The structure of stable minimal hypersurfaces in R^n

TL;DR

This paper establishes a new topological obstruction for complete stable minimal hypersurfaces in R^n, proving that for dimensions n≥4, such hypersurfaces are topologically simple with only one end.

Contribution

It introduces a novel topological obstruction and proves that complete orientable stable minimal hypersurfaces in R^n have only one end for n≥4.

Findings

Complete stable minimal hypersurfaces in R^n with n≥4 have only one end.

A new topological obstruction for such hypersurfaces is established.

An analytic theorem underpinning the topological result is proved.

Abstract

We provide a new topological obstruction for complete stable minimal hypersurfaces in R^n. For $n\geq 4$, we prove that any complete orientable stable hypersurfaces in R^n has only one end. This follows from a more general analytic theorem we prove in the paper.