MCF-stack: Special Solutions of Mean Curvature Flow

§ 5.9 Convex ancient free-boundary flows in balls (Bourni–Langford, 2021; Bourni–Langford, 2022)5E38E2470718

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–Langford — 2021-11-11 (first arXiv submission).
  2. Bourni–Langford — 2022-04-14 (first arXiv submission).
5E38E2470718
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There is a convex ancient free-boundary curve-shortening flow in the unit disk, and corresponding rotational convex ancient hypersurface flows in the unit ball $B^{n+1}$, meeting the fixed boundary sphere orthogonally. The backward limit is an equatorial disk; forward the solution contracts to a point on the boundary sphere. The uniqueness statements of Bourni–Langford are within the convex ancient classes and normalizations of their theorems, not for all free-boundary ancient flows. The slices are compact embedded disks, smooth up to their moving boundary. Their boundary is constrained to the unit sphere, unlike a freely shrinking boundary under homothety. The maximal interval is ancient with a finite forward endpoint. For $n=1$ the topology is an interval; for $n=2$ it is a disk of genus zero. These are manifolds with boundary, so completeness means completeness as compact metric spaces, not geodesic completeness without boundary. The figure keeps the unit ball fixed and shows orthogonal contact in a meridian. Its circular-arc interpolation enforces contact geometry but is not asserted to solve the ancient PDE.

Sources: [BourniLangford2023, BourniLangford2025].

Convex ancient free-boundary flows in balls. Fixed unit ball, equatorial backward limit, and later orthogonal caps. The caps are contact-angle schematics, not a circular-arc formula for the ancient solution.
Source-informed schematic. Fixed unit ball, equatorial backward limit, and later orthogonal caps. The caps are contact-angle schematics, not a circular-arc formula for the ancient solution. 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.

Curve in B² · Euclidean plane ℝ² · Embedded
Ambient space
Euclidean plane ℝ²
Topology
Closed interval with endpoints on ∂B²
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.
Disk in B³ · Euclidean space ℝ³ · Embedded
Ambient space
Euclidean space ℝ³
Topology
D² with boundary on ∂B³
Embeddedness
Embedded: The cited construction gives an embedding for this case.
Compactness
Compact
Surface genus
Genus zero.
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.
n ≥ 3 · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded
Ambient space
Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology
Dⁿ in Bⁿ⁺¹
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. [BourniLangford2023] Theodora Bourni and Mat Langford. Classification of convex ancient free boundary curve shortening flows in the disc. Analysis & PDE, 16:2225–2240, 2023. URL: https://arxiv.org/abs/2111.06051, arXiv:2111.06051, doi:10.2140/apde.2023.16.2225.All uses in this project ↩ 1
  2. [BourniLangford2025] Theodora Bourni and Mat Langford. Classification of convex ancient free boundary mean curvature flows in the ball. ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE, pages 581–603, 2025. URL: https://arxiv.org/abs/2204.06710, arXiv:2204.06710, doi:10.2422/2036-2145.202206_007.All uses in this project ↩ 1
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