Sobolev functions on varifolds

TL;DR

This paper develops Sobolev spaces on rectifiable varifolds, establishing their properties and embeddings, and connecting them to existing function spaces under curvature and density conditions.

Contribution

Introduces first order Sobolev spaces on rectifiable varifolds with key functional properties and links to classical spaces under geometric conditions.

Findings

Sobolev spaces are complete and locally convex on certain varifolds.

Embedding theorems into Lebesgue and continuous function spaces are established.

Geodesic distance on varifolds is a Sobolev function with bounded derivative.

Abstract

This paper introduces first order Sobolev spaces on certain rectifiable varifolds. These complete locally convex spaces are contained in the generally nonlinear class of generalised weakly differentiable functions and share key functional analytic properties with their Euclidean counterparts. Assuming the varifold to satisfy a uniform lower density bound and a dimensionally critical summability condition on its mean curvature, the following statements hold. Firstly, continuous and compact embeddings of Sobolev spaces into Lebesgue spaces and spaces of continuous functions are available. Secondly, the geodesic distance associated to the varifold is a continuous, not necessarily H\"older continuous Sobolev function with bounded derivative. Thirdly, if the varifold additionally has bounded mean curvature and finite measure, the present Sobolev spaces are isomorphic to those previously available for finite Radon measures yielding many new results for those classes as well. Suitable versions of the embedding results obtained for Sobolev functions hold in the larger class of generalised weakly differentiable functions.