Minimal hypersurfaces asymptotic to Simons cones

TL;DR

This paper classifies minimal hypersurfaces in Euclidean space that resemble a specific Simons cone at infinity, showing only two such hypersurfaces exist up to similarity, thus advancing understanding of their geometric structure.

Contribution

It proves the uniqueness (up to similarity) of minimal hypersurfaces asymptotic to a given Simons cone, identifying precisely two such hypersurfaces in higher-dimensional space.

Findings

Only two minimal hypersurfaces are asymptotic to the given Simons cone.

The classification holds in all dimensions where the cone exists.

The result enhances understanding of the asymptotic geometry of minimal hypersurfaces.

Abstract

In this paper, we prove that, up to similarity, there are only two minimal hypersurfaces in $\mathbb{R}^{n+2}$ that are asymptotic to a Simons cone, i.e. the minimal cone over the minimal hypersurface $\sqrt{\frac pn}\mathbb{S}^p\times \sqrt{\frac{n-p}n} \mathbb{S}^{n-p}$ of $\mathbb{S}^{n+1}$