MCF-stack: Special Solutions of Mean Curvature Flow

§ 2.41 Higher-codimension spherical shrinkers (Smoczyk, 2005; Sun–Zhu, 2020)6C99F82AA214

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. Smoczyk — 2005-07-15 (first arXiv submission).
  2. Sun–Zhu — 2020-11-03 (first arXiv submission).
6C99F82AA214

Minimal submanifolds of a round sphere give Euclidean self-shrinkers. This correspondence, and a converse with hypotheses, appear in [Smoczyk2005], p. 2985 and Theorem 1.1. We use the $\vec H=-X^\perp /2$ scaling.

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Let $n\geq 2$, $N\geq n+2$, and let $F:M^ n\to S^{N-1}(\sqrt{2n})$ be a smooth compact minimal immersion. Viewed in $\mathbb R^ N$, it satisfies

\[ \vec H=-\frac12F, \qquad F_ t=\sqrt{-t}\, F,\quad -\infty <t<0. \]

An explicit family is obtained for integers $p,q\geq 1$, $n=p+q$, by

\[ F(u,v)=(\sqrt{2p}\, u,\sqrt{2q}\, v), \quad (u,v)\in S^ p(1)\times S^ q(1), \quad N=p+q+2. \]

To avoid repeating the Clifford torus entry, one may take $p+q\geq 3$ in this explicit family.

  • For a general compact minimal immersion, the resulting shrinker is complete and proper; it is embedded when the original immersion is embedded. Topology, symmetry and further curvature properties depend on the chosen minimal submanifold and are not fixed by this construction.

  • Its mean-curvature vector has constant length $\sqrt{n/2}$, and the principal normal $\vec H/|\vec H|$ is parallel in the normal bundle. Conversely, a compact self-shrinker of dimension $n\geq 2$ is spherical if and only if its mean-curvature vector is nowhere zero and its principal normal is parallel (Smoczyk's Theorem 1.1).

  • The displayed product family is compact embedded of Euclidean codimension two, with topology $S^ p\times S^ q$ and $O(p+1)\times O(q+1)$ symmetry. It has $|A|^2=1$ and is minimal inside $S^{n+1}(\sqrt{2n})$; these identities follow directly from its two round factors.

  • For these explicit products, Gaussian factorization gives $\lambda (S^ p(\sqrt{2p})\times S^ q(\sqrt{2q}))=\lambda (S^ p(\sqrt{2p}))\lambda (S^ q(\sqrt{2q}))$. For a general compact minimal immersion $M^ n\subset S^{N-1}(\sqrt{2n})$, $F_{0,1}(M)=(4\pi )^{-n/2}e^{-n/2}\operatorname {Area}(M)$ by the constant radius.

  • The explicit spherical products are locally rigid: for fixed $p,q\geq 1$ and $0<\alpha <1$, a normalized shrinker in $\mathbb R^ N$, $N\geq p+q+2$, sufficiently close in $C^{2,\alpha }$ to the product is an ambient rotation of it. This is [SunZhu2025], Theorems 1.1–1.2, PDF p. 1. It is a local statement with center and scale fixed, not global uniqueness.

The formula determines the product geometry exactly. The figure is a factor diagram with the full ambient dimension stated.

Higher-codimension spherical shrinkers. The spherical product formula is exact. This shows its factors and ambient dimensions rather than an embedding of a higher-dimensional product in R3.
Dimension and factor diagram. The spherical product formula is exact. This shows its factors and ambient dimensions rather than an embedding of a higher-dimensional product in R3. Open drawing · Numerical references and credits
Ambient space, topology, embeddedness and entropy

Topology depends on the chosen spherical minimal immersion; the explicit product subfamily has topology Sᵖ × Sᑫ.

Surface inputs · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Depends on the construction
Ambient space
Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology
Compact minimal surface in a sphere
Embeddedness
Depends on the construction: Embedded exactly when the chosen minimal immersion into the sphere is embedded. The explicit products of round spheres are embedded; general immersed inputs need not be.
Compactness
Compact
Surface genus
Depends on parameters
Topological ends
0 (compact)
Entropy
Exact formula: λ = (4π)^(−n/2) e^(−n/2) Area(M). Spherical products have λ = λ(Sᵖ) λ(Sᑫ).
Gaussian area
Exact formula: F₀,₁ = λ for this normalized shrinker. λ = (4π)^(−n/2) e^(−n/2) Area(M). Spherical products have λ = λ(Sᵖ) λ(Sᑫ).
Higher-dimensional inputs · Euclidean dimension ≥ 5 · Depends on the construction
Ambient space
Euclidean dimension ≥ 5
Topology
Compact minimal n-manifold in a sphere; n ≥ 3
Embeddedness
Depends on the construction: Embedded exactly when the chosen minimal immersion into the sphere is embedded. The explicit products of round spheres are embedded; general immersed inputs need not be.
Compactness
Compact
Surface genus
Not a surface
Topological ends
0 (compact)
Entropy
Exact formula: λ = (4π)^(−n/2) e^(−n/2) Area(M). Spherical products have λ = λ(Sᵖ) λ(Sᑫ).
Gaussian area
Exact formula: F₀,₁ = λ for this normalized shrinker. λ = (4π)^(−n/2) e^(−n/2) Area(M). Spherical products have λ = λ(Sᵖ) λ(Sᑫ).

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References on this page

  1. [Smoczyk2005] Knut Smoczyk. Self-shrinkers of the mean curvature flow in arbitrary codimension. International Mathematics Research Notices, 2005(48):2983–3004, 2005. URL: https://arxiv.org/abs/math/0507325, doi:10.1155/IMRN.2005.2983.All uses in this project ↩ 1
  2. [SunZhu2025] Ao Sun and Jonathan J. Zhu. Rigidity and Łojasiewicz inequalities for Clifford self-shrinkers. Calculus of Variations and Partial Differential Equations, 64:163, 2025. Article 163; source locators refer to arXiv version 1, 3 November 2020, 31 pages. URL: https://arxiv.org/abs/2011.01636v1, doi:10.1007/s00526-025-03037-y.All uses in this project ↩ 1
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