MCF-stack: Special Solutions of Mean Curvature Flow

§ 5.8 Immersed ancient curves spanning a half-planeDCA2FD176411

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. Ma — 2020-03-06 (first arXiv submission).
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Ma constructs a compact immersed ancient curve-shortening flow in $\mathbb R^2$ whose time-slice convex hulls together fill a half-plane. Compactness refers to each individual immersed closed curve, not to a uniform spatial bound over ancient time. The construction joins increasingly complicated grim-reaper configurations and takes an ancient limit; it is not a homothetic curve or an explicit parametrization. The curves can have crossings at early times and become embedded later, then shrink to a round point. Hence embeddedness is time dependent. Smooth compact slices are complete and proper as immersions, have no ends, and have finite Gaussian area and entropy at each time; no uniform ancient-time bound is claimed. The source's sequence of disappearing fingers is a guide to the construction, not a formula for singular events of the final ancient limit. Our diagram separates early immersed stages from the later embedded stage and labels them as qualitative stages rather than numerically computed MCF slices.

Sources: [Ma2021].

Immersed ancient curves spanning a half-plane. Qualitative finger and crossing diagram based on Figure 1. Early, intermediate, and later stages illustrate the construction; no numerical physical times or curvature rates are claimed.
Source-informed schematic. Qualitative finger and crossing diagram based on Figure 1. Early, intermediate, and later stages illustrate the construction; no numerical physical times or curvature rates are claimed. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Compact immersed ancient curves · Euclidean plane ℝ² · Depends on the construction
Ambient space
Euclidean plane ℝ²
Topology
S¹ as parameter manifold
Embeddedness
Depends on the construction: Early slices may self-intersect; the constructed flow becomes embedded later.
Compactness
Compact
Surface genus
Not a smooth surface; surface genus is inapplicable.
Topological ends
No ends (compact parameter manifold).
Entropy
Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area
Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

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References on this page

  1. [Ma2021] John Man Shun Ma. Ancient solutions to the Curve Shortening Flow spanning the halfplane. Transactions of the American Mathematical Society, 374:4207–4226, 2021. URL: https://arxiv.org/abs/2003.03018, arXiv:2003.03018, doi:10.1090/tran/8353.All uses in this project ↩ 1
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