Mean Convex Smoothing of Mean Convex Cones

TL;DR

This paper proves that mean convex cones can be smoothly approximated by minimal hypersurfaces and self-expanders, confirming a longstanding conjecture about their geometric structure and regularity.

Contribution

It establishes the existence of smooth, properly embedded minimal hypersurfaces and self-expanders asymptotic to mean convex cones, resolving a conjecture by Lawson.

Findings

Any minimizing hypercone can be perturbed into a smooth minimal hypersurface.

Every viscosity mean convex cone admits a smooth mean convex self-expander.

The results confirm Lawson's conjecture on the structure of mean convex cones.

Abstract

We show that any minimizing hypercone can be perturbed into one side to a properly embedded smooth minimizing hypersurface in the Euclidean space, and every viscosity mean convex cone admits a properly embedded smooth mean convex self-expander asymptotic to it near infinity. These two together confirm a conjecture of Lawson \cite[Problem 5.7]{Brothers86_OpenProblems}.