On the Areas of Genus Zero Free Boundary Minimal Surfaces Embedded in the Unit $3$-ball

TL;DR

This paper establishes an area inequality for genus zero free boundary minimal surfaces in the unit 3-ball, showing the inequality is sharp and characterizing the limit surfaces as the sphere.

Contribution

It proves a sharp area inequality for genus zero free boundary minimal surfaces and describes their convergence behavior to the sphere.

Findings

The area of such surfaces is less than their radial projection area.

The inequality is asymptotically sharp.

Sequences saturating the inequality converge to the sphere.

Abstract

We prove that the area of each nonflat genus zero free boundary minimal surface embedded in the unit $3$-ball is less than the area of its radial projection to $\mathbb{S}^2$. The inequality is asymptotically sharp, and we prove any sequence of surfaces saturating it converges weakly to $\mathbb{S}^2$, as currents and as varifolds.