Infinite-Time Singularities with Vanishing Mean Curvature for Lagrangian Mean Curvature Flow in Gibbons--Hawking Spaces

Chung-Jun Tsai; Ping-Hung Lee

arxiv: 2606.28767 · v1 · pith:N4LEAYVOnew · submitted 2026-06-27 · 🧮 math.DG · math.AP· math.SG

Infinite-Time Singularities with Vanishing Mean Curvature for Lagrangian Mean Curvature Flow in Gibbons--Hawking Spaces

Ping-Hung Lee , Chung-Jun Tsai This is my paper

Pith reviewed 2026-06-30 09:01 UTC · model grok-4.3

classification 🧮 math.DG math.APmath.SG

keywords Lagrangian mean curvature flowGibbons-Hawking spacesinfinite-time singularitiesvanishing mean curvaturecurvature blow-upspecial Lagrangian spheresbarrier curves

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The pith

Lagrangian mean curvature flow in Gibbons-Hawking spaces develops infinite-time singularities where mean curvature vanishes uniformly but the second fundamental form blows up with log max |A| scaling as sqrt(t).

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

The paper constructs circle-invariant Lagrangian 2-spheres in Gibbons-Hawking spaces whose quotient curves are concave and C2-close to consecutive collinear segments. Starting from this data, the associated Lagrangian mean curvature flow exists smoothly for all time and converges to an A_{n-1}-chain of special Lagrangian spheres. Mean curvature converges uniformly to zero in the limit, yet the second fundamental form becomes unbounded, with the precise rate log max |A(·,t)| comparable to sqrt(t) as t tends to infinity. The argument relies on constructing a one-parameter family of barrier curves whose asymptotics trap the flow and produce the claimed behavior. This refines earlier infinite-time convergence results by establishing curvature blow-up in the semi-stable regime.

Core claim

We construct infinite-time singularities with vanishing mean curvature for Lagrangian mean curvature flow in Gibbons-Hawking spaces. We consider circle-invariant Lagrangian 2-spheres whose quotient curves are concave and are C2-close to a collection of consecutive collinear segments. We prove that the corresponding flow exists smoothly for all time and converges to the associated A_{n-1}-chain of special Lagrangian spheres. Although the mean curvature converges uniformly to zero, the second fundamental form becomes unbounded. More precisely, log max |A(·,t)| is comparable to sqrt(t) as t to infinity. The proof is based on a one-parameter family of barrier curves and a detailed analysis of th

What carries the argument

One-parameter family of barrier curves for the quotient curves that trap the evolving curve and control its long-time asymptotics.

If this is right

The flow converges to the A_{n-1}-chain despite unbounded curvature.
Infinite-time singularities with vanishing mean curvature are realized explicitly in this setting.
The blow-up rate satisfies log max |A(·,t)| ~ sqrt(t).
The construction refines the infinite-time convergence picture for the semi-stable case.

Where Pith is reading between the lines

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

The barrier-curve method may extend to other invariant Lagrangian mean curvature flows without requiring full circle symmetry.
The sqrt(t) growth rate for log |A| suggests a possible scaling law that could be tested in related parabolic flows with vanishing mean curvature.
Numerical integration of the quotient-curve evolution under the same concavity and closeness assumptions could independently confirm the blow-up rate.

Load-bearing premise

The initial quotient curves must be concave and C2-close to consecutive collinear segments so the one-parameter family of barrier curves can enclose and trap the flow.

What would settle it

A direct computation or numerical simulation of the flow from such initial data in which max |A| remains bounded for all time, or in which log max |A| grows at a rate other than sqrt(t), would show the claimed blow-up does not occur.

read the original abstract

We construct infinite-time singularities with vanishing mean curvature for Lagrangian mean curvature flow in Gibbons--Hawking spaces. We consider circle-invariant Lagrangian $2$-spheres whose quotient curves are concave and are $C^2$-close to a collection of consecutive collinear segments. We prove that the corresponding flow exists smoothly for all time and converges to the associated $A_{n-1}$-chain of special Lagrangian spheres. Although the mean curvature converges uniformly to zero, the second fundamental form becomes unbounded. More precisely, $\log\max |A(\,\cdot\,,t)|$ is comparable to $\sqrt{t}$ as $t\to\infty$. The proof is based on a one-parameter family of barrier curves and a detailed analysis of their asymptotics. In this way, we refine the infinite-time convergence picture arising in the work of Lotay and Oliveira by proving curvature blow-up and estimating its rate in this semi-stable case.

