Paper · Sep 9, 2019

A Bochner Formula on Path Space for the Ricci Flow

Christopher Kennedy

math.DGarXiv:1909.04193v2
View PDF View HTML arXiv page

As on arXiv

Abstract

We generalize the classical Bochner formula for the heat flow on evolving manifolds to an infinite-dimensional Bochner formula for martingales on parabolic path space of space-time . Our new Bochner formula and the inequalities that follow from it are strong enough to characterize solutions of the Ricci flow. Specifically, we obtain characterizations of the Ricci flow in terms of Bochner inequalities on parabolic path space. We also obtain gradient and Hessian estimates for martingales on parabolic path space, as well as condensed proofs of the prior characterizations of the Ricci flow from Haslhofer-Naber. Our results are parabolic counterparts of the recent results in the elliptic setting from.

View PDF View HTML arXiv page
Submitted
Sep 9, 2019
Last revised
Jul 4, 2022 (v2)
Subjects
math.DG, math.AP
Journal
Calc. Var. 62, 83 (2023)
DOI
10.1007/s00526-022-02420-3

Comments: 31 pages

Ricci flow

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 On warped product gradient η-Ricci solitons (Adara M. Blaga, 2017)8Blaga 2017 Super Ricci flows for weighted graphs (Matthias Erbar and Eva Kopfer, 2018)4Erbar 2018 New Curvature Conditions for the Bochner Technique (Peter Petersen and Matthias Wink, 2019)5Petersen 2019 Ricci limit flows and weak solutions (Beomjun Choi and Robert Haslhofer, 2021)2Choi 2021 Hessian estimates for the conjugate heat equation coupled with the Ricci flow (Hong Huang, 2021)7Huang 2021 Bochner formulas, functional inequalities and generalized Ricci flow (Eva Kopfer and Jeffrey Streets, 2022)1Kopfer 2022 On a parabolic curvature lower bound generalizing Ricci flows (Marco Flaim and Erik Hupp, 2025)3Flaim 2025 Sharp Gaussian Isoperimetry along a Ricci Flow (Robert Koirala, 2026)6Koirala 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Bochner formulas, functional inequalities and generalized Ricci flow

    Eva Kopfer and Jeffrey Streets

    Topics Ricci flow · Shared terms “Bochner formula”, “path space”, “characterization”

  2. Ricci limit flows and weak solutions

    Beomjun Choi and Robert Haslhofer

    Topics Ricci flow · Shared terms “path space”, “Haslhofer Naber”, “space time”

  3. On a parabolic curvature lower bound generalizing Ricci flows

    Marco Flaim and Erik Hupp

    Topics Ricci flow · Shared terms “Bochner”, “heat flow”, “infinite dimensional”

  4. Super Ricci flows for weighted graphs

    Matthias Erbar and Eva Kopfer

    Topics Ricci flow · Shared terms “Bochner”, “heat flow”, “characterization”

  5. New Curvature Conditions for the Bochner Technique

    Peter Petersen and Matthias Wink

    Topics Ricci flow · Shared terms “Bochner”

  6. Sharp Gaussian Isoperimetry along a Ricci Flow

    Robert Koirala

    Topics Ricci flow · Shared terms “path space”, “Naber”

  7. Hessian estimates for the conjugate heat equation coupled with the Ricci flow

    Hong Huang

    Topics Ricci flow · Shared terms “Hessian estimates”

  8. On warped product gradient -Ricci solitons

    Adara M. Blaga

    Shared terms “Bochner formula”

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)

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.