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arxiv: 2401.15432 · v2 · submitted 2024-01-27 · 🧮 math.DG · math.AP· math.SG

On the Donaldson-Scaduto conjecture

Saman Habibi Esfahani , Yang Li This is my paper

Pith reviewed 2026-05-24 04:35 UTC · model grok-4.3

classification 🧮 math.DG math.APmath.SG MSC 53C3835J96
keywords special LagrangianMonge-Ampère equationDonaldson-Scaduto conjectureA2 ALEU(1) invariantgeometric measure theoryasymptotically cylindrical endsG2 manifold
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The pith

Solving a singular real Monge-Ampère equation proves the Donaldson-Scaduto conjecture on special Lagrangians.

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

Donaldson and Scaduto conjectured the existence of associative three-holed spheres with three cylindrical ends in certain G2-manifolds, or equivalently special Lagrangians in the Calabi-Yau X × ℂ where X is an A2 ALE hyperkähler 4-manifold. The paper proves this by first solving a real Monge-Ampère equation whose right-hand side has a singularity, which gives a candidate special Lagrangian that may itself be singular. Geometric measure theory is then used to show that this candidate is actually smooth and has the correct asymptotic behavior at the ends. The same method yields additional families of U(1)-invariant special Lagrangians with similar properties.

Core claim

The central discovery is that the real Monge-Ampère equation with a singular right-hand side admits a solution whose associated special Lagrangian in X × ℂ is smooth and asymptotically cylindrical, thereby establishing the existence asserted by the Donaldson-Scaduto conjecture; the argument also constructs many further examples of this type.

What carries the argument

A real Monge-Ampère equation with singular right-hand side whose solution is shown to be a smooth special Lagrangian by geometric measure theory.

Load-bearing premise

The geometric measure theory regularity theorems apply to the potentially singular solution arising from the singular Monge-Ampère equation in this U(1)-invariant A2 ALE setting.

What would settle it

Numerical solution of the Monge-Ampère equation revealing a non-removable singularity, or analytic proof that the GMT asymptotic control fails for this specific right-hand side.

Figures

Figures reproduced from arXiv: 2401.15432 by Saman Habibi Esfahani, Yang Li.

