Paper · Feb 15, 2021

The Yamabe flow on asymptotically flat manifolds

Eric Chen, Yi Wang

math.DGWider flowsarXiv:2102.07717
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Abstract

We study the Yamabe flow starting from an asymptotically flat manifold . We show that the flow converges to an asymptotically flat, scalar flat metric in a weighted global sense if , and show that the flow does not converge otherwise. If the scalar curvature is nonnegative and integrable, then the ADM mass at time infinity drops by the limit of the total scalar curvature along the flow.

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Submitted
Feb 15, 2021
Subjects
math.DG, math.AP

Comments: 45 pages

Yamabe flowScalar curvatureBoundaries & non-compactPhysics

Its neighbourhood

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  1. The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant

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

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  7. Yamabe Flow with prescribed scalar curvature

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  8. The Exponential Convergence of the CR Yamabe Flow

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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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989) Physics Ricci flow first appeared in physics, as the leading-order renormalization-group flow of the string sigma model (Friedan, 1980). Start with: Nonlinear Models in 2+ε Dimensions (Friedan, 1980)

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.