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

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Abstract

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.

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Acknowledgements

I thank Benjamin Aslan, Jonny Evans, and Chung-Jun Tsai for helpful conversations related to this work, and thank Gavin Ball, Da Rong Cheng, Spiro Karigainnis, Wei-Bo Su, and Albert Wood for their interest and encouragement. This work was completed during the author’s postdoctoral fellowship at the National Center for Theoretical Sciences (NCTS) at National Taiwan University; I thank the Center for their support. Finally, I thank the referee for insightful comments that improved this work.

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  1. National Center for Theoretical Sciences, National Taiwan University, Taipei, Taiwan

    Jesse Madnick

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Madnick, J. Free-boundary problems for holomorphic curves in the 6-sphere. Math. Z. 303, 85 (2023). https://doi.org/10.1007/s00209-023-03234-5

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