Free-boundary problems for holomorphic curves in the 6-sphere

We remark on two free-boundary problems for holomorphic curves in nearly-Kahler 6 manifolds. First, we observe that a holomorphic curve in a geodesic ball B of the round 6-sphere that meets partial derivative B orthogonally must be totally geodesic. Consequently, we obtain rigidity results for reflection-invariant holomorphic curves in S-6 and associative cones in R-7. Second, we consider holomorphic curves with boundary on a Lagrangian submanifold in a strict nearly-Kahler 6-manifold. By deriving a suitable second variation formula for area, we observe a topological lower bound on the Morse index. In both settings, our methods are complex-geometric, closely following arguments of Fraser-Schoen and Chen-Fraser.