MCF-stack: Special Solutions of Mean Curvature Flow

§ 5.12 Ancient free-boundary curves in convex domains1D94C677A1B5

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. Bourni–Burns–Catron — 2024-04-12 (first arXiv submission).
1D94C677A1B5
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Let $\Omega \subset \mathbb R^2$ be a smooth bounded strictly convex domain, and let $D$ be an orthogonal diameter: a straight segment in $\overline\Omega $ meeting $\partial \Omega $ orthogonally at both endpoints. Bourni–Burns–Catron construct convex ancient free-boundary curve-shortening solutions approaching $D$ backward, one on each side, with the uniqueness qualifications in Theorem 1.1. The unit-disk family is a special case, cross-linked separately. Each regular slice is an embedded compact arc with endpoints moving on the fixed boundary and perpendicular contact there. Its topology is an interval, and it has no noncompact ends. The domain is not scaled with the curve. The construction is by parabolic limits, not an explicit family of elliptical arcs. The visualization uses a strictly convex elliptical domain to distinguish the fixed barrier from the evolving arc, and displays the orthogonal diameter as a backward limit. The intermediate curves are schematic; exact contact and numerical PDE convergence are not inferred from an ellipse drawing.

Sources: [BourniBurnsCatron2025].

Ancient free-boundary curves in convex domains. An orthogonal diameter in a fixed convex domain and schematic arcs on either side. The barrier remains fixed through time; the intermediate curves are not computed ancient solutions.
Source-informed schematic. An orthogonal diameter in a fixed convex domain and schematic arcs on either side. The barrier remains fixed through time; the intermediate curves are not computed ancient solutions. 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.

Planar strictly convex domain · Euclidean plane ℝ² · Embedded
Ambient space
Euclidean plane ℝ²
Topology
Compact interval with free endpoints on the fixed boundary
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.

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References on this page

  1. [BourniBurnsCatron2025] Theodora Bourni, Nathan Burns, and Spencer Catron. Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains. The Journal of Geometric Analysis, 2025. URL: https://arxiv.org/abs/2404.08824, arXiv:2404.08824, doi:10.1007/s12220-025-02036-x.All uses in this project ↩ 1
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