MCF-stack: Special Solutions of Mean Curvature Flow

§ 5.31 Mean curvature evolution of the Whitney sphere2E15631C1593

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. Viana — 2018-02-06 (first arXiv submission).
2E15631C1593
245DFF2BED27

(A collapsing Lagrangian immersed sphere) For $n\geq 2$, the initial Whitney immersion in $\mathbb R^{2n}=\mathbb C^ n$ is explicitly

\[ F_0(x_1,\ldots ,x_{n+1}) =\frac{(x_1,x_1x_{n+1},\ldots ,x_ n,x_ nx_{n+1})}{1+x_{n+1}^2}, \qquad (x_1,\ldots ,x_{n+1})\in S^ n. \]

It is a compact Lagrangian immersion, of dimension and codimension $n$, with one transverse double image point at the origin coming from the two poles. It is complete and proper as an immersion of a compact domain, and it is not embedded. Its equivariance reduces the geometry to the planar initial profile

\[ \gamma _0(u)=\left(\frac{\sin u}{1+\cos ^2u}, \frac{\sin u\cos u}{1+\cos ^2u}\right), \qquad u\in \mathbb R/2\pi \mathbb Z. \]

For $n\geq 2$ the profile evolution is the equivariant equation

\[ (\partial _ t\gamma )^\perp =\vec k-(n-1)\frac{\gamma ^\perp }{|\gamma |^2}, \]

interpreted by the smooth equivariant immersion at the double point. It is not ordinary planar curve shortening.

The maximal smooth mean curvature flow $F_ t$, $t\in [0,T)$, remains Lagrangian and equivariant. It collapses to the origin in finite time, and its tangent flow there is a Lagrangian plane with multiplicity two. The Whitney sphere is not a self-similar solution. These statements are the special case stated in [Viana18], pp. 1–3, especially the result on p. 2 and Theorem 1.3 on p. 3. The formulas above specify the initial geometry and its reduced equation, not a closed expression for $F_ t$ at positive time. The illustration specializes to $n=2$. Write $\gamma _ t=a+ib$. A single right-hand lobe, from one preimage of the origin to the other, generates the immersed two-sphere by

\[ F_ t(u,v)=(\gamma _ t(u)\cos v,\gamma _ t(u)\sin v)\in \mathbb C^2, \qquad v\in \mathbb R/2\pi \mathbb Z. \]

The circles at the two endpoints collapse to the two distinct poles of the domain, which have the same image at the origin. Using both antipodal lobes would cover the same sphere twice.

The time sequence pairs this surface, linearly projected from $\mathbb R^4$ to $\mathbb R^3$, with its planar profile. At $t=0$ the geometry is exact; the positive-time frames are numerical approximations of the equivariant PDE for $n=2$, with $\partial _ tF=\vec H$ and the initial scale fixed above. They show $0\leq t\leq 0.09$ on a common spatial scale, with no normalization by an estimated extinction time. The computation stops before the singularity: $0.09$ is not asserted to equal $T$. Spatial mesh refinement and tighter time-integration tolerances are compared, but these checks are not certified error bounds. The profile narrows and contracts rather than shrinking by a single scale factor. Its double point remains at the origin; further overlaps in the surface projection need not be intersections in $\mathbb R^4$. The multiplicity-two tangent plane is a parabolic blow-up limit, not an unscaled final surface displayed by the animation.

Whitney sphere flow at t = 0, 0.03, 0.06 and 0.09. Surface projections and figure-eight profiles contract and flatten on a common spatial scale; positive-time geometry is numerical.
Numerical PDE evolution from exact initial data. Whitney two-sphere in C² (n = 2), with trace mean-curvature speed and the initial scale in the formula above. Four snapshots share a fixed spatial scale. Each pairs a linear R⁴-to-R³ projection with the planar generating profile. Play or select computed times 0 ≤ t ≤ 0.09; this window ends before the singularity and does not specify its time T. Only t = 0 is exact. Additional overlaps may be projection artifacts. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy

Topology refers to the compact domain; the initial immersion has a transverse double point.

Whitney two-sphere · Euclidean space ℝ⁴ · Not embedded
Ambient space
Euclidean space ℝ⁴
Topology
Immersed sphere S²
Embeddedness
Not embedded: The initial Whitney immersion has a transverse double point at the origin. The regular flow remains a Lagrangian sphere and cannot become embedded.
Compactness
Compact
Surface genus
0
Topological ends
0 (compact)
Entropy
Finite; value not recorded: Finite; no value recorded.
Gaussian area
Not recorded: Not recorded in this entry.
Higher-dimensional Whitney spheres · Euclidean dimension ≥ 5 · Not embedded
Ambient space
Euclidean dimension ≥ 5
Topology
Immersed sphere Sⁿ in ℝ²ⁿ, n ≥ 3
Embeddedness
Not embedded: The initial Whitney immersion has a transverse double point at the origin. The regular flow remains a Lagrangian sphere and cannot become embedded.
Compactness
Compact
Surface genus
Not a surface
Topological ends
0 (compact)
Entropy
Finite; value not recorded: Finite; no value recorded.
Gaussian area
Not recorded: Not recorded in this entry.

Compare examples by these properties →

References on this page

  1. [Viana18] Celso Viana. A note on the evolution of the Whitney sphere along mean curvature flow. 2018. Citations use supplied arXiv version 2, December 12, 2018, 13 pages; PDF and printed pages agree. URL: https://arxiv.org/abs/1802.02108v2, arXiv:1802.02108.All uses in this project ↩ 1
Cite this page

Comments

Comments are not enabled yet.