Regularity for minimal sets near a union of two planes

TL;DR

This paper proves that 2D minimal sets in R^4 resembling two nearly orthogonal planes at infinity are actually cones, using topological properties to connect large-scale structure with local regularity.

Contribution

It establishes a global regularity result for minimal sets near a union of two planes, showing such sets are cones, which is a new insight into their geometric structure.

Findings

Minimal sets near two planes are cones.

Topological properties control local regularity.

Global structure influences local behavior.

Abstract

We discuss the global regularity of 2 dimensional minimal sets that are near a union of two planes, and prove that every global minimal set in R^4 that looks like a union of two almost orthogonal planes at infinity is a cone. The main point is to use the topological properties of a minimal set at a large scale to control its behavior at smaller scales.