Area rigidity for the equatorial disk in the ball

TL;DR

This paper establishes a gap in the area of free boundary minimal surfaces in the Euclidean ball, extending Brendle's result by comparing excesses and demonstrating a minimal area gap.

Contribution

It introduces a new method comparing excesses to prove the existence of an area gap for free boundary minimal surfaces in the ball.

Findings

Existence of a positive area gap for free boundary minimal surfaces.

Comparison of excesses to establish minimal area bounds.

Extension of Brendle's minimal area result.

Abstract

It is proved by Brendle in [4] that the equatorial disk $D^k$ has least area among $k$-dimensional free boundary minimal surfaces in the Euclidean ball $B^n$. By comparing the excess of free boundary minimal surfaces with the excess of the associated cones over the boundary, we prove the existence of a gap for the area.