Paper · Nov 27, 2024

Existence of curvature flow with forcing in a critical Sobolev space

Yuning Liu, Yoshihiro Tonegawa

math.APWider flowsarXiv:2411.18284v2
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As on arXiv

Abstract

Suppose that a closed 1-rectifiable set of finite 1-dimensional Hausdorff measure and a vector field in a dimensionally critical Sobolev space are given. It is proved that, starting from , there exists a non-trivial flow of curves with the normal velocity given by the sum of the curvature and the given vector field . The motion law is satisfied in the sense of Brakke and the flow exists through singularities.

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Submitted
Nov 27, 2024
Last revised
Jul 22, 2026 (v2)
Subjects
math.AP, math.DG

Comments: 29 pages, comment welcome!

Other curvature flowsSingularitiesWeak & singular flows

Its neighbourhood

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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.

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A note on generalized BV mean curvature flow with a critical forcing term

    Kiichi TashiroWider flows

    Topics Weak & singular flows · Shared terms “Brakke”

  2. The Bounded Diameter Conjecture and Sharp Geometric Estimates for Mean Curvature Flow

    Yiqi Huang and Wenshuai JiangWider flows

    Topics Weak & singular flows, Singularities · Shared terms “Hausdorff measure”

  3. An Intersection Principle for Mean Curvature Flow

    Tang-Kai Lee and Alec PayneWider flows

    Topics Weak & singular flows, Singularities · Shared terms “Brakke”, “Hausdorff”

  4. Infinity-harmonic functions and inverse mean curvature flow clusters

    Kai XuWider flows

    Topics Weak & singular flows, Other curvature flows, Singularities

  5. On the Multiplicity One Conjecture for Mean Curvature Flows of surfaces

    Richard H Bamler and Bruce KleinerWider flows

    Topics Weak & singular flows, Singularities

  6. Minimizing movements for the generalized power mean curvature flow

    Giovanni Bellettini and Shokhrukh Yu. KholmatovWider flows

    Topics Weak & singular flows · Shared terms “normal velocity”

  7. Mean Curvature Flow from Conical Singularities

    Otis Chodosh, J. M. Daniels-Holgate and Felix SchulzeWider flows

    Topics Weak & singular flows, Singularities

  8. The continuity of curvature flow in Fréchet distance

    Shiyi MaWider flows

    Topics Weak & singular flows, Other curvature flows

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.

Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) 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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 6A.13
    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.

  3. 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.