Symplectic geometry, special Lagrangians, and geometric flows

Thomas–Yau conjecture

Collaboration beta

Should categorical stability select a unique special Lagrangian and govern a long-time Lagrangian mean-curvature flow, while instability yields ordered special-Lagrangian factors?

E stable \Rightarrow \exists! L_{sLag} and L_{t} \to L_{sLag}

Known results and sources

The conjectural stable endpoint and unstable factorization are shown as open alternatives, not established flow behavior.

Research problem

Exact mathematical statement

Let $(X^{2n},\omega,J,\Omega)$ be a compact Calabi–Yau manifold and let $E$ be a stable Fukaya-category object with a uniformly almost-calibrated graded Lagrangian representative. The strong stable branch predicts a unique special-Lagrangian representative of phase $\operatorname{arg} Z(E)$ , together with an object-preserving Lagrangian mean-curvature flow with surgeries that exists for all time and converges to it.

E\text{ stable}\;\Longrightarrow\;\exists!L_{\mathrm{sLag}},\qquad L_t\longrightarrow L_{\mathrm{sLag}}

For an unstable object, the strong package predicts ordered semistable special-Lagrangian factors realizing a Harder–Narasimhan filtration with strictly decreasing phases. The governing source explicitly reports no complete proof of this full package.

Problem infographic

Problem at a glance

A landscape plate contrasts a stable object's interrupted path toward a special Lagrangian with an unstable object's ordered factor pieces. — Stable existence and convergence, and unstable ordered factorization, are separate conjectural branches of the Thomas–Yau–Joyce package.

Current mathematical picture

Where work on Thomas–Yau conjecture stands

Open conjecture

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureAtomwise charge and indecomposability shortcut

Revision 8 states that inherited geometry controls total atom class and charge only blockwise and forbids silently assigning that total charge to every component. Exact-charge phase blocks or repeated-phase component factorizations remain viable if the categorical equivalence and mass comparison are actually proved.

Route status · Narrowed route

Main reductionCurrent reduction

Starting from a hypothetical noncalibrated smooth tail, the source seeks an exact-charge phase-block twisted-complex model and a selected normalization whose geometric and categorical errors fall below the formal mass gap.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeConstruct a phase-ordered sectorial or stopped inverse-surgery model and prove an actual twisted-complex equivalence for the selected slice.Task status · Ready to work on

Work mapped so far

Thomas–Yau conjecture in numbers

3.4kretained lines of mathematical investigation3,443 in the current working snapshot

Argument development: 2,829 · 82%
Explored or eliminated routes: 102 · 3%
Computational analysis: 45 · 1%
Open obligations: 151 · 4%
Definitions and setup: 316 · 9%

6selected mapped statements1routes investigated6open questions6contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byRichard P. Thomas, Shing-Tung YauProposed2001Status checkedAug 14, 2026

Source recordedAug 14, 2026

Recommended next task

Construct a phase-ordered sectorial or stopped inverse-surgery model and prove an actual twisted-complex equivalence for the selected slice.

Suggested move: Specify phase blocks, charge bookkeeping, stop removal, and filtered strictification with convergence of the lower homotopies.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

13 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded

Narrowed routeAtomwise charge and indecomposability shortcut

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

6 featured tasks

Construct a phase-ordered sectorial or stopped inverse-surgery model and prove an actual twisted-complex equivalence for the selected slice.Suggested move: Specify phase blocks, charge bookkeeping, stop removal, and filtered strictification with convergence of the lower homotopies.

Ready to work on

Prove a selected normalization calibration bound comparing categorical mass with geometric volume below the effective gap.Suggested move: Bind the geometric and categorical error terms to one late normalization and verify their sum is strictly below the source's effective threshold.

Ready to work on

Stable special-Lagrangian and flow branch

Prove existence, uniqueness, all-time object-preserving flow with surgeries, and convergence for the stable branch.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.

Ready to work on

Unstable Harder–Narasimhan branch

Prove that the unstable flow produces ordered semistable special-Lagrangian factors realizing the filtration.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.

Ready to work on

Geometric realization and mass comparison

Realize the formal construction geometrically and prove the selected normalization mass comparison.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.

Ready to work on

Bridge the contradiction route to the exact special-Lagrangian existence, uniqueness, surgery, and convergence statement.Suggested move: Audit which strong-package conclusions follow from the stable-tail exclusion and identify every remaining analytic flow theorem separately.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026

Current statusOpen conjecture

Open in its general stability, special-Lagrangian existence and uniqueness, surgery, and convergence forms. The literature includes symmetric examples, an analytic and variational Stein program with explicit outstanding difficulties, and a 2025 toric-section theorem, none of which establishes the full compact Calabi–Yau or categorical package.

[1][2][3]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2025
PreprintStoppa proved a toric form for specified special-Lagrangian sections in certain Calabi–Yau fibrations mirror to toric weak Fano manifolds, under a slope-stability criterion.[4]
2022
PreprintLi formulated a Stein and almost-calibrated setting, proved Floer-theoretic obstructions under extra hypotheses, and developed a variational program while isolating unresolved Floer and…[3]
2014
PreprintJoyce updated and expanded the conjectural framework using Bridgeland stability, enlarged Fukaya objects, surgeries, and limiting semistable special-Lagrangian currents.[2]
2001
Historical sourceThomas and Yau proposed stability criteria, special-Lagrangian uniqueness and existence expectations, mean-curvature-flow behavior, and Jordan–Hölder-type decomposition ideas.[1]

Mathematical neighborhood

Related results and reusable starting points

Current focusThomas–Yau conjecture

Stronger or generalized formJoyce's Lagrangian MCF with surgeries package

Joyce's package enlarges the object class and makes detailed conjectures about surgeries, stable singularities, convergence, and semistable limits; these additions are not automatic consequences of the original statement.

