Paper · May 6, 2019

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

Shu-Cheng Chang, Ting-Jung Kuo, Takanari Saotome

math.DGWider flowsarXiv:1905.01783v2
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Abstract

In this note, we affirm the partial answer to the long open Conjecture which states that any closed embeddable strictly pseudoconvex CR -manifold admits a contact form with the vanishing CR -curvature. More precisely, we deform the contact form according to an CR analogue of %-curvature flow in a closed strictly pseudoconvex CR -manifold of the vanishing first Chern class . Suppose that is embeddable and the CR Paneitz operator is nonnegative with kernel consisting of the CR pluriharmonic functions. We show that the solution of CR -curvature flow exists for all time and has smoothly asymptotic convergence on \ As a consequence, we are able to affirm the Conjecture in a closed strictly pseudoconvex CR -manifold of the vanishing first Chern class and vanishing torsion.

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Submitted
May 6, 2019
Last revised
Jul 5, 2019 (v2)
Subjects
math.DG, math.CV

Comments: 30 pages. arXiv admin note: text overlap with arXiv:math/0510494

Other curvature flows3-manifoldsLong-time existence & convergence

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    Shu-Cheng Chang, Yingbo Han and Chin-Tung WuWider flows

    Topics 3-manifolds, Long-time existence & convergence · Same author Shu-Cheng Chang · Shared terms “asymptotic convergence”

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

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

  3. Convergence of the CR Yamabe Flow

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    Shared terms “strictly pseudoconvex”

  4. Short-Time Existence Theorem for the CR Torsion Flow

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    Same author Shu-Cheng Chang · Shared terms “pluriharmonic function”, “analogue”

  5. Contact Blow-up Analysis of the Legendrian Mean Curvature Flow and a Smale-type Theorem for Strict Contactomorphisms of Sasakian Spheres

    Shu-Cheng Chang, Chin-Tung Wu and Liuyang ZhangWider flows

    Topics Long-time existence & convergence · Same author Shu-Cheng Chang · Shared terms “flow exists”

  6. Asymptotic convergence for modified scalar curvature flow

    Ling XiaoWider flows

    Topics Long-time existence & convergence, Other curvature flows · Shared terms “asymptotic convergence”, “flow exists”

  7. Asymptotic convergence for a class of anisotropic curvature flows

    Haizhong Li, Botong Xu and Ruijia ZhangWider flows

    Topics Long-time existence & convergence, Other curvature flows · Shared terms “asymptotic convergence”, “flow exists”

  8. -curvature flow on Pseudo-Einstein CR manifolds

    Ali Maalaoui and Vittorio MartinoWider flows

    Topics 3-manifolds, Other curvature flows

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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)

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