MCF-stack: Special Solutions of Mean Curvature Flow

§ 5.36 Lagrangian singularities with prescribed orbit modelsA3EE122FF71E

Earliest recorded reference: (first arXiv submission).

Paper credits and chronology

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  1. Madnick–Wood — 2022-08-02 (first arXiv submission).
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Let $G\subset SU(n)$ be an admissible compact connected group, and let $L_\infty \subset \mu ^{-1}(0)\subset \mathbb C^ n$ be a complete connected cohomogeneity-one special Lagrangian. Here each orbit has dimension $n-1$ and the same isotropy type. Madnick–Wood construct an almost-calibrated Lagrangian MCF developing a Type II singularity at the origin for which every Type II blow-up is $L_\infty $ and every Type I blow-up is its asymptotic cone. The profile flow in a complex line is

\[ (\partial _ t\gamma )^\perp =\mathbf k-(n-1)\frac{\gamma ^\perp }{|\gamma |^2}. \]

It is not ordinary planar curve shortening. Admissible non-$SO(n)$ groups, for example the diagonal $T^{n-1}$ action, give models beyond the currently listed rotational Lawlor neck. The theorem concerns the evolving singularity-forming flow; its stationary special Lagrangian limit is not counted as an additional moving soliton. Smoothness is restricted to the source's constant-isotropy stratum. We do not infer full embeddedness or topology solely from a projected profile. The illustration uses an exact model profile with a schematic neck scale, clearly distinguished from the nonlinear flow constructed in the proof.

Sources: [MadnickWood2025].

Lagrangian singularities with prescribed orbit models. Exact special-Lagrangian profile shape with a schematic decreasing neck scale. Type I cone and Type II neck views are separate; the scaling sequence alone is not a solution of Lagrangian MCF.
Source-informed schematic. Exact special-Lagrangian profile shape with a schematic decreasing neck scale. Type I cone and Type II neck views are separate; the scaling sequence alone is not a solution of Lagrangian MCF. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Admissible non-spherical orbits, n ≥ 3 · Euclidean dimension ≥ 5 · Not recorded
Ambient space
Euclidean dimension ≥ 5
Topology
Cohomogeneity-one G-orbit flow, with constant isotropy type
Embeddedness
Not recorded: The reviewed statement gives an immersion; no general embedding assertion is recorded.
Compactness
Noncompact
Surface genus
Not a smooth surface; surface genus is inapplicable.
Topological ends
Not recorded for the full parameter manifold.
Entropy
Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area
Not recorded: No quantitative conclusion is recorded from the cited construction.

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References on this page

  1. [MadnickWood2025] Jesse Madnick and Albert Wood. Cohomogeneity-One Lagrangian Mean Curvature Flow. Mathematische Annalen, 391:4325–4388, 2025. URL: https://arxiv.org/abs/2208.01574, arXiv:2208.01574, doi:10.1007/s00208-024-03005-5.All uses in this project ↩ 1
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