Figure 1
Figure 1. Figure 1: Donaldson-Scaduto conjecture. In fact, since the vectors v1, v2, and v3 lie in a plane, say R 2×{0} ⊂ R 3 , the associative submanifold P can be equivalently interpreted as a special Lagrangian submanifold in X4 × R 2 with an appropriate Calabi-Yau structure. Our method readily generalizes to the An−1-type ALE or ALF gravitational instantons X, where n ≥ 3, and the monopole points p1, . . . , pn in the Gib… view at source ↗
Figure 2
Figure 2. Figure 2: Li,red ⊂ R 4 = R 2 (u1,u2) × R 2 (y1,y2) . Therefore, π1(∪iLi,red) is the boundary of the convex polygon with vertices p1, . . . , pn, and π2(∪iLi,red) is the union of n rays, as shown in [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: π1(∪Li,red) (left) and π2(∪Li,red) (right). Graphical case: The asymptotic cylindrical requirement motivates us to look for the reduction Lred of the conjectural special Lagrangian L, as (the closure of) the graph of a map F : U ⊂ R 2 (u1,u2) → R 2 (y1,y2) , (y1, y2) = F(u1, u2), where U is the interior of the convex polytope with vertices p1, . . . , pn. The projection of Lred to R 2 (y1,y2) is expected t… view at source ↗
Figure 4
Figure 4. Figure 4: Lred in Zred = R 4 , in the case n = 3. The equation du1 ∧ dy1 + du2 ∧ dy2 = 0 is equivalent to ∂y2 ∂u1 = ∂y1 ∂u2 . This implies that we can define a function φ : U → R such that F = ∇φ, namely y1 = ∂φ ∂u1 , y2 = ∂φ ∂u2 . 7 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Approximating domains Ut → U, in the case n = 3. The piecewise linear boundary data (3) can be extended to some Lipschitz function ϕ¯ on U. We consider the following Dirichlet problem for φt : Ut → R for each t ∈ (0, 1). ( det D2φt = V, on Ut , φt = ϕ, ¯ on ∂Ut . (4) Lemma 1 (Rauch-Taylor [8]). Let Ω ⊂ R 2 be a strictly convex domain, g : ∂Ω → R a continuous function, and µ a non-negative Borel measure on … view at source ↗
Figure 6
Figure 6. Figure 6: Standard position in the case n = 3. R(u∗, h, ε) is the shaded region. By the Monge-Amp`ere equation, and the strict positivity of V , C −1hε ≤ Z R(u∗,h,ε) V du1du2 = Z ∇φ(R(u∗,h,ε)) dy1dy2. (7) By the convexity of φ and the bound −Λ0u1 ≤ φ ≤ Cu1, we deduce |∂u2φ(u)| ≤ C(Λ0, h)u1, for all u ∈ R(u∗, h, ε). Thus by considering the gradient image, | Z ∇φ(R(u∗,h,ε)) dy1dy2| ≤ C(Λ0, h)ε sup R(u∗,h,ε) (∂u1φ(u) +… view at source ↗
Figure 7
Figure 7. Figure 7: Mapping F = ∇φ : U ⊂ R 2 (u1,u2) → R 2 (y1,y2) . Denote the subgradient sets at the vertices by Cpi = {y ∈ R 2 (y1,y2) | φ(u) − φ(pi) ≥ ⟨y, u − pi⟩ for all u ∈ U}. Lemma 3. The sets Cpi are disjoint convex closed subsets of R 2 contained in the wedge region Cpi ⊂ Wpi = {y ∈ R 2 | y · (pi+1 − pi) ≤ bi+1 − bi} ∩ {y ∈ R 2 | y · (pi−1 − pi) ≤ bi−1 − bi}. Remark 3. The wedge region is a translated copy of the w… view at source ↗
Figure 8
Figure 8. Figure 8: Subgradients Cp1 , Cp2 , and Cp3 , in the case n = 3. Lemma 4 (Image of the gradient). We have ∇φ(U) = R 2 \ (∪iCpi ). Proof. First we claim that R 2 = ∪iCpi ∪ ∇φ(U). Given any y ∈ R 2 , we consider the graph of the affine linear function a + ⟨y, u⟩ on U, where a ∈ R increases from negative infinity. There must be some a when the graph first touches the graph of the convex function φ. This shows that y is … view at source ↗
Figure 9
Figure 9. Figure 9: ∇φ(U) cannot contain an inifinite wedge W. Proof. Without loss of generality, we assume y is large compared to maxi |bi − bi+1|. The rays R≥0v˜i partition R 2 into wedge regions Wpi . We choose the direction v˜i which minimizes the angle with the direction of y. Notice v˜i is parallel to the boundary ray between Wpi and Wpi+1 , and by the choice of v˜i and the largeness of y, we see y must lie in either Wp… view at source ↗
Figure 10
Figure 10. Figure 10: Triangle T(y). We have seen that Cpi does not intersect T(y). Then Area(T(y)) ≤ Z ∇φ(U) dy1dy2 + Area(T(y) ∩ ∪jCpj ) ≤ Z U V du1du2 + C, 15 [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The π1-projection of a tangent cone at pi . 18 [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Boundary components Ci and sets U0 and Ui , in the case n = 3. Let x0 ∈ V0 ∩ V1 be the base point. We have π1(V0, x0) ∼= Z, with a generator presented by a curve encircling the U(1)-fiber based at x0. Furthermore, π1(V1, x0) = {0} and π1(V0 ∩ V1, x0) ∼= Z, and the inclusion map V0 ∩ V1 → V0 takes the generator of π1(V0 ∩ V1, x0) to the generator of π1(V0, x0). Therefore, by Van Kampen’s theorem, π1(V0 ∪ V… view at source ↗
read the original abstract

Motivated by $G_2$-manifolds with coassociative fibrations in the adiabatic limit, Donaldson and Scaduto conjectured the existence of associative submanifolds homeomorphic to a three-holed $3$-sphere with three asymptotically cylindrical ends in the $G_2$-manifold $X \times \mathbb{R}^3$, or equivalently similar special Lagrangians in the Calabi-Yau 3-fold $X \times \mathbb{C}$, where $X$ is an $A_2$-type ALE hyperk\"ahler 4-manifold. We prove this conjecture by solving a real Monge-Amp\`ere equation with a singular right-hand side, which produces a potentially singular special Lagrangian. Then, we prove the smoothness and asymptotic properties for the special Lagrangian using inputs from geometric measure theory. The method produces many other asymptotically cylindrical $U(1)$-invariant special Lagrangians in $X\times \mathbb{C}$, where $X$ arises from the Gibbons-Hawking construction.

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.

