Punctured holomorphic curves and Lagrangian embeddings

2014/11/07 by Kai Cieliebak, Cieliebak, Kai, Klaus Mohnke +1 · 4 citations
Mathematics · #53D42 #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.1411.1870

openalex publication_date 2014/11/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We use a neck stretching argument for holomorphic curves to produce symplectic disks of small area and Maslov class with boundary on Lagrangian submanifolds of nonpositive curvature. Applications include the proof of Audin's conjecture on the Maslov class of Lagrangian tori in linear symplectic space, the construction of a new symplectic capacity, obstructions to Lagrangian embeddings into uniruled symplectic manifolds, a quantitative version of Arnold's chord conjecture, and estimates on the size of Weinstein neighbourhoods. The main technical ingredient is transversality for the relevant moduli spaces of punctured holomorphic curves with tangency conditions.

Citations

• Mirror symmetry and T-duality in the complement of an anticanonical divisor

Cited by

• Legendrian barriers in prequantization bundles
• Stabilized symplectic embeddings of higher-dimensional ellipsoids
• Hamiltonian linking and Symplectic packing
• Rational Quantum Cohomology of Steenrod Uniruled Manifolds

Related

• C⁰-rigidity of Lagrangian submanifolds and punctured holomorphic discs in the cotangent bundle
• Equivariant symplectic homology, linearized contact homology and the Lagrangian capacity
• Holomorphic Disks and the Chord Conjecture
• Symplectic displacement energy for exact Lagrangian immersions
• Low-action holomorphic curves and invariant sets