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

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 $C<sup>2$-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.

• The paper establishes that, despite vanishing mean curvature, the second fundamental form blows up at a quantifiable rate in the LMCF on Gibbons–Hawking spaces.
• It employs refined ODE barrier constructions and asymptotic analysis to reduce the LMCF dynamics to a planar parabolic equation with precise curvature estimates.
• The findings challenge conventional expectations in geometric flows, refining the Thomas–Yau and Joyce conjectures by revealing smooth convergence can hide underlying curvature blow-up.

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

Introduction and Motivation

This paper rigorously analyzes the long-time dynamics of Lagrangian mean curvature flow (LMCF) in four-dimensional hyperkähler spaces endowed with a tri-Hamiltonian circle action (Gibbons–Hawking spaces). Specifically, it establishes new phenomena regarding the nature of singularity formation at infinite time for LMCF of circle-invariant Lagrangian $2$-spheres whose base quotient curves are close to consecutive collinear segments, corresponding in the quotient to an $A_{n-1}$-chain of special Lagrangian spheres. The work refines and extends previous results by Lotay and Oliveira, examining a scenario in which the mean curvature vanishes asymptotically, yet the second fundamental form blows up, providing the first precise quantification of such infinite-time singularities with vanishing mean curvature.

Background: Lagrangian Mean Curvature Flow and Gibbons–Hawking Spaces

LMCF preserves the Lagrangian condition in Calabi–Yau and hyperkähler manifolds. The long-standing Thomas–Yau and Joyce conjectures propose that LMCF serves as a tool for decomposing Lagrangian submanifolds into unions of special Lagrangians. However, generic LMCFs develop finite-time singularities, and recent work has identified both finite and infinite-time singularities in symmetric settings, notably circle-invariant Lagrangians in Gibbons–Hawking spaces, where the geometry admits a reduction of the flow to a curve shortening evolution for the quotient curve in $\mathbb{R}^{2}$.

Lotay and Oliveira verified several aspects of the conjectural picture, showing that for almost-calibrated, circle-invariant Lagrangians, the flow can pass through finitely many finite-time neck-pinch singularities and converge, in the current sense, to a special Lagrangian $A_{n-1}$-chain; their reduction to a planar graphical equation enables refined analysis.

Main Results

Geometric Setup

The Gibbons–Hawking metric is specified on a non-compact hyperkähler $4$-manifold $U$ with isolated fixed points $p_{1},\ldots,p_{n}$ under the circle action, and a harmonic potential $\phi(x,y,z)$. Circle-invariant Lagrangian $2$-spheres correspond, under projection, to curves in the $xy$-plane connecting fixed points, and the flow reduces to a nonlinear parabolic equation for the graphical function $A_{n-1}$0: $A_{n-1}$1 The preimages of consecutive collinear segments form an $A_{n-1}$2-chain of special Lagrangian spheres, with their corresponding planar curves piecewise linear and meeting at the $A_{n-1}$3.

Infinite-Time Singularity with Vanishing Mean Curvature

The principal theorem demonstrates that for initial base curves concave and $A_{n-1}$4-close to an $A_{n-1}$5-chain, the LMCF exists smoothly for all time, converging to the $A_{n-1}$6-chain. More precisely,

• The mean curvature $A_{n-1}$7 converges uniformly to zero as $A_{n-1}$8.
• The second fundamental form $A_{n-1}$9 blows up at a precise rate: $\mathbb{R}^{2}$0 as $\mathbb{R}^{2}$1.

This result is structurally robust: uniform bounds on the first derivative persist, but curvature concentration manifests in the vanishing neck regions near the points $\mathbb{R}^{2}$2. Strong control is proven via the construction of sophisticated one-parameter barrier function families, with sharp asymptotic estimates (detailed below).

Asymptotic Analysis and Barrier Construction

The paper's technical core is the precise asymptotic expansion for the shrinking necks. Graphical barriers are constructed as minimizers of certain energy functionals, yielding families $\mathbb{R}^{2}$3 solving

$\mathbb{R}^{2}$4

with detailed regularity, comparison, and differentiability in the parameter $\mathbb{R}^{2}$5 established. Asymptotic analysis reveals that as $\mathbb{R}^{2}$6 (corresponding to late times),

$\mathbb{R}^{2}$7

where $\mathbb{R}^{2}$8 is a positive eigenvalue intrinsic to the degenerate linearized operator at the $\mathbb{R}^{2}$9-chain limit.

Rescalings show that the spatial profile of the neck pinches converges to a unique, piecewise linear function explicitly determined by the configuration of projection points $A_{n-1}$0. Barrier estimates yield matching subsolutions and supersolutions for the LMCF graphical equation, guaranteeing precise sandwiched bounds on the flow and thereby quantifying the decay of the minimal neck scale.

Quantitative Curvature Blow-Up

The central quantitative assertion is: $A_{n-1}$1 or in the most symmetric cases (ALE/ALF metrics),

$A_{n-1}$2

pinning the curvature scale to an explicit geometric constant.

Implications and Context

This work demonstrates that in LMCF, even with vanishing mean curvature and global smooth existence, the formation of infinite-time singularities can exhibit controlled, quantifiable curvature blow-up. This invalidates any naive expectation that vanishing $A_{n-1}$3 and smooth long-term existence guarantee smooth convergence of the surface. Thus, for geometric flows, particularly those with non-compact symmetry reductions and higher codimension, singularity formation is subtle and can be characterized by blow-up of higher derivatives with mean curvature tending to zero.

The analysis connects with constructions of immortal flows in other works (e.g., [CS-25], [CS-26], [STW-24]), but this explicitly provides an example where the mean curvature not only remains bounded but tends to zero, separating the behavior of the first and second fundamental forms—a notable counterpoint to previous work on mean curvature flow singularities.

The techniques—precise barrier construction, tight asymptotic rescaling, and the exploitation of the reduction to ODEs or degenerate "neck" models—offer methods that are likely transferable to broader classes of geometric evolution problems, especially those exhibiting symmetry.

Theoretical and Future Directions

The findings suggest that:

• The Thomas–Yau and Joyce conjectural pictures for LMCF require careful interpretation of "smooth convergence" in light of infinite-time singularities with $A_{n-1}$4 but $A_{n-1}$5.
• The intricate relationship between topology, symmetry, and the analytic structure of singularity formation becomes crucial in both the construction and regularity theory for special Lagrangian decompositions.
• Infinite-time singularities may play a role in moduli theory for special Lagrangians, and the asymptotics developed here could inform compactification schemes for Lagrangian moduli spaces or stability conditions in mirror symmetry.

Further questions include the generality of the vanishing mean curvature/asymptotic singularity mechanism, its prevalence in less symmetric settings, and connections to stability phenomena recently studied by Székelyhidi ([Gabor-26]).

Conclusion

This paper establishes the existence and robust asymptotics of infinite-time singularities with vanishing mean curvature in Lagrangian mean curvature flow within Gibbons–Hawking spaces. By reducing the problem to ODE barrier constructions and extracting sharp curvature blow-up rates, the authors expose mechanisms whereby $A_{n-1}$6 convergence and vanishing mean curvature coexist with unbounded curvature. These results significantly refine current understanding of long-time LMCF behavior and singularity formation, and are expected to have lasting influence in geometric analysis and special Lagrangian geometry.

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Reference:

"Infinite-Time Singularities with Vanishing Mean Curvature for Lagrangian Mean Curvature Flow in Gibbons–Hawking Spaces" (2606.28767)