Hölder Regularity of Two-Dimensional Almost-Minimal Sets in ${\mathbb{R}}^n$

Abstract

We give a different and probably more elementary proof of a good part of Jean Taylor's regularity theorem for Almgren almost-minimal sets of dimension 2 in ${\mathbb{R}}^3$. We use this opportunity to settle some details about almost-minimal sets, extend a part of Taylor's result to almost-minimal sets of dimension 2 in ${\mathbb{R}}^n$, and give the expected characterization of the closed sets $E$ of dimension 2 in ${\mathbb{R}}^3$ that are minimal, in the sense that $H^2(E\setminus F)\le H^2(F\setminus E)$ for every closed set $F$ such that there is a bounded set $B$ so that $F=E$ out of $B$ and $F$ separates points of ${\mathbb{R}}^3\setminus B$ that $E$ separates.

150 pages. Submitted in May 2007

Cited by in corpus (2)

• Regularity of the singular set for Mumford-Shah minimizers in R^3 near a minimal cone
• Condensation of polyhedric structures onto soap films