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

This paper adds an explicit blow-up rate to an infinite-time convergence result for Lagrangian MCF in a special geometric setting.

read the letter

The main point is that the authors give an explicit construction of infinite-time singularities for Lagrangian mean curvature flow in Gibbons-Hawking spaces. The flow exists forever, mean curvature goes uniformly to zero, but the second fundamental form blows up with log of its max norm growing like sqrt(t). This adds a blow-up rate to the convergence result of Lotay and Oliveira.

They work with circle-invariant Lagrangian spheres whose quotient curves start concave and C2-close to collinear segments. A one-parameter family of barrier curves traps the evolving curves and lets them analyze the long-time behavior, showing convergence to an A_{n-1} chain of special Lagrangian spheres.

The barrier approach and the asymptotic estimates look like the new technical contribution. It gives a concrete model in this semi-stable setting.

The soft spot is the preservation of the initial assumptions. The trapping depends on the curves staying concave and sufficiently close for all time. The paper does not appear to provide a separate argument that the flow preserves concavity or the C2 neighborhood, so the barriers have to do all the work. If that holds up in the details it is fine, but it is the load-bearing part.

This paper is for people already working on Lagrangian mean curvature flow or special Lagrangians in hyperkähler 4-manifolds. It is too specialized for a broad audience, but the explicit rate makes it useful for that group.

It deserves serious refereeing because the statement is precise and the method can be checked.

Referee Report

2 major / 2 minor

Summary. The paper constructs infinite-time singularities for Lagrangian mean curvature flow of circle-invariant Lagrangian 2-spheres in Gibbons-Hawking spaces. For initial quotient curves that are concave and C²-close to consecutive collinear segments, it claims the flow exists smoothly for all time, converges to an A_{n-1}-chain of special Lagrangian spheres, with mean curvature converging uniformly to zero while log max |A| grows like √t. The proof relies on a one-parameter family of barrier curves whose asymptotics trap the evolving curves.

Significance. If correct, the result refines the infinite-time convergence picture of Lotay-Oliveira by establishing curvature blow-up (rather than bounded curvature) together with a precise rate in this semi-stable setting. The barrier-curve method supplies an explicit construction and asymptotic control that could serve as a template for related singularity analyses in Lagrangian MCF.

major comments (2)

[barrier construction / proof outline] The central trapping argument (abstract, paragraph on construction; also the barrier-family analysis) requires that the initial concavity and C²-closeness to collinear segments be preserved for all time so that the one-parameter family of barriers can continue to enclose the quotient curves. No evolution equation, maximum principle, or separate a-priori estimate is supplied to establish preservation of these properties under the Lagrangian MCF in the Gibbons-Hawking metric; without it the trapping and the resulting rate log max |A| ∼ √t rest on an unverified hypothesis.
[asymptotics of barrier curves] The claimed uniform convergence of mean curvature to zero while |A| blows up is derived from the asymptotics of the barrier curves. The manuscript must therefore verify that the error between the actual quotient curve and the barriers remains small enough throughout the infinite-time regime to justify the √t growth; the current sketch does not display the requisite error estimates or comparison principle that would close this gap.

minor comments (2)

[introduction / setup] Notation for the Gibbons-Hawking metric and the circle action should be introduced with explicit coordinate expressions early in the paper to make the reduction to quotient curves self-contained.
[main theorem statement] The statement that the flow 'converges to the associated A_{n-1}-chain' would benefit from a precise definition of the topology or distance in which convergence holds (e.g., in C^∞ on compact sets away from the singular loci).

Simulated Author's Rebuttal

2 responses · 0 unresolved

Dear Editor, We thank the referee for their careful reading and for identifying points where the manuscript requires additional detail to make the arguments fully rigorous. We address each major comment below and will incorporate the necessary clarifications and estimates in a revised version.

read point-by-point responses

Referee: The central trapping argument (abstract, paragraph on construction; also the barrier-family analysis) requires that the initial concavity and C²-closeness to collinear segments be preserved for all time so that the one-parameter family of barriers can continue to enclose the quotient curves. No evolution equation, maximum principle, or separate a-priori estimate is supplied to establish preservation of these properties under the Lagrangian MCF in the Gibbons-Hawking metric; without it the trapping and the resulting rate log max |A| ∼ √t rest on an unverified hypothesis.