Referee Report

2 major / 1 minor

Summary. The paper claims to prove the Donaldson-Scaduto conjecture by constructing U(1)-invariant special Lagrangians in the Calabi-Yau 3-fold X × ℂ (X an A₂-type ALE hyperkähler 4-manifold) that are homeomorphic to a three-holed 3-sphere with three asymptotically cylindrical ends. The proof proceeds in two steps: first solve a real Monge-Ampère equation with singular right-hand side to obtain a potentially singular special Lagrangian, then apply inputs from geometric measure theory to establish smoothness and the required asymptotic cylindrical ends. The same method is said to produce many other such special Lagrangians.

Significance. If the argument is complete, the result supplies the first existence proof for the conjectured associative submanifolds (or their special-Lagrangian counterparts) and introduces a PDE-plus-GMT construction that may apply to other U(1)-invariant problems on ALE backgrounds. The production of a family of examples rather than a single instance adds to the potential utility for adiabatic-limit questions in G₂-geometry.

major comments (2)
  1. [Main construction (after the Monge-Ampère step)] The central existence claim rests on the assertion that the potentially singular solution of the singular Monge-Ampère equation lies in the class to which the cited GMT regularity and asymptotic theorems apply directly. The manuscript must supply a precise verification that the U(1)-invariant A₂ ALE geometry introduces neither extra singularities nor obstructions that would invalidate the GMT inputs; without this check the weakest assumption in the argument remains unconfirmed.
  2. [Monge-Ampère analysis section] Error estimates, convergence rates, or a priori bounds for the singular Monge-Ampère solution prior to the GMT smoothing step are not visible in the argument outline. These controls are load-bearing for confirming that the output current satisfies the hypotheses of the GMT theorems used for smoothness and cylindrical asymptotics.
minor comments (1)
  1. [Introduction / setup] Notation for the singular right-hand side of the Monge-Ampère equation and for the U(1)-action should be introduced with explicit reference to the Gibbons-Hawking construction of X.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the detailed report. The comments identify places where the manuscript would benefit from additional explicit verification and expanded presentation of estimates. We address each point below and will revise the manuscript to incorporate the suggested clarifications.

read point-by-point responses

  1. Referee: [Main construction (after the Monge-Ampère step)] The central existence claim rests on the assertion that the potentially singular solution of the singular Monge-Ampère equation lies in the class to which the cited GMT regularity and asymptotic theorems apply directly. The manuscript must supply a precise verification that the U(1)-invariant A₂ ALE geometry introduces neither extra singularities nor obstructions that would invalidate the GMT inputs; without this check the weakest assumption in the argument remains unconfirmed.

    Authors: We agree that an explicit check is required. The U(1)-invariant solution produced by the singular Monge-Ampère equation is constructed so that its support and mass are controlled by the choice of the right-hand side, which is adapted to the A₂ Gibbons-Hawking metric. This ensures the current is integral and satisfies the monotonicity and density conditions of the cited GMT theorems without additional singularities arising from the ALE ends. Nevertheless, to make the verification fully transparent we will add a short dedicated paragraph immediately after the existence theorem that lists the GMT hypotheses and confirms each one holds for our current. revision: yes

  2. Referee: [Monge-Ampère analysis section] Error estimates, convergence rates, or a priori bounds for the singular Monge-Ampère solution prior to the GMT smoothing step are not visible in the argument outline. These controls are load-bearing for confirming that the output current satisfies the hypotheses of the GMT theorems used for smoothness and cylindrical asymptotics.

    Authors: The existence proof for the weak solution proceeds by approximation with smooth Monge-Ampère equations whose right-hand sides converge to the singular datum; uniform L^∞ bounds follow from a maximum principle that accounts for the singularity, and mass monotonicity supplies the necessary integral control. These estimates already guarantee that the limiting current meets the rectifiability and density hypotheses of the GMT results. We acknowledge that the estimates are distributed through the analysis section rather than collected in one place. We will therefore insert a new subsection that summarizes the a priori bounds, the convergence in the sense of currents, and the resulting verification that the GMT hypotheses are satisfied. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation proceeds by solving a real Monge-Ampère equation with singular right-hand side on the U(1)-invariant A2 ALE background to obtain a (potentially singular) special Lagrangian current, followed by an application of standard geometric measure theory results to upgrade regularity and establish the required cylindrical asymptotics. No equation or claim reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the GMT inputs are external theorems applied to the produced solution class. The construction is self-contained against external benchmarks and produces additional examples without circular renaming or ansatz smuggling.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based on abstract only; the proof rests on standard existence theory for Monge-Ampère equations and regularity theorems from geometric measure theory, with no free parameters or invented entities mentioned.

axioms (2)
  • domain assumption Existence of a (possibly singular) solution to the real Monge-Ampère equation with the prescribed singular right-hand side in the U(1)-invariant setting.
    Invoked to produce the candidate special Lagrangian before GMT smoothing.
  • domain assumption Geometric measure theory regularity and asymptotic results apply to the output of the singular Monge-Ampère equation without creating new singularities or violating the cylindrical ends.
    Central step that upgrades the potentially singular solution to a smooth special Lagrangian.

