2025/03/05 by Jia Li, Li, Jia
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Numerical methods in inverse problems #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2503.03482
openalex publication_date 2025/03/05 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28
We establish the Bonnet-Myers theorem and the Bishop-Gromov volume comparison theorem in the spectral sense for manifolds with weakly convex boundary. For n≥ 3, let (Mn,g) be a simply connected compact smooth n-manifold with weakly convex boundary ∂ M. If there exists a positive function w∈ C∞(M) that satisfies: \begincases -(n-1)/(n-2)Δw+Λ\Ric w≥ (n-1)w, \enspace in \enspace M, (∂ w)/(∂ η)=0, \enspace\enspace\enspace\enspace \enspace\enspace\enspace\enspace\enspace\enspace\enspace\enspace\enspace\enspace\enspace\enspace \enspace\enspace \enspace\enspace on \enspace∂ M, \endcases where Λ\Ric denotes the smallest eigenvalue of the Ricci tensor, η is the unit co-normal vector field of ∂ M in M, then the diameter of M satisfies \diam(M)≤ ((max w)/(min w))(n-3)/(n-1)π.\par If, in addition, w attains its minimum on the boundary ∂ M, we obtain a sharp upper bound for the volume of M: \Vol(M)≤ \Vol(\bSn+), with equality holding if and only if Mn is isometric to the unit round hemisphere \bSn+.