Paper · Apr 19, 2016

General curvature flow without singularities

Ling Xiao

math.DGWider flowsarXiv:1604.05743
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As on arXiv

Abstract

In [5], Sáez and Schnürer studied the graphical mean curvature flow of complete hypersurfaces defined on subsets of Euclidean space. They obtained long time existence. Moreover, they provided a new interpretation of weak mean curvature flow. In this paper, we generalize their results to a general curvature setting. Our key ingredient is the existence result of general curvature flow with boundary conditions, which is proved in Section 4.

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Submitted
Apr 19, 2016
Subjects
math.DG, math.AP
Mean curvature flowOther curvature flowsSingularitiesBoundaries & non-compactLong-time existence & convergence

Its neighbourhood

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 2016, 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 Shadows of graphical mean curvature flow (Wolfgang Maurer, 2016)4Maurer 2016 Neumann boundary value problem for gernera curvature flow with forcing term (Ling Xiao, 2016)1Xiao 2016 Asymptotic convergence for modified scalar curvature flow (Ling Xiao, 2020)6Xiao 2020 Graphical mean curvature flow with bounded bi-Ricci curvature (Renan Assimos, Andreas Savas-Halilaj and Knut Smoczyk, 2022)8Assimos 2022 Entire convex curvature flow in Minkowski space (Zhizhang Wang and Ling Xiao, 2022)3Wang 2022 Entire σₖ curvature flow in Minkowski space (Zhizhang Wang and Ling Xiao, 2022)2Wang 2022 Second boundary value problem for the Hessian curvature flow (Rongli Huang, Changzheng Qu, Zhizhang Wang et al., 2025)5Huang 2025 Well-posedness for the mean curvature flow on the half-space and on bounded domains (Ke Chen, Ruilin Hu and Quoc-Hung Nguyen, 2026)7Chen 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Neumann boundary value problem for gernera curvature flow with forcing term

    Ling XiaoWider flows

    Topics Long-time existence & convergence, Boundaries & non-compact, Other curvature flows · Same author Ling Xiao · Shared terms “boundary condition”

  2. Entire curvature flow in Minkowski space

    Zhizhang Wang and Ling XiaoWider flows

    Topics Long-time existence & convergence, Boundaries & non-compact, Other curvature flows · Same author Ling Xiao

  3. Entire convex curvature flow in Minkowski space

    Zhizhang Wang and Ling XiaoWider flows

    Topics Long-time existence & convergence, Boundaries & non-compact, Other curvature flows · Same author Ling Xiao

  4. Shadows of graphical mean curvature flow

    Wolfgang MaurerWider flows

    Topics Long-time existence & convergence, Boundaries & non-compact · Shared terms “graphical mean”, “boundary condition”

  5. Second boundary value problem for the Hessian curvature flow

    Rongli Huang, Changzheng Qu, Zhizhang Wang et al.Wider flows

    Topics Long-time existence & convergence, Boundaries & non-compact, Other curvature flows · Shared terms “Schnürer”, “boundary condition”

  6. Asymptotic convergence for modified scalar curvature flow

    Ling XiaoWider flows

    Topics Long-time existence & convergence, Other curvature flows · Same author Ling Xiao

  7. Well-posedness for the mean curvature flow on the half-space and on bounded domains

    Ke Chen, Ruilin Hu and Quoc-Hung NguyenWider flows

    Topics Long-time existence & convergence, Boundaries & non-compact · Shared terms “graphical mean”, “boundary condition”, “ingredient”

  8. Graphical mean curvature flow with bounded bi-Ricci curvature

    Renan Assimos, Andreas Savas-Halilaj and Knut SmoczykWider flows

    Topics Long-time existence & convergence · Shared terms “graphical mean”

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 in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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 can stop in finite time: curvature blows up, 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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  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.