Paper · Jun 6, 2021

The Prescribed -Curvature Flow for Arbitrary Even Dimension in a Critical Case

Yuchen Bi, Jiayu Li

math.APWider flowsarXiv:2106.03137
View PDF View HTML arXiv page

As on arXiv

Abstract

In this paper, we study the prescribed -curvature flow equation on a arbitrary even dimensional closed Riemannian manifold , which was introduced by S. Brendle in, where he proved the flow exists for long time and converges at infinity if the GJMS operator is weakly positive with trivial kernel and . In this paper we study the critical case that , we will prove the convergence of the flow under some geometric hypothesis. In particular, this gives a new proof of Li-Li-Liu's existence result in in dimensiona 4 and extend the work of Li-Zhu in dimension 2 to general even dimensions. In the proof, we give a explicit expression of the limit of the corresponding energy functional when the blow up occurs.

View PDF View HTML arXiv page
Submitted
Jun 6, 2021
Subjects
math.AP, math.DG

Comments: 33 pages

Other curvature flowsSingularities

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 Prescribed Q-curvature flow on closed manifolds of even dimension (Quôc Anh Ngô and Hong Zhang, 2017)1Ngô 2017 Global dynamics of a parabolic type equation arising from the curvature flow (Xueli Bai, Fang Li and Xiaoliu Wang, 2021)3Bai 2021 Translating Solitons to a Lagrangian mean curvature flow with zero Maslov class (Xiaoli Han, Jiayu Li and Jun Sun, 2024)4Han 2024 Existence of curvature flow with forcing in a critical Sobolev space (Yuning Liu and Yoshihiro Tonegawa, 2024)2Liu 2024 Prescribed T-curvature flow on the four-dimensional unit ball (Pak Tung Ho, Cheikh Birahim Ndiaye, Liming Sun et al., 2026)5Ho 2026 Infinity-harmonic functions and inverse mean curvature flow clusters (Kai Xu, 2026)6Xu 2026 Long-time existence of the Calabi flow on Kähler surfaces with bounded Sobolev constants (Haozhao Li, Bing Wang and Kai Zheng, 2026)7Li 2026 Chern scalar curvature and finite-time singularities of the conformal Chern–Calabi flow (Hongyu Chen, Li Ma and Li Sheng, 2026)8Chen 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Prescribed -curvature flow on closed manifolds of even dimension

    Quôc Anh Ngô and Hong ZhangWider flows

    Topics Other curvature flows · Shared terms “GJMS operator”, “closed Riemannian”, “Riemannian manifold”

  2. Existence of curvature flow with forcing in a critical Sobolev space

    Yuning Liu and Yoshihiro TonegawaWider flows

    Topics Other curvature flows, Singularities · Shared terms “flow exists”

  3. Global dynamics of a parabolic type equation arising from the curvature flow

    Xueli Bai, Fang Li and Xiaoliu WangWider flows

    Topics Other curvature flows, Singularities · Shared terms “explicit expression”, “energy functional”

  4. Translating Solitons to a Lagrangian mean curvature flow with zero Maslov class

    Xiaoli Han, Jiayu Li and Jun SunWider flows

    Topics Singularities · Same author Jiayu Li

  5. Prescribed -curvature flow on the four-dimensional unit ball

    Pak Tung Ho, Cheikh Birahim Ndiaye, Liming Sun et al.Wider flows

    Topics Other curvature flows · Shared terms “explicit expression”

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

    Kai XuWider flows

    Topics Other curvature flows, Singularities

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

    Haozhao Li, Bing Wang and Kai ZhengWider flows

    Topics Other curvature flows, Singularities · Shared terms “long time”

  8. Chern scalar curvature and finite-time singularities of the conformal Chern–Calabi flow

    Hongyu Chen, Li Ma and Li ShengWider flows

    Topics Other curvature flows, Singularities

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 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)

In the guidebook

  1. 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.

  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.