MCF-stack: Special Solutions of Mean Curvature Flow

§ 5.47 Immortal curve shortening with nonunique geodesic limits9C5BCCFA4320

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. Aryan–Lee — 2026-08-05 (first arXiv submission).
9C5BCCFA4320 The ambient space here is a Riemannian two-sphere, not Euclidean space.
CC2635F97DC0

(A circle of limiting geodesics) For every $C^\infty $ neighborhood of the round metric on $\mathbb S^2$, [AryanLee2026], Theorem 5.1, constructs a smooth antipodally invariant metric $g$ in that neighborhood and a smooth simple closed curve whose curve-shortening flow

\[ \partial _ t\gamma =\kappa _ gN_ g,\qquad t\in [0,\infty ), \]

stays embedded for all time. There is an embedded circle $Z\subset \mathbb {RP}^2$ and a smooth injective family $\Gamma :Z\to \mathcal E^\infty (\mathbb S^2)$ of simple closed $g$-geodesics, all of the same length, such that

\[ \omega _{C^\infty }(\gamma )=\{ \Gamma (z):z\in Z\} . \]

Convergence is convergence of unparametrized embedded curves, or smooth convergence after reparametrization. In particular there is no unique limiting geodesic.

Each regular slice is a compact complete proper embedded circle, of dimension one and codimension one in $(\mathbb S^2,g)$, with no ends. The construction uses a prescribed length function and a spiraling slow mode, not a closed time-dependent formula. It does not assert rotational symmetry or stability of the evolving curve. The diagram is a schematic in parameter space; it is not a curve-shortening simulation on the round sphere. Euclidean Gaussian area and entropy are inapplicable to this intrinsic ambient flow.

The spiral represents a slow parameter approaching a circle of critical points, each indexing a different limiting geodesic. This is not a simulation on a round sphere.
Parameter-space schematic. The spiral represents a slow parameter approaching a circle of critical points, each indexing a different limiting geodesic. This is not a simulation on a round sphere. 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.

Immortal closed curve in (S²,g) · Riemannian ambient manifold · Embedded
Ambient space
Riemannian ambient manifold
Topology
Circle S¹ at every finite time
Embeddedness
Embedded: The stated regular submanifold is embedded in its full ambient space.
Compactness
Compact
Surface genus
Curve, not surface.
Topological ends
No ends (compact).
Entropy
Not applicable: Intrinsic mean curvature flow in a Riemannian ambient manifold; Euclidean Gaussian area and entropy are not assigned.
Gaussian area
Not applicable: Intrinsic mean curvature flow in a Riemannian ambient manifold; Euclidean Gaussian area and entropy are not assigned.

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References on this page

  1. [AryanLee2026] Shrey Aryan and Tang-Kai Lee. Non-uniqueness of geodesic limits and a question of grayson and gage. 2026. Reviewed arXiv edition 2608.04855v1. Theorem 1.1 and Theorem 5.1, Section 5 proof; exact omega-limit set, smooth reparametrized convergence and antipodally invariant nonround metrics. URL: https://arxiv.org/abs/2608.04855v1.All uses in this project ↩ 1
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