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 Σ in geodesic balls B of complex space forms.We show that in CP 2 , C 2 and CH 2 , if Σ has constant Kähler angle, then its boundary curves are geodesics in ∂B. In fact, if Σ is Lagrangian and has genus zero, or if Σ is a ±-holomorphic disk, then Σ is totally geodesic. In CP n , C n and CH n for n ≥ 2, we show that if Σ is totally real and of genus zero, then Σ is superminimal.