Paper · Feb 7, 2023

Existence of the -Ricci-Yamabe flow on closed manifolds

Liangdi Zhang

math.DGarXiv:2302.03199
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Abstract

On a smooth closed Riemannian manifold, we show short time existence of smooth solutions to the -Ricci-Yamabe flow, which is a natural generalization of the Ricci flow and the Yamabe flow. We also establish some long time existence theorems for the closed -Ricci-Yamabe flow by estimating its curvatures.

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Submitted
Feb 7, 2023
Subjects
math.DG, math.AP
Ricci flowYamabe flowLong-time existence & convergenceHarmonic maps & DeTurck trick

Its neighbourhood

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  1. The existence and uniqueness of infinite combinatorial Yamabe flows

    Bohao JiWider flows

    Topics Yamabe flow, Harmonic maps & DeTurck trick, Long-time existence & convergence · Shared terms “short time”

  2. Generalized Yamabe Flows

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  3. Infinite-time incompleteness of noncompact Yamabe flow

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  4. Infinite Combinatorial Yamabe Flows in Three Dimensions

    Bohao JiWider flows

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  5. Combinatorial Yamabe flow on infinitely triangulated hyperbolic surfaces

    Yuerong Bian and Xu XuWider flows

    Topics Yamabe flow, Harmonic maps & DeTurck trick, Long-time existence & convergence · Shared terms “smooth solutions”, “short time”

  6. Convergence of the Weighted Yamabe Flow

    Zetian YanWider flows

    Topics Yamabe flow, Long-time existence & convergence

  7. A non-convergence phenomenon for the CR Yamabe flow

    Claudio Afeltra, Andrea Pinamonti and Pak Tung HoWider flows

    Topics Yamabe flow

  8. On the existence and classification of -Yamabe gradient solitons

    Maria Fernanda Espinal and Mariel Sáez

    Topics Yamabe flow

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.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Yamabe flow Wider flows The Yamabe flow deforms a metric within its conformal class, driven by scalar curvature. It is a parabolic approach to the Yamabe problem: find a… Start with: The yamabe flow on locally conformally flat manifolds with positive ricci curvature (Chow, 1992) 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) Harmonic maps & DeTurck trick Ricci flow is only weakly parabolic, because it is invariant under diffeomorphisms. The DeTurck trick adds a Lie-derivative term that makes it… Start with: Harmonic Mappings of Riemannian Manifolds (Eells et al., 1964)

In the guidebook

  1. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  2. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  3. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.