Uniqueness results for free-boundary minimal hypersurfaces in conformally Euclidean balls and annular domains

TL;DR

This paper establishes uniqueness results for free-boundary minimal hypersurfaces in Euclidean balls and annular domains, showing that under certain curvature bounds, the flat disk is the only such hypersurface.

Contribution

It proves new uniqueness theorems for free-boundary minimal hypersurfaces with curvature bounds in Euclidean and conformally Euclidean settings.

Findings

Uniqueness of the flat free-boundary minimal disk under curvature constraints.

Extension of results to hypersurfaces in annular conformally Euclidean domains.

Curvature bounds involving the second fundamental form are critical for the uniqueness.

Abstract

In this paper we prove that a flat free-boundary minimal $n$-disk, $n\geq3$, in the unit Euclidean ball $B^{n+1}$ is the unique compact free boundary minimal hypersurface in the unit Euclidean ball which the squared norm of the second fundamental form is less than either $\frac{n^2}{4}$ or $\frac{(n-2)^2}{4|x|^2}$. Moreover, we prove analogous results for compact free boundary minimal hypersurfaces in annular domains with a conformally Euclidean metric.