Weakly differentiable functions on varifolds

TL;DR

This paper develops a framework for weakly differentiable functions on rectifiable varifolds, establishing key properties, embeddings, and inequalities, with applications to geodesic distances and curvature varifolds.

Contribution

It introduces a new concept of weak differentiability on varifolds using integration by parts, and proves fundamental properties and applications in geometric measure theory.

Findings

Sobolev Poincaré embeddings for varifolds

Pointwise differentiability results for weakly differentiable functions

Finiteness of geodesic distance and characterization of curvature varifolds

Abstract

The present paper is intended to provide the basis for the study of weakly differentiable functions on rectifiable varifolds with locally bounded first variation. The concept proposed here is defined by means of integration by parts identities for certain compositions with smooth functions. In this class the idea of zero boundary values is realised using the relative perimeter of superlevel sets. Results include a variety of Sobolev Poincar\'e type embeddings, embeddings into spaces of continuous and sometimes H\"older continuous functions, pointwise differentiability results both of approximate and integral type as well as coarea formulae. As prerequisite for this study decomposition properties of such varifolds and a relative isoperimetric inequality are established. Both involve a concept of distributional boundary of a set introduced for this purpose. As applications the finiteness of the geodesic distance associated to varifolds with suitable summability of the mean curvature and a characterisation of curvature varifolds are obtained.