MCF-stack: Special Solutions of Mean Curvature Flow

§ 3.25 Su–Wood Lagrangian translating helicoids405ACD56C839

Earliest recorded reference: (first arXiv submission).

Paper credits and chronology

Paper credits are chronological and include every author of the cited papers. They identify sources, not exclusive discovery credit. Background-only citations do not determine the date of this example. A classical example may predate the references recorded here.

  1. Su–Wood — 2025-07-02 (first arXiv submission).
405ACD56C839
6386C95CB0EA

Use the Su–Wood Ansatz with $a=1$ and an admissible compact group $G\subset SU(m-1)$ acting on $\mathbb C^{m-1}$ with isotropic orbits of dimension $m-2$. If $\gamma ,\xi $ are the regular (with the smooth extension at a line profile's origin understood) profile curves satisfying equation (1) of the source, its profile map is

\[ \check F(x,y)=\left(\gamma (x)\xi (y),\; \int ^ y \xi '(v)\overline{\xi (v)}\, dv- \int ^ x\gamma '(u)\overline{\gamma (u)}\, du\right). \]

The full Lagrangian is the $G$-orbit of the specified linear inclusion of $\check F$ into $\mathbb C^ m$ and translates in $\operatorname {Re}z_ m$. The source uses opposite profile soliton constants for the two factors. It is the induced orbit geometry, not ordinary planar curve shortening, that gives the Anciaux profile equation.

Example 1 takes a line and a nonconstant closed Anciaux shrinker profile, with $G=SO(m-1)$. The resulting image is a smooth embedded translating helicoid, periodic in $\operatorname {Im}z_ m$; the nonconstant profile distinguishes it from the already recorded constant-radius rotating translator. For other groups the analogous construction can have a one-dimensional singular set and is not included in this smooth case.

The associated motion is translation for every real time. The plotted object is the two-dimensional profile projection used in Figures 1–2, not the full $m$-dimensional Lagrangian. The source explicitly does not claim new complete smooth embedded almost-calibrated translators from this Ansatz.

Sources: [SuWood2026].

Su--Wood Lagrangian translating helicoids. Numerical Anciaux profiles in the exact Su–Wood orbit Ansatz; a projection of the two-dimensional profile is shown at three translation times. The hidden orbit directions and uncomputed global identifications are stated explicitly.
Numerical ODE profile projection. Numerical Anciaux profiles in the exact Su–Wood orbit Ansatz; a projection of the two-dimensional profile is shown at three translation times. The hidden orbit directions and uncomputed global identifications are stated explicitly. 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.

SO(m−1) case, m ≥ 3 · Euclidean dimension ≥ 5 · Embedded
Ambient space
Euclidean dimension ≥ 5
Topology
G-orbits of a two-dimensional profile; quotient identifications as in source
Embeddedness
Embedded: The cited construction gives an embedding for this case.
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. [SuWood2026] Wei-Bo Su and Albert Wood. Lagrangian translating solitons and special lagrangians in cm with symmetries. 2026. Reviewed edition 2507.01345v1; selected statements checked. URL: https://arxiv.org/abs/2507.01345v1.All uses in this project ↩ 1
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