Paper · Feb 17, 2017

Weak and smooth solutions for a fractional Yamabe flow: the case of general compact and locally conformally flat manifolds

Panagiota Daskalopoulos, Yannick Sire, Juan-Luis Vázquez

math.APWider flowsarXiv:1702.05221
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Abstract

As a counterpart of the classical Yamabe problem, a fractional Yamabe flow has been introduced by Jin and Xiong (2014) on the sphere. Here we pursue its study in the context of general compact smooth manifolds with positive fractional curvature. First, we prove that the flow is locally well posed in the weak sense on any compact manifold. If the manifold is locally conformally flat with positive Yamabe invariant, we also prove that the flow is smooth and converges to a constant scalar curvature metric. We provide different proofs using extension properties introduced by Chang and González (2011) for the conformally covariant fractional order operators.

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Submitted
Feb 17, 2017
Subjects
math.AP
Yamabe flowAnisotropic & nonlocal flowsScalar curvature

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  1. Convergence of the fractional Yamabe flow for a class of initial data

    Hardy Chan, Yannick Sire and Liming SunWider flows

    Topics Yamabe flow, Anisotropic & nonlocal flows · Same author Yannick Sire · Shared terms “fractional Yamabe”, “fractional curvature”

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    Topics Yamabe flow · Shared terms “locally conformally”, “conformally flat”, “flat manifolds”

  3. Gap Theorem on locally conformally flat manifold

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    Topics Yamabe flow · Shared terms “locally conformally”, “conformally flat”, “flat manifolds”

  4. Yamabe Flow: steady solitons and type II singularities

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    Topics Yamabe flow · Same author Panagiota Daskalopoulos · Shared terms “conformally flat”

  5. Convergence of the fractional Yamabe flow for arbitrary initial energy

    Jingeon An-Lacroix, Hardy Chan and Pak Tung HoWider flows

    Topics Yamabe flow · Shared terms “fractional Yamabe”

  6. Type II Singularities on complete non-compact Yamabe flow

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    Topics Yamabe flow · Same author Panagiota Daskalopoulos · Shared terms “conformally flat”

  7. New type I ancient compact solutions of the Yamabe flow

    Panagiota Daskalopoulos, Manuel del Pino, John King et al.Wider flows

    Topics Yamabe flow · Same author Panagiota Daskalopoulos

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

    Mario B. SchulzWider flows

    Topics Yamabe flow, Scalar curvature · Shared terms “compact manifolds”, “positive Yamabe”, “Yamabe invariant”

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) Anisotropic & nonlocal flows Wider flows When surface energy depends on direction, as in crystals, mean curvature flow becomes anisotropic, and the Wulff shape replaces the round sphere. Start with: Crystalline variational problems (Taylor, 1978) 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. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

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