Paper · Dec 21, 2017

Brownian motion on Perelman's almost Ricci-flat manifold

Esther Cabezas-Rivas, Robert Haslhofer

math.DGarXiv:1712.08184
View PDF View HTML arXiv page

As on arXiv

Abstract

We study Brownian motion and stochastic parallel transport on Perelman's almost Ricci flat manifold , whose dimension depends on a parameter unbounded from above. We construct sequences of projected Brownian motions and stochastic parallel transports which for converge to the corresponding objects for the Ricci flow. In order to make precise this process of passing to the limit, we study the martingale problems for the Laplace operator on and for the horizontal Laplacian on the orthonormal frame bundle . As an application, we see how the characterizations of two-sided bounds on the Ricci curvature established by A. Naber applied to Perelman's manifold lead to the inequalities that characterize solutions of the Ricci flow discovered by Naber and the second author.

View PDF View HTML arXiv page
Submitted
Dec 21, 2017
Subjects
math.DG, math.AP, math.PR
Ricci flowRicci-flat & Calabi–Yau

Its neighbourhood

Connected papers

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.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Stability of ALE Ricci-flat manifolds under Ricci flow (Alix Deruelle and Klaus Kroencke, 2017)1Deruelle 2017 From the Ricci flow evolution equation to vanishing theorems for Ricci solitons (Vladimir Rovenski, Sergey Stepanov and Irina Tsyganok, 2020)8Rovenski 2020 Convergence of the Ricci flow to Ricci-flat ALE manifolds and positive scalar curvature rigidity (Klaus Kroencke and Oliver Lindblad Petersen, 2020)7Kroencke 2020 Dynamical (in)stability of Ricci-flat ALE metrics along Ricci flow (Alix Deruelle and Tristan Ozuch, 2021)5Deruelle 2021 Ricci limit flows and weak solutions (Beomjun Choi and Robert Haslhofer, 2021)2Choi 2021 Weak scalar curvature lower bounds along Ricci flow (Wenshuai Jiang, Weimin Sheng and Huaiyu Zhang, 2021)3Jiang 2021 Bismut Ricci flat manifolds with symmetries (Fabio Podestà and Alberto Raffero, 2022)4Podestà 2022 Ricci Flow on ALF manifolds (Dain Kim and Tristan Ozuch, 2025)6Kim 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Stability of ALE Ricci-flat manifolds under Ricci flow

    Alix Deruelle and Klaus Kroencke

    Topics Ricci-flat & Calabi–Yau

  2. Ricci limit flows and weak solutions

    Beomjun Choi and Robert Haslhofer

    Topics Ricci flow · Same author Robert Haslhofer · Shared terms “Brownian motion”, “Naber”

  3. Weak scalar curvature lower bounds along Ricci flow

    Wenshuai Jiang, Weimin Sheng and Huaiyu Zhang

    Topics Ricci-flat & Calabi–Yau

  4. Bismut Ricci flat manifolds with symmetries

    Fabio Podestà and Alberto Raffero

    Topics Ricci-flat & Calabi–Yau

  5. Dynamical (in)stability of Ricci-flat ALE metrics along Ricci flow

    Alix Deruelle and Tristan Ozuch

    Topics Ricci-flat & Calabi–Yau · Shared terms “Perelman”

  6. Ricci Flow on ALF manifolds

    Dain Kim and Tristan Ozuch

    Topics Ricci-flat & Calabi–Yau · Shared terms “Perelman”

  7. Convergence of the Ricci flow to Ricci-flat ALE manifolds and positive scalar curvature rigidity

    Klaus Kroencke and Oliver Lindblad Petersen

    Topics Ricci-flat & Calabi–Yau · Shared terms “parallel”

  8. From the Ricci flow evolution equation to vanishing theorems for Ricci solitons

    Vladimir Rovenski, Sergey Stepanov and Irina Tsyganok

    Topics Ricci-flat & Calabi–Yau

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) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

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.