On Lagrangian Tori in $S^2\times S^2$

Han Lou

REVIEW 3 major objections 3 minor 3 cited by

On Lagrangian Tori in $S^2\times S^2$

T0 review · 3 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper classifies all interior toric fibers in the resolved toric degeneration of S^2×S^2: those with x+y≠1 are Hamiltonian isotopic to standard toric fibers, and those with x+y=1 are not Hamiltonian isotopic to any product torus.

desk verdict A solid classification paper that settles FOOO's question, with one under-proved step in Proposition 2.6 that is load-bearing but likely fixable. read the letter →

arxiv 2412.16356 v1 pith:37VMM3NG submitted 2024-12-20 math.SG math.GT

Han Lou This is my paper · ORCID

classification math.SGmath.GT MSC 53D1257K43

keywords LagrangiansubmanifoldstoricfiberHamiltonianisotopysymmetricprobesdisplacementenergygermdegenerationmonotonesymplecticmanifold

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies the Lagrangian tori that arise in a standard way from a toric degeneration that resolves into the symplectic four-manifold $S^{2}$×$S^{2}$. Its central result is a classification: for any interior fiber L(x,y), the value of x+y decides everything. If x+y≠1, the fiber is Hamiltonian isotopic (deformable by a time-dependent Hamiltonian flow) to a product of two circles in the two factors of $S^{2}$×$S^{2}$, and the paper says exactly which product. If x+y=1, the fiber is not Hamiltonian isotopic to any product of circles; these diagonal fibers form a whole family of exotic Lagrangian tori, including displaceable ones that earlier arguments could not handle.

What carries the argument

The proof starts from the explicit description of L(x,y) as the set {(v,w)∈$S^{2}$×$S^{2}$ : v_1+w_1=2p, v·w=$2q^{2}$−1} in coordinates p=x+y−1, q=1−y. This set is the orbit of an embedded curve Γ under the diagonal circle action (R_t,R_t), and after a symplectic change of coordinates it becomes $F^{{-1}}$(Γ)∩$H^{{-1}}$({−p}) in $C^{2}$, where F(z_1,z_2)=z_1z_2 and H=|z_1|^2−|z_2|^2. The curve Γ lies in the unit disk, and the paper computes its area with respect to the modified symplectic form ω_p=2rp/√($p^{2}$+$4r^{2}$) dr∧dφ; for p≠0 the area is 2π−2πq, which forces Γ to be Hamiltonian isotopic to a round circle $S^{1}$(r). The preimage of that circle is exactly a standard toric fiber, giving Theorem A in the range 0<$p^{2}$<$q^{4}$; the remaining range $p^{2}$≥$q^{4}$ is handled by symmetric probes, which move the fiber into the first range.

What would settle it

Check the curve Γ given in the Appendix: if it self-intersects for any allowed (p,q) with 0<$p^{2}$<$q^{4}$, or if its ω_p-area is not 2π−2πq, then the Hamiltonian-isotopy step in Proposition 2.6 fails and the classification in Theorem A would be wrong for that fiber.

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Extended reading notes

Core claim

The central claim is Theorem A: for every interior point (x,y) in the moment polytope P2 with x+y≠1, the Lagrangian torus L(x,y) is Hamiltonian isotopic to the standard toric fiber T(1/2−y, 3/2−2y−x) when 1−y<x<2−2y, and to T(−1/2+x, 1/2−y) when 0<x<1−y. Here T(ξ,ζ) is the fiber over (ξ,ζ) in the standard moment square, i.e. a product of two latitude circles. Theorem B states that for interior points with x+y=1, L(x,y) is not Hamiltonian isotopic to any product torus. Together the theorems give a complete Hamiltonian-isotopy classification of the interior toric fibers of the resolved degeneration.

Load-bearing premise

The load-bearing premise is that the explicitly parametrized curve Γ is a simple closed curve and that two simple closed curves in the open unit disk enclosing the same ω_p-area can be moved to one another by a Hamiltonian isotopy; the paper asserts the latter and leaves the former implicit, and Theorem A collapses without them.

