The Branched Deformations of the Special Lagrangian Submanifolds

In this paper, we investigate the branched deformations of immersed compact special Lagrangian submanifolds. If there exists a nondegenerate Z 2 harmonic 1-form over a special Lagrangian submanifold L , we construct a family of immersed special Lagrangian submanifolds L ~ t , that are diffeomorphic...

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Bibliographic Details
Published inGeometric and functional analysis Vol. 33; no. 5; pp. 1266 - 1321
Main Author He, Siqi
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.10.2023
Springer Nature B.V
Subjects
Analysis
Deformation
Harmonic functions
Manifolds (mathematics)
Mathematics
Mathematics and Statistics
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Summary:In this paper, we investigate the branched deformations of immersed compact special Lagrangian submanifolds. If there exists a nondegenerate Z 2 harmonic 1-form over a special Lagrangian submanifold L , we construct a family of immersed special Lagrangian submanifolds L ~ t , that are diffeomorphic to a branched covering of L and L ~ t converge to 2 L as current. This answers a question suggested by Donaldson (Deformations of multivalued harmonic functions, 2019. arXiv:1912.08274 ). As a corollary, we discover examples of special Lagrangian submanifolds that are rigid in the classical sense but exhibit branched deformations. In conjunction with the work of Abouzaid and Imagi in Nearby special lagrangians, 2021. arXiv:2112.10385 , we derive constraints on the existence of nondegenerate Z 2 harmonic 1-forms.
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ISSN:1016-443X
1420-8970
DOI:10.1007/s00039-023-00645-8