Uniqueness theorems for free boundary minimal disks in space forms

TL;DR

This paper proves that free boundary minimal disks in space forms are totally geodesic or totally umbilic under certain conditions, extending previous three-dimensional results to higher dimensions.

Contribution

It generalizes known results about free boundary minimal disks from three dimensions to higher-dimensional space forms, introducing weaker conditions for classification.

Findings

Minimal disks with free boundary are totally geodesic in space forms.

Under weaker conditions, disks are contained in 3D submanifolds and are totally umbilic.

Results extend classical theorems to higher dimensions.

Abstract

We show that a minimal disk satisfying the free boundary condition in a constant curvature ball of any dimension is totally geodesic. We weaken the condition to parallel mean curvature vector in which case we show that the disk lies in a three dimensional constant curvature submanifold and is totally umbilic. These results extend to higher dimensions earlier three dimensional work of J. C. C. Nitsche and R. Souam.