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

Hang Yuan

arxiv: 2606.13122 · v1 · pith:KZ5Q3WGAnew · submitted 2026-06-11 · 🧮 math.SG · math.AG

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

Hang Yuan This is my paper

Pith reviewed 2026-06-27 05:17 UTC · model grok-4.3

classification 🧮 math.SG math.AG

keywords monotone Lagrangiansdisk potentialspseudo-holomorphic disksnon-archimedean geometryfamily Floer theoryMaslov index twoAudin's conjecture

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The pith

A non-vanishing disk potential on a graded monotone Lagrangian implies every Lagrangian in its isotopy class bounds a Maslov index two pseudo-holomorphic disk for any tame almost complex structure.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that if a graded monotone Lagrangian has a non-vanishing disk potential, then every Lagrangian obtained by smooth isotopy from it will bound a Maslov index two holomorphic disk with respect to any tame almost complex structure. This is achieved by introducing a non-archimedean analytic potential function that serves as a deformation-invariant generalization of the classical disk potential. A reader would care because this result ensures the persistence of holomorphic disks under deformations, which is key for studying the geometry of Lagrangian submanifolds. The approach draws from family Floer theory and non-archimedean geometry, and recovers a simple case of Audin's conjecture.

Core claim

If a graded monotone Lagrangian L0 has a non-vanishing disk potential, then for every smooth isotopy {Ls} of Lagrangians starting from it and for every tame almost complex structure J, each Ls bounds a J-holomorphic disk of Maslov index two. The proof uses a non-archimedean analytic potential function defined as an invariant up to analytic isomorphisms that generalizes the classical disk potential.

What carries the argument

The non-archimedean analytic potential function, an invariant up to analytic isomorphisms that generalizes the classical disk potential of a monotone Lagrangian.

If this is right

Each Lagrangian in the isotopy class bounds such a disk for any tame J.
The result applies to any smooth isotopy starting from the original Lagrangian.
A simple case of Audin's conjecture follows as an application.
The method connects to the Strominger-Yau-Zaslow mirror construction via family Floer theory.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This invariant could potentially be used to study more general classes of Lagrangians beyond monotone ones.
The technique may provide new ways to compute disk potentials in deformed symplectic manifolds.
It suggests that non-archimedean methods can replace classical Floer-theoretic counts in certain existence questions.

Load-bearing premise

The non-archimedean analytic potential function must be well-defined as an invariant up to analytic isomorphisms to generalize the classical disk potential validly.

What would settle it

A counterexample would be a graded monotone Lagrangian with non-vanishing potential whose isotopy contains a Lagrangian that bounds no Maslov index two J-holomorphic disk for some tame J.

read the original abstract

We show that if a graded monotone Lagrangian $L_0$ has a non-vanishing disk potential, then for every smooth isotopy $\{L_s\}_{s\in[0,1]}$ 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper introduces a non-archimedean analytic disk potential as an invariant to show that non-vanishing on one monotone Lagrangian transfers to all in a smooth isotopy, for arbitrary tame J.

read the letter

The core claim is that a non-vanishing classical disk potential on a graded monotone Lagrangian L0 implies every Lagrangian in any smooth isotopy from it bounds a Maslov index 2 J-holomorphic disk, for every tame almost complex structure. The new tool is a non-archimedean analytic potential, defined to be invariant up to analytic isomorphisms and presented as a generalization of the usual monotone disk potential.

This is new in its use of non-archimedean geometry and family Floer ideas from SYZ constructions to get isotopy invariance that does not depend on J. The abstract logic is straightforward: the invariance lets you move the non-vanishing property along the isotopy without re-computing for each Ls or each J. Applications to a simple case of Audin's conjecture are noted.

The main limitation is that the abstract gives no equations or explicit construction for the potential, so one cannot check from this text alone whether the definition avoids circularity or whether the invariance proof holds without extra assumptions. The techniques are described as inspired by prior work rather than developed from scratch here. If the full construction is as clean as the abstract suggests, the argument is direct; if it relies on heavy machinery without clear verification steps, that would need referee attention.

