MCF-stack: Special Solutions of Mean Curvature Flow

§ 2.42 An embedded self-shrinking Möbius bundle (Lee–Mu-Tao Wang, 2007; Braxton–Lee–Zhu, 2026)5D9025FCD9CD

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. Lee–Mu-Tao Wang — 2007-07-02 (first arXiv submission).
  2. Braxton–Lee–Zhu — 2026-09-17 (first arXiv submission).
5D9025FCD9CD This example is a surface in $\mathbb C^2=\mathbb R^4$, of codimension two.
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(An embedded nonorientable shrinker) Let $M=(\mathbb S^1\times \mathbb R)/((\theta ,r)\sim (\theta +\pi ,-r))$ and set

\[ X([\theta ,r])=(e^{2i\theta }\cosh r,\sqrt2 e^{-i\theta }\sinh r). \]

This is an exact parametrization with $\vec H=-X^\perp /2$; its ancient flow is $\sqrt{-t}\, X(M)$, $t<0$. Lee–Wang constructed the surface, up to a unitary change of coordinates, in [LeeWang2009], Section 2.2. The quotient map is an embedding; the map on $\mathbb S^1\times \mathbb R$ is a double cover, not an embedding.

The surface is Lagrangian, Hamiltonian stationary, noncompact, and asymptotic to a Schoen–Wolfson $(2,1)$ cone. Its topology is the open Mobius line bundle, with one end and one crosscap. Orientable surface genus does not apply. The identity $|X|^2=1+3\sinh ^2r$ proves properness, and the induced metric

\[ g=(\sinh ^2r+2\cosh ^2r)(2\, d\theta ^2+dr^2) \]

proves completeness. The diagonal circle action has weights $(2,-1)$.

For this single-covered embedding, [BraxtonLeeZhu2026], Theorem 1.2 and Section 7, prove $F$-stability under arbitrary normal variations modulo translations and dilation. This does not establish nonlinear stability or realization as a generic singularity. Remark 7.4 gives

\[ F_{0,1}(M)=\lambda (M)=\frac{\sqrt2 e^{1/8}}8 \left[K_0(3/8)+3K_1(3/8)\right],\qquad K_ j(z)=\int _0^\infty e^{-z\cosh v}\cosh (jv)\, dv. \]

The illustration projects the exact four-dimensional parametrization; projected crossings are not self-intersections of $M$.

A finite patch of the exact embedded Möbius quotient in C². Rotation changes the projection; projected crossings do not imply self-intersection.
Exact four-dimensional projection. A finite patch of the exact embedded Möbius quotient in C². Rotation changes the projection; projected crossings do not imply self-intersection. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy

Dimensions, topology, and embeddedness refer to the stated full ambient geometry and regular slices. See the source hypotheses and illustration conventions on this page.

Single-covered Möbius bundle in C² · Euclidean space ℝ⁴ · Embedded
Ambient space
Euclidean space ℝ⁴
Topology
Open Möbius line bundle; one crosscap
Embeddedness
Embedded: Embedded after (θ + π, −r) ~ (θ, r). The cylindrical parametrization before quotient is a nonembedding double cover.
Compactness
Noncompact
Surface genus
Nonorientable surface with one crosscap; orientable genus is inapplicable.
Topological ends
One end; the two r-directions are identified by the quotient.
Entropy
Exact formula: λ = (√2 e^(1/8)/8)[K₀(3/8) + 3K₁(3/8)], for the embedded quotient counted once. Kⱼ is the modified Bessel function of the second kind. The formula is exact; no rounded scalar is used in proved-bound filtering.
Gaussian area
Exact formula: F₀,₁ = λ for this normalized proper asymptotically conical shrinker. λ = (√2 e^(1/8)/8)[K₀(3/8) + 3K₁(3/8)], for the embedded quotient counted once. Kⱼ is the modified Bessel function of the second kind. The formula is exact; no rounded scalar is used in proved-bound filtering.

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References on this page

  1. [LeeWang2009] Yng-Ing Lee and Mu-Tao Wang. Hamiltonian stationary shrinkers and expanders for lagrangian mean curvature flows. 2009. Reviewed arXiv edition 0707.0239v1. Section 2.2, equations for S and its metric; Proposition 2.2. Unit convention checked against the normalized Möbius map. URL: https://arxiv.org/abs/0707.0239v1.All uses in this project ↩ 1
  2. [BraxtonLeeZhu2026] Kahnrad Braxton, Tang-Kai Lee, and Jonathan J. Zhu. A stable self-shrinking mobius bundle in r4. 2026. Reviewed arXiv edition 2609.19648v1. Theorem 1.2; Section 2.1, quotient (2.2), metric and shrinker identity; Section 7.2 and Remarks 7.3–7.6, including exact entropy. No genericity theorem inferred. URL: https://arxiv.org/abs/2609.19648v1.All uses in this project ↩ 1
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