Global regularity for minimal sets near a $\T$ set and counterexamples

TL;DR

This paper investigates the global regularity of 2D minimal sets near a d set, showing conditions under which such sets are cones and addressing potential counterexamples with complex topologies.

Contribution

It demonstrates that minimal sets close to a d set are cones, refutes a proposed counterexample with simpler topology, and constructs a potential counterexample with a Klein bottle topology.

Findings

All 1D Almgren-minimal sets in n are cones.

All 2D Mumford-Shah minimal sets in  are cones.

Potential counterexamples require complex topological structures.

Abstract

We discuss the global regularity for 2 dimensional minimal sets that are near a $\T$ set, that is, whether every global minimal set in $\R^n$ that looks like a $\T$ set at infinity is a $\T$ set or not. The main point is to use the topological properties of a minimal set at large scale to control its topology at smaller scales. This is the idea to prove that all 1-dimensional Almgren-minimal sets in $\R^n$, and all 2-dimensional Mumford-Shah minimal sets in $\R^3$ are cones. In this article we discuss two types of 2-dimensional minimal sets: Almgren-minimal set in $\R^3$ whose blow-in limit is a $\T$ set; topological minimal sets in $\R^4$ whose blow-in limit is a $\T$ set. For the first one we eliminate an existing potential counterexample that was proposed by several people, and show that a real counterexample should have a more complicated topological structure; for the second we construct a potential example using a Klein bottle.