Paper · Feb 5, 2016

Motion Of Level Set By General Curvature

Ling Xiao

math.DGWider flowsarXiv:1602.02111v2
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Abstract

In this paper, we study the motion of level sets by general curvature. The difficulty of this setting is that a general curvature function is only well defined in an admissible cone. In order to extend the existence of a weak solution of a general curvature flow to outside the cone we introduce a new approximation function (see (3.1)). Moreover, using the idea in [4], we give an elliptic approach for the Ben-Andrews' non-collapsing result in fully nonlinear curvature flows; we hope this approach can be generalized to a wider class of elliptic equations.

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Submitted
Feb 5, 2016
Last revised
Sep 12, 2016 (v2)
Subjects
math.DG, math.AP
Other curvature flows(Non)collapsingWeak & singular flows

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 A level set crystalline mean curvature flow of surfaces (Yoshikazu Giga and Norbert Požár, 2016)2Giga 2016 Regularity and stability results for the level set flow via the mean curvature flow with surgery (Alexander Mramor, 2017)8Mramor 2017 Motion of level sets by inverse anisotropic mean curvature (Francesco Della Pietra, Nunzia Gavitone and Chao Xia, 2018)1Della Pietra 2018 The streamlines of ∞-harmonic functions obey the inverse mean curvature flow (Roger Moser, 2021)6Moser 2021 Entire convex curvature flow in Minkowski space (Zhizhang Wang and Ling Xiao, 2022)7Wang 2022 Rotational symmetry of ancient solutions to fully nonlinear curvature flows (A. Cogo, S. Lynch and O. Vičánek Martínez, 2023)4Cogo 2023 Noncollapsing for Curvature Flows with Inhomogeneous Speeds (Weimin Sheng and Ye Zhu, 2025)3Sheng 2025 Finite-Shift Inverse-Concavity and Exterior Noncollapsing for Fully Nonlinear Curvature Flows of Nonconvex Hypersurfaces (Weimin Sheng and Ye Zhu, 2026)5Sheng 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Motion of level sets by inverse anisotropic mean curvature

    Francesco Della Pietra, Nunzia Gavitone and Chao XiaWider flows

    Topics Weak & singular flows · Shared terms “level set”, “approximation”

  2. A level set crystalline mean curvature flow of surfaces

    Yoshikazu Giga and Norbert PožárWider flows

    Topics Weak & singular flows · Shared terms “level set”, “approximation”

  3. Noncollapsing for Curvature Flows with Inhomogeneous Speeds

    Weimin Sheng and Ye ZhuWider flows

    Topics (Non)collapsing, Other curvature flows · Shared terms “nonlinear curvature”, “fully nonlinear”, “Andrews”

  4. Rotational symmetry of ancient solutions to fully nonlinear curvature flows

    A. Cogo, S. Lynch and O. Vičánek MartínezWider flows

    Topics (Non)collapsing, Other curvature flows · Shared terms “nonlinear curvature”, “fully nonlinear”

  5. Finite-Shift Inverse-Concavity and Exterior Noncollapsing for Fully Nonlinear Curvature Flows of Nonconvex Hypersurfaces

    Weimin Sheng and Ye ZhuWider flows

    Topics (Non)collapsing, Other curvature flows · Shared terms “nonlinear curvature”, “fully nonlinear”, “Andrews”

  6. The streamlines of -harmonic functions obey the inverse mean curvature flow

    Roger MoserWider flows

    Topics Weak & singular flows, Other curvature flows · Shared terms “level set”, “approximation”

  7. Entire convex curvature flow in Minkowski space

    Zhizhang Wang and Ling XiaoWider flows

    Topics Other curvature flows · Same author Ling Xiao · Shared terms “nonlinear curvature”, “fully nonlinear”

  8. Regularity and stability results for the level set flow via the mean curvature flow with surgery

    Alexander MramorWider flows

    Topics Weak & singular flows · Shared terms “level set”

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) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982) 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. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  2. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.

  3. 12A.8
    Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing

    Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.