MCF-stack: Special Solutions of Mean Curvature Flow

§ 5.38 Stolarski–Su finite-time Lagrangian neck pinches71FD28B23406

Earliest recorded reference: (first arXiv submission).

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  1. Stolarski–Su — 2026-07-03 (first arXiv submission).
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(Prescribed Type II blow-up rates) For $n\geq 4$ and every integer $K\geq 2$, and for $n=3$ with $K\in \{ 2,3,4\} $, [StolarskiSu2026], Theorem 1.1 and Remark 1, construct smooth noncompact properly embedded exact almost-calibrated Lagrangian mean curvature flows $L(t)\subset \mathbb C^ n$, $0\leq t<T<\infty $. Here we record the $SO(n)$-invariant subfamily, of dimension and codimension $n$. Its topology is $S^{n-1}\times \mathbb R$, with two ends, and every regular slice is complete. The topology follows from the global profile graph and its spherical group orbits.

The origin is a finite-time Type II singularity with

\[ 0<\liminf _{t\uparrow T}(T-t)^{K/2}\sup _{L(t)}|A| \leq \limsup _{t\uparrow T}(T-t)^{K/2}\sup _{L(t)}|A|<\infty . \]

The unique tangent flow is a stationary multiplicity-one pair of transverse special Lagrangian planes. In contrast, the finer rescaling satisfies

\[ (T-t)^{-K/2}L(t)\longrightarrow aL \quad \hbox{smoothly on compact sets},\qquad a>0, \]

where $L$ is the rotational Lawlor neck. Up to a fixed unitary rotation, that limiting neck has the exact parametrization

\[ X(u,\theta )=(\sin (n\theta ))^{-1/n}e^{i\theta }u, \quad u\in S^{n-1},\quad 0<\theta <\pi /n. \]

This is a formula for the blow-up limit, not for $L(t)$. The actual flow is constructed by solving a nonlinear parabolic problem near a modulated neck of scale comparable to $(T-t)^{K/2}$, and has no closed formula. The initial Lagrangian angle oscillation can be arbitrarily small. These existence and convergence statements do not assert dynamical stability, uniqueness of the initial data, or a numerical entropy value. The illustration uses the exact Lawlor profile above with a prescribed scale to show the near-singularity dynamics. With a reference time $t_0<T$, write

\[ \tau =\frac{t-t_0}{T-t_0},\qquad q=1-\tau ,\qquad a_{\mathrm{model}}(\tau )=a_0q^{K/2}. \]

Here $a_0$ and the equality defining $a_{\mathrm{model}}$ are choices for the drawing, not an exact formula for the modulation parameter of the flow. The actual profile includes a correction obtained by solving the parabolic equation; see [StolarskiSu2026], Theorem 1.1(3) and Sections 2.2–2.3. A shrinking family of these minimal necks alone does not solve mean curvature flow.

The time comparison keeps the same spatial scale. Under parabolic magnification by $q^{-1/2}$ the model neck has scale $a_0q^{(K-1)/2}\to 0$; under the finer magnification by $q^{-K/2}$ its scale remains $a_0$. These illustrate the distinct tangent-plane and smooth-neck limits. The surface view restricts the spherical orbit to $u=(\cos \phi ,\sin \phi ,0,\ldots ,0)$ and projects this two-dimensional slice from $\mathbb C^2$ into $\mathbb R^3$. Projection may create apparent intersections. The planar view shows the exact profiles and their limiting rays. None of these drawings is a numerical time integration or a continuation through $T$. The compact immersed variant mentioned in the source's Remark 3 is outside this entry.

Three times of a shrinking Lawlor-neck model and their planar profiles, on identical spatial scales. The purple waist collapses; the finer neck zoom preserves its shape. Asymptotic illustration, not computed flow slices.
Asymptotic time illustration. Exact Lawlor-neck geometry with the illustrative scale a/a₀ = (1 − τ)^(K/2), where τ = (t − t₀)/(T − t₀). Compare three times, play the collapse, or switch between physical, parabolic, and neck zooms. The surface is a projected 2D slice of the n-manifold in Cⁿ; apparent crossings are projection artifacts. The PDE correction of the actual flow is not computed. Open drawing · Numerical references and credits

Three shrinking neck profiles approach four rays under parabolic rescaling; under neck rescaling the three profiles coincide.
Two rescalings of the asymptotic model. At the parabolic scale the neck disappears into the tangent pair of planes. At the finer Type II scale a smooth Lawlor neck remains. The profiles describe the limiting geometry; the time-dependent family itself is not a mean curvature flow. Open drawing · Numerical references and credits
Ambient space, topology, embeddedness and entropy

Full Euclidean ambient dimension is 2n or 2m, at least six. Auxiliary slices and curves do not determine the genus of this higher-dimensional example.

SO(n)-invariant Type II flows: n = 3, K = 2,3,4; n ≥ 4, integer K ≥ 2 · Euclidean dimension ≥ 5 · Embedded
Ambient space
Euclidean dimension ≥ 5
Topology
Sⁿ⁻¹ × R at every regular time
Embeddedness
Embedded: Embedded for this explicitly restricted case; no claim about other weights, groups, or singular limits.
Compactness
Noncompact
Surface genus
Intrinsic dimension at least three; surface genus is inapplicable.
Topological ends
Two ends; the singular pair of planes is only a limit.
Entropy
Not recorded: No value or bound recorded from the checked source; entropy is not inferred from a single centered Gaussian integral.
Gaussian area
Not recorded: No value or bound recorded from the checked source; entropy is not inferred from a single centered Gaussian integral.

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References on this page

  1. [StolarskiSu2026] Maxwell Stolarski and Wei-Bo Su. Finite-time singularities of lagrangian mean curvature flow with precise dynamics. 2026. Reviewed edition 2607.03152v1; selected statements checked. URL: https://arxiv.org/abs/2607.03152v1.All uses in this project ↩ 1 ↩ 2
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