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Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

  1. [1]

    A localization property of viscosity solutions to the Monge-Amp` ere equation and their strict convexity

    L. A. Caffarelli. “A localization property of viscosity solutions to the Monge-Amp` ere equation and their strict convexity”. In: Ann. of Math. (2) 131.1 (1990), pp. 129–134. issn: 0003-486X,1939-8980. doi: 10.2307/1971509. url: https://doi.org/10.23 07/1971509

    work page doi:10.2307/1971509 1990

  2. [2]

    Adiabatic limits of co-associative Kovalev-Lefschetz fibrations

    Simon Donaldson. “Adiabatic limits of co-associative Kovalev-Lefschetz fibrations”. In: Algebra, geometry, and physics in the 21st century . Vol. 324. Progr. Math. Birkh¨ auser/Springer, Cham, 2017, pp. 1–29. isbn: 978-3-319-59938-0; 978-3-319- 59939-7. doi: 10.1007/978-3-319-59939-7\_1 . url: https://doi.org/10.1007 /978-3-319-59939-7_1 . 24

    work page doi:10.1007/978-3-319-59939-7 2017

  3. [3]

    Associative submanifolds and gradient cycles

    Simon Donaldson and Christopher Scaduto. “Associative submanifolds and gradient cycles”. In: Surveys in differential geometry 2019. Differential geometry, Calabi-Yau theory, and general relativity. Part 2 . Vol. 24. Surv. Differ. Geom. Int. Press, Boston, MA, [2022] ©2022, pp. 39–65. isbn: 978-1-57146-413-2

    work page 2019

  4. [4]

    Monopoles, Singularities and Hyperkahler Geometry

    Saman Habibi Esfahani. Monopoles, Singularities and Hyperkahler Geometry . Thesis (Ph.D.)–State University of New York at Stony Brook. ProQuest LLC, Ann Arbor, MI, 2022. isbn: 979-8351-45288-3. url: http://gateway.proquest.com/openurl ?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&r es_dat=xri:pqm&rft_dat=xri:pqdiss:29327731

    work page 2022

  5. [5]

    Special Lagrangian cones

    Mark Haskins. “Special Lagrangian cones”. In: Amer. J. Math. 126.4 (2004), pp. 845–

    work page 2004

  6. [6]

    url: http://muse.jhu.edu/journals/american _journal_of_mathematics/v126/126.4haskins.pdf

    issn: 0002-9327,1080-6377. url: http://muse.jhu.edu/journals/american _journal_of_mathematics/v126/126.4haskins.pdf

    work page

  7. [7]

    Joyce, U(1) -invariant special Lagrangian 3-folds

    Dominic Joyce. “U(1)-invariant special Lagrangian 3-folds. III. Properties of singular solutions”. In: Adv. Math. 192.1 (2005), pp. 135–182. issn: 0001-8708,1090-2082. doi: 10.1016/j.aim.2004.03.016 . url: https://doi.org/10.1016/j.aim.2004.03 .016

    work page doi:10.1016/j.aim.2004.03.016 2005

  8. [8]

    Partial regularity for singular solutions to the Monge-Amp` ere equation

    Connor Mooney. “Partial regularity for singular solutions to the Monge-Amp` ere equation”. In: Comm. Pure Appl. Math. 68.6 (2015), pp. 1066–1084. issn: 0010- 3640,1097-0312. doi: 10.1002/cpa.21534. url: https://doi.org/10.1002/cpa.2 1534

    work page doi:10.1002/cpa.21534 2015

  9. [9]

    The Dirichlet problem for the multidimensional Monge-Amp` ere equation

    Jeffrey Rauch and B. A. Taylor. “The Dirichlet problem for the multidimensional Monge-Amp` ere equation”. In:Rocky Mountain J. Math. 7.2 (1977), pp. 345–364. issn: 0035-7596,1945-3795. doi: 10.1216/RMJ-1977-7-2-345 . url: https://doi .org/10.1216/RMJ-1977-7-2-345 . Department of Mathematics, Duke University, 120 Science Dr, Durham, NC 27708-0320 E-mail ad...

    work page doi:10.1216/rmj-1977-7-2-345 1977