Existence of pseudo-holomorphic disks via non-archimedean disk potentials

2026/06/11 by Hang Yuan · 1 voice
Mathematics · #math.SG #math.AG

paper · pdf

Abstract

We show that if a graded monotone Lagrangian L₀ has a non-vanishing disk potential, then for every smooth isotopy \L_s\_s∈[0,1] of Lagrangians starting from it and for every tame almost complex structure J, each L_s bounds a J-holomorphic disk of Maslov index two. The main input is a non-archimedean analytic potential function, defined as an invariant up to analytic isomorphisms, generalizing the classical disk potential of a monotone Lagrangian. The techniques are inspired by recent developments in the Strominger-Yau-Zaslow mirror construction via family Floer theory and non-archimedean geometry. We also discuss applications such as recovering a simple case of Audin's conjecture.

Citations

• Non-archimedean analytic continuation of unobstructedness
• Aspherical Lagrangian submanifolds, Audin's conjecture and cyclic dilations
• Infinitely many monotone Lagrangian tori in higher projective spaces
• Family Floer SYZ conjecture for A_n singularity
• Family Floer SYZ singularities for the conifold transition
• Disk potential functions for quadrics
• Construction of Kuranishi structures on the moduli spaces of pseudo holomorphic disks: II
• The wall-crossing formula and Lagrangian mutations
• Infinitely many exotic monotone Lagrangian tori in CP²
• On a result by Y. Groman and J. P. Solomon
• Punctured holomorphic curves and Lagrangian embeddings
• On Exotic Lagrangian Tori in CP²
• A reverse isoperimetric inequality for J-holomorphic curves
• Formes différentielles réelles et courants sur les espaces de Berkovich
• Cyclic symmetry and adic convergence in LagrangianFloer theory
• Toric degenerations of Gelfand-Cetlin systems and potential functions
• Lagrangian Floer theory on compact toric manifolds II : Bulk deformations
• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• A biased view of symplectic cohomology
• A General Fredholm Theory I: A Splicing-Based Differential Geometry
• A general Fredholm theory I: a splicing-based differential geometry
• The Maslov class of Lagrangian tori and quantum products in Floer cohomology
• Non-archimedean amoebas and tropical varieties
• Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds
• Mirror Symmetry
• Mirror symmetry is T-duality

Discussions

• Hang Yuan: Existence of pseudo-holomorphic disks via non-archimedean disk potentials https://arxiv.org/abs/2606.13122 https://arxiv.org/pdf/2606.13122 https://arxiv.org/html/2606.13122 [bsky, 0 points, 0 comments]

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