$C^{1+\alpha}$-Regularity for Two-Dimensional Almost-Minimal Sets in $\R^n$

TL;DR

This paper presents a new, elementary proof that two-dimensional almost-minimal sets in ^n are locally ^{1+b1}-equivalent to minimal cones, extending prior results with a focus on regularity and deformation techniques.

Contribution

It provides a simplified proof and partial generalization of Jean Taylor's regularity result for almost-minimal sets, utilizing harmonic functions and a local separation theorem.

Findings

Almost-minimal sets are locally ^{1+b1}-equivalent to minimal cones.

The proof extends to higher dimensions with dependence on blow-up limits.

A new deformation method reduces the area of certain cones over Lipschitz arcs.

Abstract

We give a new proof and a partial generalization of Jean Taylor's result [Ta] that says that Almgren almost-minimal sets of dimension 2 in $\R^3$ are locally $C^{1+\alpha}$-equivalent to minimal cones. The proof is rather elementary, but uses a local separation result proved in [D3] and an extension of Reifenberg's parameterization theorem [DDT]. The key idea is still that if $X$ is the cone over an arc of small Lipschitz graph in the unit sphere, but $X$ is not contained in a disk, we can use the graph of a harmonic function to deform $X$ and diminish substantially its area. The local separation result is used to reduce to unions of cones over arcs of Lipschitz graphs. A good part of the proof extends to minimal sets of dimension 2 in $\R^n$, but in this setting our final regularity result on $E$ may depend on the list of minimal cones obtained as blow-up limits of $E$ at a point.