Free-Boundary Minimal Surfaces of Constant Kahler Angle in Complex Space Forms

Abstract

In real space forms, Fraser and Schoen proved that a free-boundary minimal disk in a geodesic ball is totally geodesic. In this note, we consider free-boundary minimal surfaces $\Sigma$ (of any genus) in geodesic balls of complex space forms. In ${\mathbb{CP}}^2$, ${\mathbb{C}}^2$ and ${\mathbb{CH}}^2$, we show that if $\Sigma$ is Lagrangian, then $\Sigma$ is totally geodesic. In ${\mathbb{CP}}^n$, ${\mathbb{C}}^n$ and ${\mathbb{CH}}^n$ for $n\ge 2$, we show that if $\Sigma$ has Kähler angle $\pi /2$, then $\Sigma$ is superminimal.

7 pages