# Minimal hypersurfaces in the ball with free boundary

## TL;DR

This paper proves that minimal hypersurfaces with free boundary on a sphere are flat disks if graphical with respect to any Killing field, and establishes curvature bounds for disk-shaped hypersurfaces, enhancing understanding of their geometric properties.

## Contribution

It introduces new results on minimal hypersurfaces with free boundary, showing flatness under certain conditions and providing curvature bounds, independent of topology.

## Key findings

- Graphical minimal hypersurfaces are flat disks.
- Curvature squared interior supremum is bounded below by n times the boundary infimum.
- Provides insights into the curvature behavior of free boundary minimal hyperdisks.

## Abstract

In this note we use the strong maximum principle and integral estimates prove two results on minimal hypersurfaces $F:M^n\rightarrow\mathbb{R}^{n+1}$ with free boundary on the standard unit sphere. First we show that if $F$ is graphical with respect to any Killing field, then $F(M^n)$ is a flat disk. This result is independent of the topology or number or boundaries. Second, if $M^n = \mathbb{D}^n$ is a disk, we show the supremum of the curvature squared on the interior is bounded below by $n$ times the infimum of the curvature squared on the boundary. These may be combined the give an impression of the curvature of non-flat minimal hyperdisks with free boundary.