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

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 \({\mathbb {S}}^2\) ${\mathbb{S}}^2$ . The inequality is asymptotically sharp, and we prove any sequence of surfaces saturating it converges weakly to \({\mathbb {S}}^2\) ${\mathbb{S}}^2$ , as currents and as varifolds.