Paper · Apr 24, 2019

Stochastic Ricci Flow on Compact Surfaces

Julien Dubédat, Hao Shen

math.PRarXiv:1904.10909v2
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Abstract

In this paper we introduce the stochastic Ricci flow (SRF) in two spatial dimensions. The flow is symmetric with respect to a measure induced by Liouville Conformal Field Theory. Using the theory of Dirichlet forms, we construct a weak solution to the associated equation of the area measure on a flat torus, in the full " regime" where is the noise strength. We also describe the main necessary modifications needed for the SRF on general compact surfaces, and list some open questions.

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Submitted
Apr 24, 2019
Last revised
Jan 22, 2021 (v2)
Subjects
math.PR, math-ph, math.DG

Comments: 42 pages

Ricci flowWeak & singular flows

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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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    Matteo Lulli, Antonino Marciano and Kristian Piscicchia

    Topics Ricci flow · Shared terms “stochastic Ricci”

  2. Noise-induced stabilization of Schwarzschild–AdS black holes under stochastic Ricci flow

    Jihui Wang, Matteo Lulli and Antonino Marciano

    Topics Ricci flow · Shared terms “stochastic Ricci”

  3. Weak Solutions of the Chern-Ricci flow on compact complex surfaces

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    Topics Weak & singular flows

  4. Singular Ricci Flows on surfaces with boundary and positive scalar curvature

    Jean C. Cortissoz and Juan J. Villamarín

    Topics Weak & singular flows · Shared terms “compact surfaces”

  5. Preserving curvature lower bounds when Ricci flowing non-smooth initial data

    Miles Simon

    Topics Weak & singular flows

  6. On weak formulations of (super) Ricci flows

    Sajjad Lakzian

    Topics Weak & singular flows

  7. Lectures on mean curvature flow of surfaces

    Robert Haslhofer

    Topics Weak & singular flows

  8. Ricci Solitons, Conical Singularities, and Nonuniqueness

    Sigurd B. Angenent and Dan Knopf

    Topics Weak & singular flows

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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 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.

  3. 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.