2017/09/16 by Ulrich Menne, Menne, Ulrich, Christian Scharrer +1
Mathematics · #46E35 #49Q05 #49Q15 #53A07 (Primary) #53A10 #53C22 (Secondary) #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1709.05504
openalex publication_date 2017/09/16 · openalex created_date 2024/03/29 · openalex updated_date 2026/07/28
Based on a novel type of Sobolev-Poincaré inequality (for generalised weakly differentiable functions on varifolds), we establish a finite upper bound of the geodesic diameter of generalised compact connected surfaces-with-boundary of arbitrary dimension in Euclidean space in terms of the mean curvatures of the surface and its boundary. Our varifold setting includes smooth immersions, surfaces with finite Willmore energy, two-convex hypersurfaces in level-set mean curvature flow, integral currents with prescribed mean curvature vector, area minimising integral chains with coefficients in a complete normed commutative group, varifold solutions to Plateau's problem furnished by min-max methods or by Brakke flow, and compact sets solving Plateau problems based on Čech homology. Due to the generally inevitable presence of singularities, path-connectedness was previously known neither for the class of varifolds (even in the absence of boundary) nor for the solutions to the Plateau problems considered.