Paper · Dec 24, 2020

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

Li Ma

math.DGWider flowsarXiv:2012.13069
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Abstract

In this paper, we study the existence of complete Yamabe metric with zero scalar curvature on an n-dimensional complete Riemannian manifold , . Under suitable conditions about the initial metric, we show that there is a global fine solution to the Yamabe flow. The interesting point here is that we have no curvature assumption about the initial metric. We show that on an n-dimensional complete Riemannian manifold with non-negative Ricci curvature, , the Yamabe flow enjoys the local -stability property from the view-point of the porous media equation. Complete Yamabe metrics with zero scalar curvature on an n-dimensional Riemannian model space are also discussed.

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Submitted
Dec 24, 2020
Subjects
math.DG, math.AP

Comments: submiited, 20 pages

Yamabe flowScalar curvatureStability & uniqueness

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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  5. Unconditional existence of conformally hyperbolic Yamabe flows

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  6. Long-time existence of the edge Yamabe flow

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  7. Convergence of the CR Yamabe Flow

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  8. Yamabe flow on non-compact manifolds with unbounded initial curvature

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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.

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) 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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.