Semicontinuity of capacity under pointed intrinsic flat convergence

TL;DR

This paper studies how the capacity of sets in Riemannian and metric spaces behaves under pointed intrinsic flat convergence, proving upper semicontinuity results and exploring implications for geometric analysis and general relativity.

Contribution

It establishes upper semicontinuity of capacity under volume-preserving intrinsic flat convergence, extending previous concepts to non-smooth spaces and connecting to mass in general relativity.

Findings

Capacity is upper semicontinuous under VF convergence for fixed-radius balls.

Lipschitz sublevel sets also exhibit upper semicontinuity.

Examples show strict semicontinuity and necessity of volume-preserving assumptions.

Abstract

The concept of the capacity of a compact set in $\mathbb R^n$ generalizes readily to noncompact Riemannian manifolds and, with more substantial work, to metric spaces (where multiple natural definitions of capacity are possible). Motivated by analytic and geometric considerations, and in particular Jauregui's definition of capacity-volume mass and Jauregui and Lee's results on the lower semicontinuity of the ADM mass and Huisken's isoperimetric mass, we investigate how the capacity functional behaves when the background spaces vary. Specifically, we allow the background spaces to consist of a sequence of local integral current spaces converging in the pointed Sormani--Wenger intrinsic flat sense. For the case of volume-preserving ($\mathcal{VF}$) convergence, we prove two theorems that demonstrate an upper semicontinuity phenomenon for the capacity: one version is for balls of a fixed radius centered about converging points; the other is for Lipschitz sublevel sets. Our approach is motivated by Portegies' investigation of the semicontinuity of eigenvalues under $\mathcal{VF}$ convergence. We include examples to show the semicontinuity may be strict, and that the volume-preserving hypothesis is necessary. Finally, there is a discussion on how capacity and our results may be used towards understanding the general relativistic total mass in non-smooth settings.