MCF-stack: Special Solutions of Mean Curvature Flow

§ 5.27 Immortal annuli converging to a double plane00966387D6EF

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. Chen–Sun — 2023-12-29 (first arXiv submission).
00966387D6EF
A50018836027

(Chen–Sun's noncompact multiplicity-two limit) Chen and Sun construct a nonstatic immortal mean curvature flow $M_ t\subset \mathbb R^3$, $t\in [0,\infty )$, smooth for $t>0$, of connected embedded rotational surfaces. It is also symmetric across the limiting plane. Each positive-time slice consists of two increasing radial graphs joined smoothly at a neck of radius $a(t)>0$:

\[ M_ t=\{ (\rho \cos \theta ,\rho \sin \theta ,\pm f(\rho ,t)): \rho \geq a(t),\ \theta \in \mathbb R/2\pi \mathbb Z\} . \]

This is a description of the constructed PDE solution, not an explicit formula for $f$. Away from the neck its upper graph satisfies

\[ f_ t=\frac{f_{\rho \rho }}{1+f_\rho ^2}+\frac{f_\rho }{\rho }. \]

Near the neck, the inverse graph $\rho =u(z,t)$ instead satisfies

\[ u_ t=\frac{u_{zz}}{1+u_ z^2}-\frac1u,\qquad u_ z(0,t)=0. \]

The proof selects a critical initial surface by interpolation among rotational surfaces in finite cylinders and then takes a limit as the cylinder radius tends to infinity. Its initial trace is locally Lipschitz; choosing any positive time as the initial time gives an entirely smooth immortal flow. No closed formula, canonical parameter value, or uniqueness of the constructed critical flow is supplied.

The two graphical ends and the smooth connecting neck give topology $S^1\times \mathbb R$: genus zero and two ends. Properness follows from the graphs escaping to infinite radius; the smooth connected surface is complete. It remains in a fixed slab, and the two ends stay asymptotic to their original parallel planes. The neck radius tends to zero only as $t\to \infty $. On compact regions the flow converges to the middle plane with multiplicity two; the two sheets converge smoothly away from the origin. The convergence is not uniform over the entire noncompact surface. This is an infinite-time limit of the unrescaled flow, not a multiplicity-two finite-time tangent flow of a closed embedded surface.

Source: [CS26R3], Theorems 1.1 and 4.13, Section 1.1, equation (2.5), Proposition 4.6 and Remark 4.7, and Appendix A, arXiv v3 PDF pp. 1, 3, 6, 23–24, and 27–28. The topology and properness statements above are direct consequences of the source's two-graph parametrization. The paper gives no exact entropy or fixed-center Gaussian-area value for this constructed family. The interactive picture compares ordered finite-time stages and a separate infinite-time limit at a fixed viewing scale. A cutaway exposes the connecting neck; the compressed radial profile records the fixed far-end heights. The illustrated shapes and stage spacing are schematic, not computed critical PDE time slices or an asserted convergence rate.

Three schematic finite-time annuli with decreasing positive neck radii and locally flattening sheets, followed by one limiting plane counted twice. Far ends retain their asymptotic planes.
Schematic time sequence. Rotate the annulus, use the cutaway, and compare earlier, intermediate, and later finite stages with the separate limit t → ∞. The viewing scale is fixed. The compressed radial profile shows the fixed asymptotic heights ±h. Shapes and stage spacing are illustrative, not computed PDE time slices or a convergence rate. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy

Smooth for positive time; the initial trace in the construction is locally Lipschitz. The double-plane limit occurs locally as t→∞.

Annulus S¹ × ℝ · Euclidean space ℝ³ · Embedded
Ambient space
Euclidean space ℝ³
Topology
Annulus S¹ × ℝ
Embeddedness
Embedded: Every smooth positive-time slice is embedded. The multiplicity-two plane is an infinite-time limit, not a regular slice.
Compactness
Noncompact
Surface genus
0
Topological ends
2 graphical ends
Entropy
Not recorded: Not recorded in this entry.
Gaussian area
Not recorded: Not recorded for the stated center and scale.

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References on this page

  1. [CS26R3] Jingwen Chen and Ao Sun. Mean curvature flow with multiplicity 2 convergence in ℝ³. Analysis & PDE, 19:1029–1060, 2026. Source locators use arXiv v3, August 20, 2025, 30 pages, rather than the final journal pagination. URL: https://arxiv.org/abs/2312.17457v3, doi:10.2140/apde.2026.19.1029.All uses in this project ↩ 1
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