MCF-stack: Special Solutions of Mean Curvature Flow

§ 3.26 Su–Wood paired-profile translators31FE1060BFF7

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).
31FE1060BFF7
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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 3 uses a shrinker and an expander profile and gives immersed $SO(m-1)$- or $T^{m-2}$-invariant translators. They are unbounded in $\operatorname {Im}z_ m$ and are not almost calibrated. Full embeddedness and properness are not asserted uniformly here. Non-spherical orbit choices must retain the smoothness restrictions of Section 5.4.

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 paired-profile translators. 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 · Not recorded
Ambient space
Euclidean dimension ≥ 5
Topology
G-orbits of a two-dimensional profile; quotient identifications as in source
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.
T^(m−2) case, m ≥ 3 · Euclidean dimension ≥ 5 · Not recorded
Ambient space
Euclidean dimension ≥ 5
Topology
Torus-orbit profile construction
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. [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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