Paper · Jun 28, 2017

Ricci-Yamabe maps for Riemannian flows and their volume variation and volume entropy

Mircea Crasmareanu, Sinem Güler

math.DGarXiv:1706.09368
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Abstract

The aim of this short note is to produce new examples of geometrical flows associated to a given Riemannian flow . The considered flow in covariant symmetric -tensor fields will be called Ricci-Yamabe map since it involves a scalar combination of Ricci tensor and scalar curvature of . Due to the signs of considered scalars the Ricci-Yamabe flow can be also a Riemannian or semi-Riemannian or singular Riemannian flow. We study the associated function of volume variation as well as the volume entropy. Finally, since the two-dimensional case was the most handled situation we express the Ricci flow equation in all four orthogonal separable coordinate systems of the plane.

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Submitted
Jun 28, 2017
Subjects
math.DG
Ricci flowYamabe flowSurfacesScalar curvature

Its neighbourhood

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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.

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Existence of the -Ricci-Yamabe flow on closed manifolds

    Liangdi Zhang

    Topics Yamabe flow

  2. Convergence of the Yamabe flow on manifolds with minimal boundary

    Sergio Almaraz and Liming SunWider flows

    Topics Yamabe flow, Scalar curvature

  3. Yamabe Flow with prescribed scalar curvature

    Inas Amacha and Rachid RegbaouiWider flows

    Topics Yamabe flow, Scalar curvature

  4. Incomplete Yamabe flows and removable singularities

    Mario B. SchulzWider flows

    Topics Yamabe flow, Surfaces

  5. Yamabe metrics, Fine solutions to the Yamabe flow, and local L1-stability

    Li MaWider flows

    Topics Yamabe flow, Scalar curvature

  6. On the Chern-Yamabe flow

    Mehdi Lejmi and Ali MaalaouiWider flows

    Topics Yamabe flow, Scalar curvature

  7. A note on the Chern-Yamabe flow on compact Hermitian manifolds with negative Gauduchon degree

    Weike YuWider flows

    Topics Yamabe flow, Scalar curvature

  8. Convergence of the CR Yamabe Flow

    Pak Tung Ho, Weimin Sheng and Kunbo WangWider flows

    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 inside its conformal class at a rate set by scalar curvature. It is a parabolic approach to the Yamabe problem… Start with: The yamabe flow on locally conformally flat manifolds with positive ricci curvature (Chow, 1992) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

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. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.