Paper · Mar 17, 2016

Ricci flow on asymptotically Euclidean manifolds

Yu Li

math.DGarXiv:1603.05336v4
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Abstract

In this paper, we prove that if an asymptotically Euclidean manifold under the condition that has long time existence of Ricci flow, the mass of is nonnegative. In addition, we give an independent proof of positive mass theorem in dimension .

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Submitted
Mar 17, 2016
Last revised
Dec 15, 2017 (v4)
Subjects
math.DG
Journal
Geom. Topol. 22 (2018) 1837-1891
DOI
10.2140/gt.2018.22.1837
Ricci flowLong-time existence & convergencePhysics

Its neighbourhood

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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. A continuous Positive Mass Theorem for perturbations of Euclidean space

    Paula Burkhardt-Guim

    Topics Physics · Shared terms “mass theorem”, “positive mass”, “Euclidean”

  2. The Pohozaev-Schoen Identity on Asymptotically Euclidean Manifolds: Conservation Identities and their applications

    Allan Freitas and Rodrigo Ávalos

    Shared terms “Euclidean manifolds”, “asymptotically Euclidean”

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

    Gilles Carron, Eric Chen and Yi WangWider flows

    Shared terms “Euclidean manifolds”, “asymptotically Euclidean”

  4. The spinorial energy for asymptotically Euclidean Ricci flow

    Julius Baldauf and Tristan Ozuch

    Topics Ricci flow · Shared terms “asymptotically Euclidean”, “Euclidean manifolds”

  5. Spinors and mass on weighted manifolds

    Julius Baldauf and Tristan Ozuch

    Topics Physics · Shared terms “asymptotically Euclidean”, “mass theorem”

  6. Spaces with distributional scalar curvature bounded from below: Optimal regularity and positive mass

    Man-Chun Lee, Florian Litzinger and Miles Simon

    Topics Physics · Shared terms “positive mass”, “mass theorem”, “Euclidean”

  7. Ricci flow and the scalar curvature rigidity of Einstein manifolds

    Klaus Kroencke

    Topics Physics · Shared terms “mass theorem”, “positive mass”, “Euclidean”

  8. Parabolic Dimensional Reductions of 11D Supergravity

    Teng Fei, Bin Guo and Duong H. Phong

    Topics Physics, Long-time existence & convergence

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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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. 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.

  2. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.