Editorial extensions

If this is right

Every non-diagonal interior fiber is Hamiltonian isotopic to a product torus, so its displacement energy, monotonicity class, and other Hamiltonian-isotopy invariants coincide with those of the explicit standard fiber T(ξ,ζ).
The diagonal x+y=1 is a full family of exotic Lagrangian tori: none is a product, even though the subfamily 0<y<1/2 is displaceable.
The Hamiltonian-isotopy classes of toric fibers agree between the resolved degeneration and the standard toric structure on S^2×S^2 everywhere except the diagonal, so the difference between the two structures is concentrated on that single line.
The monotone fiber L(1/2,1/2), known to coincide with several previously constructed non-standard tori, is certified by Theorem B not to be a product torus.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The displacement-energy-germ comparison used for Theorem B should detect non-product tori in other monotone toric degenerations: whenever the germ near a fiber is a single affine function rather than a minimum of two independent functions, that fiber is likely exotic.
As a non-diagonal fiber approaches the diagonal x+y=1, its assigned standard fiber converges to the diagonal standard fiber T(1/2−y,1/2−y), while Theorem B says the diagonal fiber itself is exotic; this suggests the Hamiltonian-isotopy classification has a genuine discontinuity, or wall, along the diagonal.
A numerical check of the explicit curve Γ for a few (p,q) values would quickly test the implicit embeddedness assumption and identify any edge cases before full rigor is supplied.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Core result: I think the classification is correct, and it's a real step forward. Theorem A gives an explicit Hamiltonian isotopy from each nondiagonal FOOO fiber to a standard toric fiber, and Theorem B handles the diagonal fibers, including the displaceable interval that FOOO couldn't touch. The novelty is the synthesis and the explicit mapping, not a new method; the tools are symmetric probes, the OU16 symplectomorphism, the EP93 lemma, and displacement energy germs.

The paper does a lot of good work. The appendix computes the ω_p-area of Γ in full detail, and the displacement energy germ computation in Theorem B is clean and convincing. The author is honest about what's from FOOO and what's new.

The soft spot is Proposition 2.6. The proof says that because Γ and S^1(r) enclose the same ω_p-area, they are Hamiltonian isotopic, with no argument or citation. For embedded loops in a disk with an exact area form, that's a standard Moser-type fact, but it should be stated and justified. More importantly, the argument silently assumes Γ is an embedded simple closed curve. The paper calls Γ1 embedded, and Γ2 is its image under a diffeomorphism, so that's fine. But Γ is the image of Γ2 under F(z1,z2)=z1z2, and while the explicit parametrization in the appendix strongly suggests it's injective, injectivity is never proved. The EP93 lemma requires Γ embedded, and the area computation assumes Γ bounds a disk. This is load-bearing because every off-diagonal fiber goes through this construction.

My read is that both missing pieces are true. The curve looks like a simple loop winding once, and the equal-area-to-isotopy statement is a known fact. So the fix is small: add a lemma or a citation for the area-isotopy assertion, and prove or explicitly cite embeddedness of Γ. Until that's done, the proof of Theorem A has a gap.

For the rest, I have no concerns. The symmetric probe argument in Case 2 is sound, the case split checks out, and Theorem B's displacement energy germ argument is the right tool to rule out the diagonal. The citation pattern is fine.

This paper is for specialists in symplectic topology of 4-manifolds. It deserves a serious referee; after the small gaps are patched, it's publishable. I'd send it to a good journal and ask the author to address the two missing justifications.

Referee Report

3 major / 3 minor

Summary. This paper gives a complete Hamiltonian classification of the Lagrangian toric fibers arising in Fukaya--Oh--Ohta--Ono's resolved toric degeneration of S^2 x S^2. The main result, Theorem A, asserts that every such fiber L(x,y) with x+y != 1 is Hamiltonian isotopic to an explicitly identified standard toric fiber T(xi,zeta) in S^2 x S^2, with two formulas according to the sign of x+y-1. Theorem B asserts that the remaining fibers, those with x+y = 1, are not Hamiltonian isotopic to any product torus. The proof of Theorem A uses the Oakley--Usher symplectomorphism, reduces to a curve Gamma in the unit disk with an auxiliary area form omega_p, deforms Gamma to a circle by a Hamiltonian flow, and then identifies the resulting Lagrangian with a standard toric fiber. Theorem B is proved by combining the classification with a displacement-energy-germ computation.