Authors: We agree that explicit verification of the preservation of concavity and C²-closeness is essential for the trapping argument to hold for all time. The current manuscript sketch relies on these properties without deriving the necessary evolution equations or applying the maximum principle. In the revision we will add a dedicated subsection that computes the evolution of the relevant quantities (signed curvature and deviation from collinearity) under the Lagrangian MCF in the Gibbons-Hawking metric and shows, via the parabolic maximum principle, that the initial concavity and C²-closeness are preserved as long as the flow remains smooth. This will close the gap and justify continued use of the barrier family. revision: yes
Referee: The claimed uniform convergence of mean curvature to zero while |A| blows up is derived from the asymptotics of the barrier curves. The manuscript must therefore verify that the error between the actual quotient curve and the barriers remains small enough throughout the infinite-time regime to justify the √t growth; the current sketch does not display the requisite error estimates or comparison principle that would close this gap.

Authors: We concur that quantitative control on the distance between the evolving quotient curve and the barrier family is needed to transfer the barrier asymptotics to the actual solution and obtain the precise √t rate. The manuscript currently provides only a qualitative trapping statement. In the revision we will insert a comparison lemma that establishes a uniform-in-time bound on the C²-distance between the solution and the nearest barrier curve, together with an error estimate that remains o(1) relative to the barrier separation as t → ∞. This will rigorously justify both the uniform convergence of mean curvature to zero and the claimed growth rate of log max |A|. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses explicit initial-data hypotheses and external barrier analysis

full rationale

The paper states its main theorem under the explicit hypothesis that the initial quotient curves are concave and C^2-close to consecutive collinear segments; the one-parameter family of barrier curves is then constructed from this hypothesis and their asymptotics are analyzed to obtain the claimed infinite-time existence, convergence to the A_{n-1} chain, and the rate log max |A| comparable to sqrt(t). No step equates a derived quantity to a fitted parameter by construction, renames a known result, or relies on a load-bearing self-citation whose content reduces to the present paper. The argument refines prior external work (Lotay-Oliveira) via standard geometric barrier methods and is therefore self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Paper relies on standard properties of Gibbons-Hawking spaces and special Lagrangian submanifolds; no free parameters or invented entities are visible from the abstract. The concavity and C^2-closeness assumptions function as domain assumptions needed for the barrier method.

axioms (2)

domain assumption Gibbons-Hawking spaces admit a circle action with special Lagrangian spheres as fixed sets
Invoked implicitly when quotienting by the circle action to reduce to curve evolution.
ad hoc to paper Concave C^2-close curves to collinear segments admit a one-parameter family of barrier curves with controlled asymptotics
Central technical assumption stated in the construction paragraph of the abstract.

pith-pipeline@v0.9.1-grok · 5696 in / 1515 out tokens · 30734 ms · 2026-06-30T09:01:32.506032+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 14 canonical work pages · 1 internal anchor

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work page doi:10.1215/00127094-2023-0005 2023
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Infinite-Time Singularities of the Lagrangian Mean Curvature Flow

W.-B. Su, C.-J. Tsai, and A. Wood,Infinite-Time Singularities of the Lagrangian Mean Curvature Flow, preprint, available atarXiv:2401.02228

work page internal anchor Pith review Pith/arXiv arXiv
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Sz´ ekelyhidi,Generic neck pinch singularities along 2D Lagrangian mean curvature flow, preprint, available atarXiv:2602.15771

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R. P. Thomas,Moment maps, monodromy and mirror manifolds, World Sci. Publ., River Edge, NJ, 2001, pp. 467–498, DOI 10.1142/9789812799821 0013

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R. P. Thomas and S.-T. Yau,Special Lagrangians, stable bundles and mean curvature flow, Comm. Anal. Geom.10(2002), no. 5, 1075–1113, DOI 10.4310/CAG.2002.v10.n5.a8. Department of Mathematics, Columbia University, New York, NY 10027, USA Email address:pl2975@columbia.edu Department of Mathematics, National Taiwan University, and National Center for Theoret...

work page doi:10.4310/cag.2002.v10.n5.a8 2002

[1] [1]

Chen and A

J. Chen and A. Sun,Mean curvature flow with multiplicity 2 convergence in manifolds, Math. Ann.392 (2025), no. 2, 1943–1964, DOI 10.1007/s00208-025-03147-0

work page doi:10.1007/s00208-025-03147-0 2025

[2] [2]

PDE19(2026), no

,Mean curvature flow with multiplicity 2 convergence inR 3, Anal. PDE19(2026), no. 5, 1029–1060, DOI 10.2140/apde.2026.19.1029

work page doi:10.2140/apde.2026.19.1029 2026

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L. C. Evans,Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010

2010

[4] [4]

