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

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We remark on two free-boundary problems for holomorphic curves in nearly-Kähler 6-manifolds. First, we observe that a holomorphic curve in a geodesic ball B of the round 6-sphere that meets $$\partial B$$ orthogonally must be totally geodesic. Consequently, we obtain rigidity results for reflection-invariant holomorphic curves in $$\mathbb {S}^6$$ and associative cones in $$\mathbb {R}^7$$ . Second, we consider holomorphic curves with boundary on a Lagrangian submanifold in a strict nearly-Kähler 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.