated_date 2026/08/05
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≤ 1 in terms of vector fields. As consequences, we give the optimal conditions for modified m-Bakry-Émery Ricci tensor under m≤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≥ n if the vector field is a gradient type. When m<1, our results are new in the literature.