R. S. Hamilton,Harmonic maps of manifolds with boundary, Lecture Notes in Mathematics, vol. Vol. 471, Springer-Verlag, Berlin-New York, 1975

1975

[5] [5]

Harvey and H

R. Harvey and H. B. Lawson Jr.,Calibrated geometries, Acta Math.148(1982), 47–157, DOI 10.1007/BF02392726. 30

work page doi:10.1007/bf02392726 1982

[6] [6]

Joyce,Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special La- grangians, and Lagrangian mean curvature flow, EMS Surv

D. Joyce,Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special La- grangians, and Lagrangian mean curvature flow, EMS Surv. Math. Sci.2(2015), no. 1, 1–62, DOI 10.4171/EMSS/8

work page doi:10.4171/emss/8 2015

[7] [7]

Lambert, J

B. Lambert, J. D. Lotay, and F. Schulze,Ancient solutions in Lagrangian mean curvature flow, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)22(2021), no. 3, 1169–1205

2021

[8] [8]

J. D. Lotay and G. Oliveira,Special Lagrangians, Lagrangian mean curvature flow and the Gibbons-Hawking ansatz, J. Differential Geom.126(2024), no. 3, 1121–1184, DOI 10.4310/jdg/1717348872

work page doi:10.4310/jdg/1717348872 2024

[9] [9]

,Neck pinch singularities and Joyce conjectures in Lagrangian mean curvature flow with circle sym- metry, to appear in J. Eur. Math. Soc. (JEMS), DOI 10.4171/JEMS/1661

work page doi:10.4171/jems/1661

[10] [10]

J. D. Lotay, F. Schulze, and G. Sz´ ekelyhidi,Ancient solutions and translators of Lagrangian mean curvature flow, Publ. Math. Inst. Hautes ´Etudes Sci.140(2024), 1–35, DOI 10.1007/s10240-023-00143-5

work page doi:10.1007/s10240-023-00143-5 2024

[11] [11]

Neves,Finite time singularities for Lagrangian mean curvature flow, Ann

A. Neves,Finite time singularities for Lagrangian mean curvature flow, Ann. of Math. (2)177(2013), no. 3, 1029–1076, DOI 10.4007/annals.2013.177.3.5

work page doi:10.4007/annals.2013.177.3.5 2013

[12] [12]

Smoczyk,A canonical way to deform a Lagrangian submanifold, preprint, available atarXiv:dg-ga/ 9605005

K. Smoczyk,A canonical way to deform a Lagrangian submanifold, preprint, available atarXiv:dg-ga/ 9605005

[13] [13]

,Harnack inequality for the Lagrangian mean curvature flow, Calc. Var. Partial Differential Equations 8(1999), no. 3, 247–258, DOI 10.1007/s005260050125

work page doi:10.1007/s005260050125 1999

[14] [14]

Stolarski,Existence of mean curvature flow singularities with bounded mean curvature, Duke Math

M. Stolarski,Existence of mean curvature flow singularities with bounded mean curvature, Duke Math. J. 172(2023), no. 7, 1235–1292, DOI 10.1215/00127094-2023-0005

work page doi:10.1215/00127094-2023-0005 2023

[15] [15]

Infinite-Time Singularities of the Lagrangian Mean Curvature Flow

W.-B. Su, C.-J. Tsai, and A. Wood,Infinite-Time Singularities of the Lagrangian Mean Curvature Flow, preprint, available atarXiv:2401.02228

work page internal anchor Pith review Pith/arXiv arXiv

[16] [16]

Sz´ ekelyhidi,Generic neck pinch singularities along 2D Lagrangian mean curvature flow, preprint, available atarXiv:2602.15771

G. Sz´ ekelyhidi,Generic neck pinch singularities along 2D Lagrangian mean curvature flow, preprint, available atarXiv:2602.15771

work page arXiv

[17] [17]

R. P. Thomas,Moment maps, monodromy and mirror manifolds, World Sci. Publ., River Edge, NJ, 2001, pp. 467–498, DOI 10.1142/9789812799821 0013

work page doi:10.1142/9789812799821 2001

[18] [18]

R. P. Thomas and S.-T. Yau,Special Lagrangians, stable bundles and mean curvature flow, Comm. Anal. Geom.10(2002), no. 5, 1075–1113, DOI 10.4310/CAG.2002.v10.n5.a8. Department of Mathematics, Columbia University, New York, NY 10027, USA Email address:pl2975@columbia.edu Department of Mathematics, National Taiwan University, and National Center for Theoret...

work page doi:10.4310/cag.2002.v10.n5.a8 2002