Optimal regularity for quasiminimal sets of codimension one in $\R^2$ and $\R^3$

TL;DR

This paper characterizes the regularity and separation properties of quasiminimal sets in dimensions 2 and 3, establishing their connection with local John domains and their local topological structure.

Contribution

It proves that quasiminimal sets in dimensions 2 and 3 separate space into local John domains and characterizes their local topological structure, providing a new understanding of their regularity.

Findings

Quasiminimal sets in $ ^2$ and $ ^3$ separate space into local John domains.

Such sets locally separate space into two components, except at isolated points or lower-dimensional subsets.

The property of separating space into John domains is both necessary and sufficient for quasiminimality.

Abstract

Quasiminimal sets are sets for which a pertubation can decrease the area but only in a controlled manner. We prove that in dimensions $2$ and $3$, such sets separate a locally finite family of local John domains. Reciprocally, we show that this property is a sufficient for quasiminimality. In addition, we show that quasiminimal sets locally separate the space in two components, except at isolated points in $\R^2$ or out a of subset of dimension strictly less than $N-1$ in $\R^N$.