Free-Boundary Problems for Holomorphic Curves in the 6-Sphere

DOI10.1007/S00209-023-03234-5arXiv2105.10562MaRDI QIDQ6368258

Jesse Madnick

Publication date: 21 May 2021

Abstract: We remark on two different free-boundary problems for holomorphic curves in nearly-K"{a}hler 6-manifolds. First, we observe that a holomorphic curve in a geodesic ball $B$ of the round 6-sphere that meets $partialB$ 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-K"{a}hler 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.

Mathematics Subject Classification ID

Global differential geometry of Hermitian and Kählerian manifolds (53C55) Kähler manifolds (32Q15) Local differential geometry of Hermitian and Kählerian structures (53B35) Pseudoholomorphic curves (32Q65)

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