2019/10/05 by Kwok‐Kun Kwong, Kwong, Kwok-Kun
Mathematics · #49Q20 #53C23 #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1910.02331
openalex publication_date 2019/10/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove some sharp isoperimetric type inequalities for domains with smooth boundary on Riemannian manifolds. For example, using generalized convexity, we show that among all domains with a lower bound l for the cut distance and Ricci curvature lower bound (n-1)k, the geodesic ball of radius l in the space form of curvature k has the largest area-to-volume ratio. A similar but reversed inequality holds if we replace a lower bound on the cut distance by a lower bound of the mean curvature. As an application we show that C2 isoperimetric domains in standard space forms are balls. Generalized convexity also provides a simple proof of Toponogov theorem. We also prove another isoperimetric inequality involving the extrinsic radius of a domain when the curvature of the ambient space is bounded above. We then extend this inequality in two directions: one involves the higher order mean curvatures, and the other involves the Hausdorff measure of the cut locus.