Paper · Jul 14, 2019

Functional inequalities for the heat flow on time-dependent metric measure spaces

Eva Kopfer, Karl-Theodor Sturm

math.AParXiv:1907.06184v2
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Abstract

We prove that synthetic lower Ricci bounds for metric measure spaces – both in the sense of Bakry-Émery and in the sense of Lott-Sturm-Villani – can be characterized by various functional inequalities including local Poincaré inequalities, local logarithmic Sobolev inequalities, dimension independent Harnack inequality, and logarithmic Harnack inequality. More generally, these equivalences will be proven in the setting of time-dependent metric measure spaces and will provide a characterization of super-Ricci flows of metric measure spaces.

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Submitted
Jul 14, 2019
Last revised
Jan 8, 2021 (v2)
Subjects
math.AP
DOI
10.1112/jlms.12452

Comments: Revised version

Ricci flowHarnack inequalitiesWeak & singular flowsLimit spaces & RCD

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 Super-Ricci Flows for Metric Measure Spaces (Karl-Theodor Sturm, 2016)2Sturm 2016 Heat Flows on Time-dependent Metric Measure Spaces and Super-Ricci Flows (Eva Kopfer and Karl-Theodor Sturm, 2016)1Kopfer 2016 Super Ricci flows for weighted graphs (Matthias Erbar and Eva Kopfer, 2018)7Erbar 2018 Metric Measure Spaces and Synthetic Ricci Bounds – Fundamental Concepts and Recent Developments (Karl-Theodor Sturm, 2024)3Sturm 2024 Synthetic notions of Ricci flow for metric measure spaces (Matthias Erbar, Zhenhao Li and Timo Schultz, 2025)6Erbar 2025 On Perelman's W-entropy and Shannon entropy power for super Ricci flows on metric measure spaces (Xiang-Dong Li, 2025)4Li 2025 Synthetic approaches to Ricci flows (Matthias Erbar, Marco Flaim, Eric Hupp et al., 2025)5Erbar 2025 Gaussian Volume Functional, Integral Scalar Curvature, and Minimal Super-Ricci Flows (Marco Flaim, Erik Hupp and Karl-Theodor Sturm, 2026)8Flaim 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Heat Flows on Time-dependent Metric Measure Spaces and Super-Ricci Flows

    Eva Kopfer and Karl-Theodor Sturm

    Topics Limit spaces & RCD, Weak & singular flows · Same authors Eva Kopfer, Karl-Theodor Sturm · Shared terms “dependent metric”, “metric measure”, “measure spaces”

  2. Super-Ricci Flows for Metric Measure Spaces

    Karl-Theodor Sturm

    Topics Limit spaces & RCD, Weak & singular flows · Same author Karl-Theodor Sturm · Shared terms “metric measure”, “measure spaces”, “time dependent”

  3. Metric Measure Spaces and Synthetic Ricci Bounds – Fundamental Concepts and Recent Developments

    Karl-Theodor Sturm

    Topics Harnack inequalities, Limit spaces & RCD, Weak & singular flows · Same author Karl-Theodor Sturm · Shared terms “metric measure”, “measure spaces”, “Ricci bounds”

  4. On Perelman's -entropy and Shannon entropy power for super Ricci flows on metric measure spaces

    Xiang-Dong Li

    Topics Harnack inequalities, Limit spaces & RCD, Weak & singular flows · Shared terms “metric measure”, “measure spaces”, “super Ricci”

  5. Synthetic approaches to Ricci flows

    Matthias Erbar, Marco Flaim, Eric Hupp et al.

    Topics Limit spaces & RCD, Weak & singular flows · Same author Karl-Theodor Sturm · Shared terms “metric measure”, “measure spaces”, “time dependent”

  6. Synthetic notions of Ricci flow for metric measure spaces

    Matthias Erbar, Zhenhao Li and Timo Schultz

    Topics Limit spaces & RCD, Weak & singular flows · Shared terms “dependent metric”, “metric measure”, “measure spaces”

  7. Super Ricci flows for weighted graphs

    Matthias Erbar and Eva Kopfer

    Topics Limit spaces & RCD, Weak & singular flows · Same author Eva Kopfer · Shared terms “time dependent”, “super Ricci”, “heat flow”

  8. Gaussian Volume Functional, Integral Scalar Curvature, and Minimal Super-Ricci Flows

    Marco Flaim, Erik Hupp and Karl-Theodor Sturm

    Topics Limit spaces & RCD, Weak & singular flows · Same author Karl-Theodor Sturm · Shared terms “metric measure”, “measure spaces”, “super Ricci”

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) Harnack inequalities Differential Harnack inequalities compare a solution with itself at different points and times. Hamilton's matrix Harnack inequality says curvature… Start with: On the parabolic kernel of the Schrödinger operator (Li et al., 1986) 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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

In the guidebook

  1. 6D.5
    Diffusions on Manifolds and CurvatureBook 6D · Diffusions, Finance and Computation

    Brownian motion on a Riemannian manifold, the Bakry–Émery calculus, curvature-dimension inequalities, gradient estimates and log-Sobolev, coupling, and Brownian motion along a Ricci flow.

  2. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  3. 12A.6
    𝓛-Geodesics and the Variational FormulasBook 12A · Entropy and Noncollapsing

    Perelman's §7.1 derived line by line: the first and second variation of 𝓛-length, 𝓛-Jacobi fields, adapted fields, Hamilton's Harnack expressions, and the differential inequalities for the reduced distance.