Free boundary minimal hypersurfaces outside of the ball

TL;DR

This paper classifies free boundary minimal hypersurfaces outside the unit ball in Euclidean space, showing that stable examples with certain ends are catenoids and providing a complete description of these hypersurfaces.

Contribution

It establishes classification theorems for exterior free boundary minimal hypersurfaces, identifying catenoids as the only stable examples with specific end conditions.

Findings

Stable exterior free boundary minimal hypersurfaces with parallel ends are catenoids.

Any such hypersurface with one regular end is a catenoid.

Complete description and index calculation of catenoidal hypersurfaces.

Abstract

In this paper we obtain two classification theorems for free boundary minimal hypersurfaces outside of the unit ball (exterior FBMH for short) in Euclidean space. The first result states that the only exterior stable FBMH with parallel embedded regular ends are the catenoidal hypersurfaces. To achieve this we prove a B\^ocher type result for positive Jacobi functions on regular minimal ends in $\mathbb{R}^{n+1}$ which, after some calculations, implies the first theorem. The second theorem states that any exterior FBMH $\Sigma$ with one regular end is a catenoidal hypersurface. Its proof is based on a symmetrization procedure similar to R. Schoen [14]. We also give a complete description of the catenoidal hypersurfaces, including the calculation of their indices.