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

TL;DR

This paper investigates free-boundary minimal surfaces with constant Kähler angle in complex space forms, establishing conditions under which they are totally geodesic or superminimal, extending classical results to higher dimensions and complex geometries.

Contribution

It generalizes Fraser and Schoen's results to complex space forms, characterizing free-boundary minimal surfaces with specific Kähler angles as totally geodesic or superminimal.

Findings

Lagrangian free-boundary minimal surfaces in complex space forms are totally geodesic.

Surfaces with Kähler angle π/2 are superminimal in higher-dimensional 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 $n \geq 2$, we show that if $\Sigma$ has K\"{a}hler angle $\pi/2$, then $\Sigma$ is superminimal.