Significance. If correct, the paper resolves the natural follow-up questions to FOOO and Oakley--Usher, giving a complete answer for the FOOO toric fibers: all non-diagonal fibers are standard product tori up to Hamiltonian isotopy, and the diagonal family consists of genuinely non-product tori. The proof is largely self-contained and includes an explicit, detailed computation of the area enclosed by the auxiliary curve Gamma in the appendix. The main caveat is that the pivotal step in Theorem A relies on two geometric facts -- an equal-area Hamiltonian-isotopy criterion and the embeddedness of Gamma -- that are asserted but not proved; both are plausible and likely standard, but until they are supplied the central claim is not fully established.

major comments (3)

[§2.1, Prop. 2.6] The proof of Proposition 2.6 asserts that when Gamma and S^1(r) enclose the same omega_p-area, they are Hamiltonian isotopic in (B^2(1), omega_p), and then concludes that there is a Hamiltonian K with phi^{1,p,K}(Gamma) = S^1(r). No proof or citation is given for the equal-area implication. This is a genuine statement about Hamiltonian isotopies for the non-standard exact area form omega_p (and p may be negative, so orientation conventions matter), and it is the step that produces the Hamiltonian deformation to a toric fiber. Please add a proof via a Moser-type argument, or an exact citation, that applies to embedded loops in the open unit disk with the given area form.
[§2.1, before Lemma 2.7; Appendix, Prop. 4.2] The application of [EP93, Lemma 4.2A] requires that Gamma = F(tilde_Gamma_2) be an embedded simple closed curve in B^2(1). The paper asserts this informally ('since Gamma is an embedded curve') but never proves it; the explicit parametrization in the appendix is long and injectivity is not checked. The area computation in Proposition 4.2 also presupposes that Gamma bounds a disk, i.e., that it is a simple closed curve. Because every off-diagonal fiber in Theorem A is handled by this one construction, embeddedness of Gamma is load-bearing and must be proved.
[§3, displacement energy germ comparison] The final comparison of displacement energy germs in the proof of Theorem B is stated informally: the germ of T(xi,xi) is said to be 'determined by two linearly independent functions' while that of L1(0,q) is 'determined by a single function.' To be fully rigorous one should show directly that no invertible linear map can take S^e_{T(xi,xi)} to S^e_{L1(0,q)}, for example by comparing the non-smooth locus (the line delta'_1 = delta'_2 for the toric germ) with the smoothness of the computed germ on all small vectors with delta_1 != 0. As written the argument is plausible but not precise.

minor comments (3)

[§2.2, Prop. 2.10] In Proposition 2.10, the phrase 'the omega_p-area of S^1(r) is pi|p| - pi sqrt(p^2+4r^2)' is inconsistent with the positive area 2pi-2piq reported for Gamma in Proposition 4.2; the later formula for r^2 corresponds to pi sqrt(p^2+4r^2) - pi|p|. Please state the orientation convention explicitly.
[§2.2, Prop. 2.12] In the proof of Proposition 2.12, the step 'S_{q1} and S_{q2} bound disks with the same area' follows from the equal-distance condition but is not spelled out; adding one sentence would improve readability.
[General] There are minor typographical errors throughout (e.g., 'Buliding' in the introduction and several misspellings in the text), and a careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: all load-bearing inputs are external theorems or explicit coordinate computations; the flagged gaps are omitted proofs, not circular reductions.

full rationale

The derivation is not circular. Theorem A is proved by taking the known symplectomorphism from FOOO's resolved degeneration to S^2 x S^2 (from OU16), expressing L(p,q) as F^{-1}(Gamma) cap H^{-1}({-p}) via the external lemma [EP93, Lemma 4.2A], and then moving Gamma by a Hamiltonian flow on the model ball. Proposition 2.6 chooses the auxiliary radius r by an area equality in the ball; that area equality is a geometric input, not a restatement of the Hamiltonian-isotopy conclusion, and the subsequent identification with T(xi,zeta) in Lemma 2.7 is an explicit coordinate computation. No fitted parameter is later renamed as a prediction, and no definition of L(x,y) is given in terms of a standard toric fiber. The external citations [OU16], [EP93], [FOOO12], [Bre23], [ABM14], and [CS10] are used as tools, and none is authored by Han Lou, so there is no self-citation chain. The equal-area-to-Hamiltonian-isotopy assertion in Proposition 2.6 and the unstated embeddedness of Gamma are genuine exposition gaps and would be correctness risks, but they are not circular steps: a failure of either would make the proof incomplete, not make the theorem equivalent to its own input. Theorem B invokes Theorem A for nearby tori with p != 0, which is legitimate because Theorem A was already established independently. Thus the central claims do not reduce by construction to their inputs.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim uses no fitted parameters and introduces no new entities. It rests on cited external theorems and a couple of unproved but standard geometric assertions; the most fragile is the equal-area Hamiltonian isotopy in Proposition 2.6.