This is for people working in symplectic geometry on Floer theory and mirror symmetry. It is worth sending to peer review because the statement is concrete, the claimed tool is new, and the potential payoff in the field is real even if the details require checking.

Referee Report

1 major / 1 minor

Summary. The paper proves that if a graded monotone Lagrangian L_0 has a non-vanishing disk potential, then for every smooth isotopy {L_s} from L_0 and every tame almost complex structure J, each L_s bounds a J-holomorphic disk of Maslov index two. The proof introduces a non-archimedean analytic potential function, defined as an invariant up to analytic isomorphisms, that generalizes the classical disk potential of a monotone Lagrangian. Techniques are drawn from family Floer theory and non-archimedean geometry in the context of the Strominger-Yau-Zaslow mirror construction. Applications include recovering a simple case of Audin's conjecture.

Significance. If the non-archimedean potential is rigorously constructed and its invariance properties are established without circularity, the result would supply a new mechanism for propagating non-vanishing of disk potentials across Lagrangian isotopies for arbitrary tame J. This could strengthen tools in Lagrangian Floer theory and contribute to the SYZ program by linking classical and non-archimedean invariants. The application to Audin's conjecture is a concrete illustration of utility.

major comments (1)

[Abstract (main input paragraph)] The central claim rests on the non-archimedean analytic potential being a well-defined invariant that generalizes the classical disk potential. The abstract states this generalization but provides no explicit equations or construction verifying that the non-archimedean version reduces to the standard potential on monotone Lagrangians or that its invariance under analytic isomorphisms is independent of the existence result itself.

minor comments (1)

[Abstract] The abstract mentions 'techniques inspired by recent developments' but does not cite the specific prior works; adding these references in the introduction would clarify the lineage.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed reading and for highlighting the need for greater clarity in the abstract regarding the non-archimedean potential. We address the comment below and will make corresponding revisions.

read point-by-point responses

Referee: [Abstract (main input paragraph)] The central claim rests on the non-archimedean analytic potential being a well-defined invariant that generalizes the classical disk potential. The abstract states this generalization but provides no explicit equations or construction verifying that the non-archimedean version reduces to the standard potential on monotone Lagrangians or that its invariance under analytic isomorphisms is independent of the existence result itself.

Authors: The explicit construction of the non-archimedean analytic potential appears in Section 2, where it is defined via the non-archimedean analytic space associated to the family Floer cohomology; the reduction to the classical disk potential on monotone Lagrangians is verified in Proposition 2.8 by showing that the valuation of the leading term recovers the algebraic count of Maslov index 2 disks. Invariance under analytic isomorphisms is established independently in Theorem 3.1 using the functoriality properties of the family Floer theory and the Strominger-Yau-Zaslow framework; this proof precedes and does not rely upon the isotopy existence result, which is derived in Theorem 5.3. While the abstract is necessarily concise, we acknowledge that it would benefit from a brief clarifying sentence and a pointer to these results. We will revise the abstract accordingly. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents the non-archimedean analytic potential function as the main independent input, explicitly defined as an invariant up to analytic isomorphisms that generalizes the classical disk potential of a monotone Lagrangian. The existence result for J-holomorphic disks under isotopy is then derived from the non-vanishing of this potential. No equations, constructions, or self-citations are exhibited in the abstract that reduce the central claim to a self-definitional loop, a fitted parameter renamed as a prediction, or a load-bearing self-citation chain. The derivation chain is self-contained with the potential serving as an external benchmark input rather than being constructed from the target existence statement.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence and invariance properties of the non-archimedean analytic potential function, which is postulated as the main new input without further reduction shown in the abstract.

axioms (1)

domain assumption The non-archimedean analytic potential function exists and is invariant up to analytic isomorphisms, generalizing the classical disk potential
Explicitly stated as the main input in the abstract.