[2]

Solved special casesymmetric special-Lagrangian flow cases

The original paper proves uniqueness under mild conditions and verifies convergence in specified symmetric Shapere–Vafa examples, not in general.

[1]

Related problemThomas–Yau semistability in Calabi–Yau Stein manifolds

Li's holomorphic-curve and variational program interprets semistability versus special-Lagrangian existence in an exact Stein setting and identifies concrete obstructions and analytic gaps.

[3]

Solved special casetoric Thomas–Yau conjecture for Lagrangian sections

A toric stability criterion is proved for a class of Lagrangian sections in specified mirror fibrations; arbitrary objects and the full flow package remain outside the theorem.

[4]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

Formalization targetA formal statement needs Calabi–Yau and symplectic geometry, graded Lagrangian branes, central charge, special-Lagrangian phase, and uniform almost-calibration.
Formalization targetThe categorical form requires Fukaya categories, bounding cochains, Bridgeland stability, semistable objects, and Harder–Narasimhan filtrations.
Formalization targetThe flow form requires weak or surgery solutions of Lagrangian mean-curvature flow, object preservation, singular-time semantics, and a topology of convergence to integral currents.
Formalization targetUniqueness must specify Hamiltonian isotopy or categorical object scope, while unstable factorization must preserve phase ordering and filtration data.
Formalization targetNo scoped problem-level formal statement or proof was recorded in this bounded pass.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

4 standing statements2 proposed statements6 open questions1 narrowed routes

Statements by mathematical role6 selected mapped statements

theorem candidate1 of 61
reduction2 of 62
lemma2 of 62
negative result1 of 61

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.19 displayed rows · 1 route included

retained route statementDoes stability govern special Lagrangians and long-time Lagrangian flow?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementFormal categorical mass gapintermediate
retained route statementZero-charge factor forkintermediate
retained route statementCharge bookkeeping is blockwiseintermediate
Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Useful failureAtomwise charge and indecomposability shortcutreported failure
Research targetConstruct a phase-ordered sectorial or stopped inverse-surgery model and prove an actual twisted-complex equivalence for the selected slice.open
Research targetProve a selected normalization calibration bound comparing categorical mass with geometric volume below the effective gap.open
Research targetBridge the contradiction route to the exact special-Lagrangian existence, uniqueness, surgery, and convergence statement.open
Research targetStable special-Lagrangian and flow branchopen
Research targetUnstable Harder–Narasimhan branchopen
Research targetGeometric realization and mass comparisonopen
Narrowed routeAtomwise charge and indecomposability shortcutRevision 8 states that inherited geometry controls total atom class and charge only blockwise and forbids silently assigning that total charge to every component. Exact-charge phase blocks or repeated-phase component factorizations remain viable if the categorical equivalence and mass comparison are actually proved.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeConstruct a phase-ordered sectorial or stopped inverse-surgery model and prove an actual twisted-complex equivalence for the selected slice.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

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Prepared starting pointConstruct a phase-ordered sectorial or stopped inverse-surgery model and prove an actual twisted-complex equivalence for the selected slice.

Thomas–Yau conjecture · ready to start

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Research contextPrepared context for any AI agent

Should categorical stability select a unique special Lagrangian and govern a long-time Lagrangian mean-curvature flow, while instability yields ordered special-Lagrangian factors?

Exact question and boundaries
Current routes and known obstacles
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Sources and references4 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

1
Special Lagrangians, stable bundles and mean curvature floworiginal source · Richard P. Thomas, Shing-Tung Yau · Communications in Analysis and Geometry · 2001 preprint; 2002 journal · ARXIV math/0104197 · DOI 10.4310/CAG.2002.v10.n5.a8 · accessed Aug 14, 2026
2
Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature flowpreprint · Dominic Joyce · arXiv · 2014 · ARXIV 1401.4949 · accessed Aug 14, 2026
3
Thomas-Yau conjecture and holomorphic curvespreprint · Yang Li · arXiv · 2022 · ARXIV 2203.01467 · accessed Aug 14, 2026
4
Nakai-Moishezon criteria and the toric Thomas-Yau conjecturepreprint · Jacopo Stoppa · arXiv · 2025 · ARXIV 2505.07228 · accessed Aug 14, 2026

Important qualifications

Thomas–Yau refers to related formulations about stability, existence and uniqueness of special Lagrangians, and Lagrangian mean-curvature flow. Joyce's later package adds enlarged objects, surgeries, and semistable-factor limits; this record does not silently treat every formulation as identical.
The full stability-to-special-Lagrangian and all-time flow package remains open. Symmetry, Stein, and toric-section results are retained only at their stated scopes.
Stoppa's 2025 result proves a form for specified Lagrangian sections in mirrors of toric weak Fano manifolds; it is not a proof for arbitrary compact Calabi–Yau manifolds or arbitrary Fukaya objects.
No current problem-level formalization or canonical computation was identified, but this bounded pass did not exhaust every proof assistant or private project.
No intake-packet claim was used as external status authority, and no proof, review, acceptance, credit, publication, or deployment authority is granted.

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