Paper · Oct 8, 2026

Compact Mean Curvature Flows from Non-Compact Initial Data

Theodora Bourni, Nathan Burns, Marcus Flook

math.DGWider flowsarXiv:2610.12817
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Abstract

We construct compact, embedded solutions to the mean curvature flow in that arise from various non-compact initial data and exist for a finite time.

Mean curvature flowSingularitiesBoundaries & non-compact

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Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Mean curvature flow with generic initial data II (Otis Chodosh, Kyeongsu Choi and Felix Schulze, 2023)6Chodosh 2023 Compact curve shortening flow solutions out of non compact curve (Theodora Bourni and Martin Reiris, 2023)1Bourni 2023 Mean curvature flow with generic low-entropy initial data II (Otis Chodosh, Christos Mantoulidis and Felix Schulze, 2023)7Chodosh 2023 Blow Up of Compact Mean Curvature Flow Solutions with Bounded Mean Curvature (Zichang Liu, 2024)3Liu 2024 Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains (Theodora Bourni, Nathan Burns and Spencer Catron, 2024)4Bourni 2024 Aleksandrov reflection for Geometric Flows in Hyperbolic Spaces (Theodora Bourni, José M. Espinar and Aakash Mishra, 2026)8Bourni 2026 Stability of the Shrinking Semi-Circle Under the Free Boundary Curve Shortening Flow (Theodora Bourni, Nathan Burns and Mat Langford, 2026)2Bourni 2026 Generic mean curvature flow has finitely many singularities (J. M. Daniels-Holgate, Or Hershkovits and Gábor Székelyhidi, 2026)5Daniels-Holg 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Compact curve shortening flow solutions out of non compact curve

    Theodora Bourni and Martin ReirisWider flows PDF

    Topics Boundaries & non-compact, Singularities · Same author Theodora Bourni · Shared terms “embedded solution”, “finite time”

  2. Stability of the Shrinking Semi-Circle Under the Free Boundary Curve Shortening Flow

    Theodora Bourni, Nathan Burns and Mat LangfordWider flows PDF

    Topics Singularities · Same authors Theodora Bourni, Nathan Burns · Shared terms “finite time”

  3. Blow Up of Compact Mean Curvature Flow Solutions with Bounded Mean Curvature

    Zichang LiuWider flows PDF

    Topics Singularities · Shared terms “compact mean”, “construct compact”, “finite time”

  4. Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains

    Theodora Bourni, Nathan Burns and Spencer CatronWider flows PDF

    Same authors Theodora Bourni, Nathan Burns

  5. Generic mean curvature flow has finitely many singularities

    J. M. Daniels-Holgate, Or Hershkovits and Gábor SzékelyhidiWider flows PDF

    Topics Singularities · Shared terms “compact initial”

  6. Mean curvature flow with generic initial data II

    Otis Chodosh, Kyeongsu Choi and Felix SchulzeWider flows PDF

    Topics Singularities · Shared terms “initial data”

  7. Mean curvature flow with generic low-entropy initial data II

    Otis Chodosh, Christos Mantoulidis and Felix SchulzeWider flows PDF

    Topics Singularities · Shared terms “initial data”

  8. Aleksandrov reflection for Geometric Flows in Hyperbolic Spaces

    Theodora Bourni, José M. Espinar and Aakash MishraWider flows PDF

    Topics Boundaries & non-compact · Same author Theodora Bourni

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

In the guidebook

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    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  2. 6C.8
    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.

  3. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.