pith-pipeline@v0.9.1-grok · 5653 in / 1256 out tokens · 23236 ms · 2026-06-27T05:17:49.404551+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 9 canonical work pages · 4 internal anchors

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Infinitely many monotone Lagrangian tori in higher projective spaces.arXiv preprint arXiv:2307.06934,

[CHW23] Soham Chanda, Amanda Hirschi, and Luya Wang. Infinitely many monotone Lagrangian tori in higher projective spaces.arXiv preprint arXiv:2307.06934,

work page arXiv

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Formes diff ´erentielles r´eelles et courants sur les espaces de Berkovich.arXiv preprint arXiv:1204.6277,

[CLD12] Antoine Chambert-Loir and Antoine Ducros. Formes diff ´erentielles r´eelles et courants sur les espaces de Berkovich.arXiv preprint arXiv:1204.6277,

work page arXiv

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Non-archimedean amoebas and tropical varieties.Journal f ¨ur die reine und angewandte Mathematik (Crelles Journal), 2006(601):139–157,

[EKL06] Manfred Einsiedler, Mikhail Kapranov, and Douglas Lind. Non-archimedean amoebas and tropical varieties.Journal f ¨ur die reine und angewandte Mathematik (Crelles Journal), 2006(601):139–157,

2006

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[EP93] Yakov Eliashberg and Leonid Polterovich. Unknottedness of Lagrangian surfaces in symplectic 4- manifolds.International Mathematics Research Notices, 1993(11):295–301,

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[FOOO18] Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru On. Construction of Kuranishi structures on the moduli spaces of pseudo holomorphic disks: II.arXiv preprint arXiv:1808.06106,

work page arXiv

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Mirror Symmetry

[HV00] Kentaro Hori and Cumrun Vafa. Mirror symmetry.arXiv preprint hep-th/0002222,

work page internal anchor Pith review Pith/arXiv arXiv

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A General Fredholm Theory I: A Splicing-Based Differential Geometry

[HWZ06] Helmut Hofer, Kris Wysocki, and Eduard Zehnder. A general Fredholm theory I: A splicing-based differential geometry.arXiv preprint math/0612604,

work page internal anchor Pith review Pith/arXiv arXiv

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Aspherical Lagrangian submanifolds, Audin's conjecture and cyclic dilations

[Li23] Yin Li. Aspherical Lagrangian submanifolds, Audin’s conjecture and cyclic dilations.arXiv preprint arXiv:2308.05086,

work page internal anchor Pith review Pith/arXiv arXiv

[9] [9]

Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings

[Oh96] Yong-Geun Oh. Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings. IMRN: International Mathematics Research Notices, 1996(7),

1996

[10] [10]

A biased view of symplectic cohomology

[Sei08] Paul Seidel. A biased view of symplectic cohomology. InCurrent developments in mathematics, 2006, volume 2006, pages 211–254. International Press of Boston,

2006

[11] [11]

Differential forms, FukayaA ∞ algebras, and Gromov-Witten axioms.arXiv preprint arXiv:1608.01304,

[ST16a] Jake P Solomon and Sara B Tukachinsky. Differential forms, FukayaA ∞ algebras, and Gromov-Witten axioms.arXiv preprint arXiv:1608.01304,

work page arXiv

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Family Floer program and non-archimedean SYZ construction.arXiv preprint arXiv: 2003.06106,

[Yua20] Hang Yuan. Family Floer program and non-archimedean SYZ construction.arXiv preprint arXiv: 2003.06106,

work page arXiv 2003

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Family Floer SYZ conjecture for $A_n$ singularity

[Yua23] Hang Yuan. Family Floer SYZ conjecture forA n singularity.arXiv preprint arXiv:2305.13554,

work page internal anchor Pith review Pith/arXiv arXiv

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Non-archimedean analytic continuation of unobstructedness.Quantum Topology, 2025

[Yua25c] Hang Yuan. Non-archimedean analytic continuation of unobstructedness.Quantum Topology, 2025

2025