MCF-stack: Special Solutions of Mean Curvature Flow

§ 5.11 Ancient curves with mixed boundary conditions4450DF902745

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. Langford–Liu–McNamara — 2024-04-01 (first arXiv submission).
4450DF902745
FB8BC9B105B9

In the unit disk fix $o=(-d,0)$, with $0<d\le 1$. Langford–Liu–McNamara construct ancient curve-shortening flows with one endpoint fixed at $o$ (Dirichlet condition) and the other moving on the unit circle with orthogonal contact (Neumann condition). For $0<d<1$, the distinguished flow is eternal and connects the longer unstable stationary segment backward to the shorter stable stationary segment forward. At $d=1$ the forward endpoint is a finite-time singularity with a half-grim-reaper Type II model. The regular curves are embedded compact intervals. Their endpoints obey different conditions, so the construction is neither the two-free-endpoint ball flow nor a homothetic shrinker. The figure marks the fixed endpoint, moving endpoint and boundary normal at each stage; it separates the interior-point eternal case from the boundary-point singular case. Intermediate arcs are schematic. The half-grim reaper is a rescaled limit, not a formula for the whole curve or for continuation through the singular time.

Sources: [LangfordLiuMcNamara2025].

Ancient curves with mixed boundary conditions. Fixed Dirichlet endpoint and moving orthogonal Neumann endpoint, with the two stationary segments distinguished. Stages are schematic; the fixed disk is never rescaled.
Source-informed schematic. Fixed Dirichlet endpoint and moving orthogonal Neumann endpoint, with the two stationary segments distinguished. Stages are schematic; the fixed disk is never rescaled. 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.

0 < d < 1, eternal case · Euclidean plane ℝ² · Embedded
Ambient space
Euclidean plane ℝ²
Topology
Interval with one fixed and one free endpoint
Embeddedness
Embedded: The cited construction gives an embedding for this case.
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.
d = 1, finite-time endpoint singularity · Euclidean plane ℝ² · Embedded
Ambient space
Euclidean plane ℝ²
Topology
Interval before the singular time
Embeddedness
Embedded: The cited construction gives an embedding for this case.
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.

Compare examples by these properties →

References on this page

  1. [LangfordLiuMcNamara2025] Mat Langford, Yuxing Liu, and George McNamara. Ancient curve shortening flow in the disc with mixed boundary condition. Arnold Mathematical Journal, 011:93, 2025. URL: https://arxiv.org/abs/2404.01525, arXiv:2404.01525, doi:10.56994/armj.011.003.004.All uses in this project ↩ 1
Cite this page

Comments

Comments are not enabled yet.