Abstract:

In a Riemannian manifold with boundary, a "free-boundary minimal surface" is a (2-dimensional) minimal surface whose boundary intersects the boundary of orthogonally. If is a geodesic ball in a real space form and is a free-boundary minimal disk in , Fraser and Schoen proved that is totally-geodesic. In this talk, we consider geodesic balls in the complex space forms , , and . If , we show that a Lagrangian free-boundary minimal surface of genus zero is totally geodesic. In particular, there is no "free-boundary Lagrangian catenoid" in . If ≥3, we show that a free-boundary minimal surface of genus zero and Kahler angle must be superminimal, implying that such minimal surfaces in can be constructed from holomorphic curves.