assumptions (7)

domain assumption The explicit Oakley-Usher formula for L(x,y) in S^2×S^2: L(x,y) = {(v,w) | v1+w1 = 2(x+y-1), v·w = 2(1-y)^2-1}.
Invoked at the start of Section 2 as the model for all computations. It is taken from [OU16, Proof of Proposition 2.1] and not reproven.
standard math EP93 Lemma 4.2 A: for p≠0 and an embedded curve Γ in B^2(1), F^{-1}(Γ)∩H^{-1}({-p}) is a Lagrangian torus.
Used in Section 2.1 to identify the torus tilde L2(p,q) and in Lemma 2.5 and Proposition 2.8.
domain assumption Any two embedded closed curves in (B^2(1), ω_p) enclosing the same ω_p-area are Hamiltonian isotopic.
Used in Proposition 2.6 to construct K with φ^{1,p,K}(Γ)=S^1(r). This is asserted without proof or citation and also assumes Γ is a simple closed curve.
domain assumption Symmetric probes: for the probe σ={p=a}, points at equal distance from the boundary give Hamiltonian isotopic L2 fibers; the isotopy lifts from the reduced space via a cutoff function.
Proved in Proposition 2.12 for the special case, but the lifting step relies on [AM13] and [Bre20]. Used to reduce p^2 ≥ q^4 to |p| < q^2.
standard math The displacement energy of a toric fiber T(a,b) is min(1/2 - |a|, 1/2 - |b|).
Used in Section 3 for the displacement energy germ computations of L1(0,q) and T(ξ,ξ); cited from [Bre20, Example 4.1].
domain assumption FOOO's result that L1(0,q) with 0 < q ≤ 1/2 are nondisplaceable and thus not Hamiltonian isotopic to standard toric fibers.
Used in Theorem B to restrict attention to q > 1/2; the nondisplaceability is cited from [FOOO12], and the fact that the only nondisplaceable standard toric fiber is the Clifford torus is standard.
standard math Weinstein's Lagrangian neighborhood theorem.
Used in Theorem B to represent nearby Lagrangians as graphs of closed 1-forms over a toric fiber.

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Pith. "Pith review of On Lagrangian Tori in $S^2\times S^2$." pith.science (2026). https://pith.science/paper/37VMM3NG

@misc{pith2026241216356,
  author       = {Pith},
  title        = {Pith review of: On Lagrangian Tori in $S^2\times S^2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/37VMM3NG}},
  note         = {Machine review of arXiv:2412.16356}
}

abstract

In [FOOO12], K. Fukaya, Y. Oh, H. Ohta, and K. Ono (FOOO) obtained the monotone symplectic manifold $S^2\times S^2$ by resolving the singularity of a toric degeneration of a Hirzebruch surface. They identified a continuum of toric fibers in the resolved toric degeneration that are not Hamiltonian isotopic to the toric fibers of the standard toric structure on $S^2\times S^2$. In this paper, we provide a comprehensive classification: for any toric fiber in FOOO's construction of $S^2\times S^2$, we determine whether it is Hamiltonian isotopic to a toric fiber of the standard toric structure of $S^2\times S^2$.

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Forward citations

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Reference graph

Works this paper leans on

17 extracted references · 16 canonical work pages · cited by 3 Pith papers

[1]

Miguel Abreu, Matthew Storm Borman, and Dusa McDuff, Displacing lagrangian toric fibers by extended probes, Algebraic & Geometric Topology 14 (2014), 687--752

work page 2014
[2]

8, 1046--1051

Peter Albers and Urs Frauenfelder, A nondisplaceable lagrangian torus in T^*S^2 , Communications on Pure and Applied Mathematics 61 (2007), no. 8, 1046--1051

work page 2007
[3]

7, 3851--3875

Miguel Abreu and Leonardo Macarini, Remarks on lagrangian intersections in toric manifolds, Transactions of the American Mathematical Society 365 (2013), no. 7, 3851--3875

work page 2013
[4]

