A priori bounds for geodesic diameter. Part III. A Sobolev-Poincar\'e inequality and applications to a variety of geometric variational problems

TL;DR

This paper introduces a new Sobolev-Poincaré inequality applicable to varifolds, enabling the derivation of finite upper bounds for the geodesic diameter of complex geometric surfaces with boundary, including singular cases.

Contribution

It establishes a novel Sobolev-Poincaré inequality for varifolds and applies it to bound the geodesic diameter of diverse geometric surfaces with boundary, even with singularities.

Findings

Finite upper bounds for geodesic diameter in terms of mean curvature.

Applicability to a wide class of varifold-based surfaces.

Path-connectedness established for complex varifold solutions.

Abstract

Based on a novel type of Sobolev-Poincar\'e 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 \v{C}ech 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.