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

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 ∂ B orthogonally must be totally geodesic. Consequently, we obtain rigidity results for reflection-invariant h...

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Bibliographic Details
Published inMathematische Zeitschrift Vol. 303; no. 4
Main Author Madnick, Jesse
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2023
Springer Nature B.V
Subjects
Lower bounds
Manifolds (mathematics)
Mathematical analysis
Mathematics
Mathematics and Statistics
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Summary: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 ∂ 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-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.
ISSN:0025-5874
1432-1823
DOI:10.1007/s00209-023-03234-5