Paper · Aug 23, 2016

Long Time Existence of the Cross Curvature Flow in 3-Manifolds with Negative Sectional Curvature

Wei-Hung Liao

math.DGWider flowsarXiv:1608.06599v2
View PDF View HTML arXiv page

As on arXiv

Abstract

Given a closed 3-manifold with an initial Riemannian metric of negative sec- tional curvature, we consider the cross curvature flow an evolution equation of metric on M3. We prove long-time existence of a solution to the cross curvature flow via the maximum principle theorem. Besides, we demonstrate the solution exists for all time and converges pointwisely to a hyperbolic metric

View PDF View HTML arXiv page
Submitted
Aug 23, 2016
Last revised
Sep 9, 2016 (v2)
Subjects
math.DG, math.AP

Comments: A calculation error in theorem 4.1

Other curvature flows3-manifoldsLong-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 General curvature flow without singularities (Ling Xiao, 2016)8Xiao 2016 ODE Maximum Principle at Infinity and Non-Compact Solutions of IMCF in Hyperbolic Space (Brian Allen, 2016)4Allen 2016 Harnack inequalities for curvature flows in Riemannian and Lorentzian manifolds (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2017)6Bryan 2017 Global existence and convergence for the CR Q-curvature flow in a closed strictly pseudoconvex CR 3-manifold (Shu-Cheng Chang, Ting-Jung Kuo and Takanari Saotome, 2019)2Chang 2019 Negatively Curved Three-Manifolds, Hyperbolic Metrics, Isometric Embeddings In Minkowski Space And The Cross Curvature Flow (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2019)1Bryan 2019 Entropy in A Closed Manifold and Partial Regularity of Mean Curvature Flow Limit of Surfaces (Ao Sun, 2019)3Sun 2019 Existence of twisted Calabi flow and deformation from the J-flow to Calabi flow (Jie He and Haozhao Li, 2025)5He 2025 Long-time existence of the Calabi flow on Kähler surfaces with bounded Sobolev constants (Haozhao Li, Bing Wang and Kai Zheng, 2026)7Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Negatively Curved Three-Manifolds, Hyperbolic Metrics, Isometric Embeddings In Minkowski Space And The Cross Curvature Flow

    Paul Bryan, Mohammad N. Ivaki and Julian Scheuer

    Topics 3-manifolds, Other curvature flows · Shared terms “cross curvature”

  2. Global existence and convergence for the CR Q-curvature flow in a closed strictly pseudoconvex CR 3-manifold

    Shu-Cheng Chang, Ting-Jung Kuo and Takanari SaotomeWider flows

    Topics 3-manifolds, Long-time existence & convergence, Other curvature flows

  3. Entropy in A Closed Manifold and Partial Regularity of Mean Curvature Flow Limit of Surfaces

    Ao SunWider flows

    Topics 3-manifolds · Shared terms “negative sectional”, “sectional curvature”

  4. ODE Maximum Principle at Infinity and Non-Compact Solutions of IMCF in Hyperbolic Space

    Brian AllenWider flows

    Topics Long-time existence & convergence, Other curvature flows · Shared terms “maximum principle”

  5. Existence of twisted Calabi flow and deformation from the -flow to Calabi flow

    Jie He and Haozhao LiWider flows

    Topics Long-time existence & convergence, Other curvature flows

  6. Harnack inequalities for curvature flows in Riemannian and Lorentzian manifolds

    Paul Bryan, Mohammad N. Ivaki and Julian ScheuerWider flows

    Topics Other curvature flows · Shared terms “negative sectional”, “sectional curvature”

  7. Long-time existence of the Calabi flow on Kähler surfaces with bounded Sobolev constants

    Haozhao Li, Bing Wang and Kai ZhengWider flows

    Topics Long-time existence & convergence, Other curvature flows

  8. General curvature flow without singularities

    Ling XiaoWider flows

    Topics Long-time existence & convergence, 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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

  2. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.