V. I. Arnol'd, First steps in symplectic topology, Russian Mathematical Surveys 41 (1986), no. 6, 3--18

work page 1986
[5]

Biran, Lagrangian non-intersections, Geometric & Functional Analysis GAFA 16 (2006), 279--326

P. Biran, Lagrangian non-intersections, Geometric & Functional Analysis GAFA 16 (2006), 279--326

work page 2006
[6]

Jo e Brendel, Real lagrangian tori and versal deformation, arXiv: 2002.03696 (2020)

arXiv 2020
[7]

, Hamiltonian classification of toric fibres and symmetric probes, arXiv:2302.00334 (2023)

work page Pith review arXiv 2023
[8]

Yu. V. Chekanov, Lagrangian tori in symplectic vector space and global symplectomorphism, Mathematische Zeitschrift 223 (1996), 547--559

work page 1996

Show all 17 references

[9]

Yuri Chekanov and Felix Schlenk, Notes on monotone lagrangian twist tori, Electronic Research Announcements in Mathematical Sciences 17 (2010), 104--121

2010
[10]

Yakov Eliashberg and Leonid Polterovich, The problem of lagrangian knots in four-manifolds, 1993 Georgia International Topology Conference, August2-13, 1993: Geometric Topology (1993), 313--327

1993
[11]

Michael Entov and Leonid Polterovich, Rigid subsets of symplectic manifolds, Compositio Mathematica 145 (2009), 773--826

2009
[12]

13, 2942--2993

Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, Toric degeneration and nondisplaceable lagrangian tori, International Mathematics Research Notices 2012 (2012), no. 13, 2942--2993

2012
[13]

2, 343--361

Agn e s Gadbled, On exotic monotone lagrangian tori in CP ^2 and S ^2 S ^2 , Journal of Symplectic Geometry 11 (2013), no. 2, 343--361

2013
[14]

Misha Gromov, Pseudo holomorphic curves in symplectic manifolds, Inventiones Mathematicae 82 (1985), 307–347

1985
[15]

Dusa McDuff, Displacing lagrangian toric fibers via probes, Low-dimensional and Symplectic Topology, Proceedings of Symposia in Pure Mathematics 82 (2011), 131--160

2011
[16]

Joel Oakley and Michael Usher, On certain lagrangian submanifolds of S^2 S^2 and C P^n , Algebraic & Geometric Topology 16 (2016), 149--209

2016
[17]

Georgios Dimitroglou Rizell, Elizabeth Goodman, and Alexander Ivrii, Lagrangian isotopy of tori in S^2 S^2 and C P^2 , Geometric & Functional Analysis GAFA 26 (2016), 1297--1358

2016

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Miguel Abreu, Matthew Storm Borman, and Dusa McDuff, Displacing lagrangian toric fibers by extended probes, Algebraic & Geometric Topology 14 (2014), 687--752

work page 2014

[2] [2]

8, 1046--1051

Peter Albers and Urs Frauenfelder, A nondisplaceable lagrangian torus in T^*S^2 , Communications on Pure and Applied Mathematics 61 (2007), no. 8, 1046--1051

work page 2007

[3] [3]

7, 3851--3875

Miguel Abreu and Leonardo Macarini, Remarks on lagrangian intersections in toric manifolds, Transactions of the American Mathematical Society 365 (2013), no. 7, 3851--3875

work page 2013

[4] [4]

V. I. Arnol'd, First steps in symplectic topology, Russian Mathematical Surveys 41 (1986), no. 6, 3--18

work page 1986

[5] [5]

Biran, Lagrangian non-intersections, Geometric & Functional Analysis GAFA 16 (2006), 279--326

P. Biran, Lagrangian non-intersections, Geometric & Functional Analysis GAFA 16 (2006), 279--326

work page 2006

[6] [6]

Jo e Brendel, Real lagrangian tori and versal deformation, arXiv: 2002.03696 (2020)

arXiv 2020

[7] [7]

, Hamiltonian classification of toric fibres and symmetric probes, arXiv:2302.00334 (2023)

work page Pith review arXiv 2023

[8] [8]

Yu. V. Chekanov, Lagrangian tori in symplectic vector space and global symplectomorphism, Mathematische Zeitschrift 223 (1996), 547--559

work page 1996