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

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.

arXiv commentUpdated with varifold and current convergence results. We thank Misha Karpukhin for suggesting that we prove these results