Paper · Jul 11, 2018

Conic manifolds under the Yamabe flow

Nikolaos Roidos

math.APWider flowsarXiv:1807.04139v2
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Abstract

We consider the unnormalized Yamabe flow on manifolds with conical singularities. Under certain geometric assumption on the initial cross-section we show well posedness of the short time solution in the -setting. Moreover, we give a picture of the deformation of the conical tips under the flow by providing an asymptotic expansion of the evolving metric close to the boundary in terms of the initial local geometry. Due to the blow up of the scalar curvature close to the singularities we use maximal -regularity theory for conically degenerate operators.

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Submitted
Jul 11, 2018
Last revised
Nov 27, 2019 (v2)
Subjects
math.AP, math.DG, math.FA
Journal
J. Evol. Equ. 20, no. 2, 321-334 (2020)

Comments: 11 pages

Yamabe flowSingularitiesScalar 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
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    Sergio Almaraz and Liming SunWider flows

    Topics Yamabe flow, Scalar curvature

  2. First eigenvalues of geometric operators under the Yamabe flow

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    Topics Yamabe flow

  4. On the Chern-Yamabe flow

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    Topics Yamabe flow, Scalar curvature

  5. The Exponential Convergence of the CR Yamabe Flow

    Weimin Sheng and Kunbo WangWider flows

    Topics Yamabe flow, Scalar curvature · Shared terms “time solution”

  6. Yamabe Flow with prescribed scalar curvature

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    Topics Yamabe flow, Scalar curvature

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

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    Topics Yamabe flow, Scalar curvature

  8. Normalized Yamabe flow on manifolds with bounded geometry

    Bruno Caldeira, Luiz Hartmann and Boris VertmanWider flows

    Topics Yamabe flow, Scalar curvature · Shared terms “time solution”, “short time”

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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.

  3. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.