2012/09/05 by Haizhong Li, Yong Wei · 1 citation
Mathematics · #Boundary (topology) #Combinatorics #Curvature #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Laplace operator #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Point processes and geometric inequalities #Pure mathematics #Ricci curvature #Topology (electrical circuits) #Upper and lower bounds #math.DG
paper · pdf · doi:10.1007/s12220-013-9434-5
published as The Journal of Geometric Analysis, 2015, 25(1): 421-435 · 15 pages
arxiv created 2012/09/05 · openalex publication_date 2013/08/15 · arxiv updated 2017/05/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In this paper, we first prove a compactness theorem for the space of closed embedded f-minimal surfaces of fixed topology in a closed three-manifold with positive Bakry-Émery Ricci curvature. Then we give a Lichnerowicz type lower bound of the first eigenvalue of the f-Laplacian on compact manifold with positive m-Bakry-Émery Ricci curvature, and prove that the lower bound is achieved only if the manifold is isometric to the n-shpere, or the n-dimensional hemisphere. Finally, for compact manifold with positive m-Bakry-Émery Ricci curvature and f-mean convex boundary, we prove an upper bound for the distance function to the boundary, and the upper bound is achieved if only if the manifold is isometric to an Euclidean ball.