MCF-stack: Special Solutions of Mean Curvature Flow

§ 2.45 The Clifford shrinking torus in four-spaceCDFAEA3DDEF8

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. Evans–Lotay–Schulze — 2018-02-05 (first arXiv submission).
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We follow the supplied June 2020 version of [ELS2020]; its Sections 2, 4 and 5 distinguish exact geometry, Hamiltonian stability, flow instability and local rigidity.

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The exact normalized immersion is

\[ X(\theta _1,\theta _2)=\sqrt2 (\cos \theta _1,\sin \theta _1,\cos \theta _2,\sin \theta _2), \qquad (\theta _1,\theta _2)\in (\mathbb R/2\pi \mathbb Z)^2. \]

It lies in $\mathbb R^4=\mathbb C^2$, and $X_ t=\sqrt{-t}\, X$ gives a smooth flow for $t<0$.

  • This is a compact embedded Lagrangian torus of dimension two and codimension two, complete and proper. It is minimal in $S^3(2)$ and has the product rotation symmetry $U(1)\times U(1)$.

  • The induced metric is $2(d\theta _1^2+d\theta _2^2)$; it is intrinsically flat, has area $8\pi ^2$, $|\vec H|=1$, and $\vec H=-X/2$. Its entropy is exactly $2\pi /e$ ([ELS2020], Section 2 and Lemma 2.4, PDF pp. 3–6).

  • It is $F$-unstable under Lagrangian variations but $F$-stable under Hamiltonian variations after accounting for translations. It is locally volume minimizing under Hamiltonian variations (Lemmas 4.1–4.2 and Theorem 4.3, PDF p. 13).

  • Nevertheless it is not a local entropy minimum, even within Hamiltonian variations. For every $\varepsilon >0$ and $k\in \mathbb N$ there is a compact embedded Lagrangian torus Hamiltonian isotopic and $\varepsilon $-close in $C^ k$ whose first finite-time singularity has a blow-up outside the $U(2)$-orbit of the Clifford torus (Corollary 4.8 and Theorem 4.9, PDF p. 16).

  • A compact embedded two-dimensional self-shrinker in $\mathbb C^2$ sufficiently $C^{2,\alpha }$-close, for $0<\alpha <1$, is the Clifford torus up to translation, dilation and rotation (Theorem 5.7, PDF p. 29). This is a local rigidity statement.

The interactive picture rotates the second and fourth coordinates in $\mathbb R^4$ and then projects onto the first three coordinates. Apparent intersections in a projection do not mean that the original torus is immersed nonembedded.

The Clifford shrinking torus in four-space. A linear projection of the Clifford torus. The 4D rotation control changes the projection by rotating the second and fourth coordinates. Projected overlaps are not self-intersections of the embedded torus in R4.
Exact R4 geometry, projected to R3. A linear projection of the Clifford torus. The 4D rotation control changes the projection by rotating the second and fourth coordinates. Projected overlaps are not self-intersections of the embedded torus in R4. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy
Torus S¹ × S¹ · Euclidean space ℝ⁴ · Embedded
Ambient space
Euclidean space ℝ⁴
Topology
Torus S¹ × S¹
Embeddedness
Embedded: The product of two singly covered circles is embedded in ℝ⁴. Crossings in a three-dimensional projection are not self-intersections of this torus.
Compactness
Compact
Surface genus
1
Topological ends
0 (compact)
Entropy
Exact formula: λ = 2π/e ≈ 2.31145.
Gaussian area
Exact formula: F₀,₁ = λ for the normalized shrinker. λ = 2π/e ≈ 2.31145.

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References on this page

  1. [ELS2020] Christopher G. Evans, Jason D. Lotay, and Felix Schulze. Remarks on the self-shrinking Clifford torus. 2020. Version 3, 10 June 2020; theorem and page locators refer to this supplied version. URL: https://arxiv.org/abs/1802.01423v3, arXiv:1802.01423.All uses in this project ↩ 1 ↩ 2
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