Regularity of sets with quasiminimal boundary surfaces in metric spaces

TL;DR

This paper investigates the boundary regularity of quasiminimizing sets in metric spaces, establishing boundary coincidence, finite Minkowski content, and rectifiability properties, advancing understanding of geometric measure theory in these contexts.

Contribution

It proves that the measure-theoretic boundary matches the topological boundary for quasiminimizing sets and explores rectifiability in weighted Euclidean spaces.

Findings

Boundary of quasiminimizing sets coincides with topological boundary

Sets have finite Minkowski content

Results apply to rectifiability in weighted Euclidean spaces

Abstract

This paper studies regularity of perimiter quasiminimizing sets in metric measure spaces with a doubling measure and a Poincare inequality. The main result shows that the measure theoretic boundary of a quasiminimizing set coincides with the topological boundary. We also show that such a set has finite Minkowski content and apply the regularity theory to study rectifiability issues related to quasiminimal sets in strong A_{\infty}-weighted Euclidean case.