Paper · Nov 8, 2016

Heat Flows on Time-dependent Metric Measure Spaces and Super-Ricci Flows

Eva Kopfer, Karl-Theodor Sturm

math.DGarXiv:1611.02570v4
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Abstract

We study the heat equation on time-dependent metric measure spaces (as well as the dual and the adjoint heat equation) and prove existence, uniqueness and regularity. Of particular interest are properties which characterize the underlying space as a super Ricci flow as previously introduced by the second author. Our main result yields the equivalence of (i) dynamic convexity of the Boltzmann entropy on the (time-dependent) -Wasserstein space; (ii) monotonicity of -Kantorovich-Wasserstein distances under the dual heat flow acting on probability measures (backward in time); (iii) gradient estimates for the heat flow acting on functions (forward in time); (iv) a Bochner inequality involving the time-derivative of the metric. Moreover, we characterize the heat flow on functions as the unique forward EVI-flow for the (time-dependent) energy in -Hilbert space and the dual heat flow on probability measures as the unique backward EVI-flow for the (time-dependent) Boltzmann entropy in -Wasserstein space.

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Submitted
Nov 8, 2016
Last revised
Dec 20, 2017 (v4)
Subjects
math.DG, math.AP, math.FA, math.PR

Comments: This is an enhanced version of the previous paper "Super-Ricci flows for Metric Measure Spaces. II". We changed the title and included two examples in the introduction. Moreover we revised the arguments in Section 2. To appear in Comm. Pure Appl. Math

Ricci flowWeak & singular flowsLimit spaces & RCDStability & uniqueness

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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 Super Ricci flows for weighted graphs (Matthias Erbar and Eva Kopfer, 2018)3Erbar 2018 Functional inequalities for the heat flow on time-dependent metric measure spaces (Eva Kopfer and Karl-Theodor Sturm, 2019)1Kopfer 2019 Heat flow on time-dependent manifolds (Beomjun Choi, Jianhui Gao, Robert Haslhofer et al., 2020)8Choi 2020 Metric Measure Spaces and Synthetic Ricci Bounds – Fundamental Concepts and Recent Developments (Karl-Theodor Sturm, 2024)7Sturm 2024 Synthetic notions of Ricci flow for metric measure spaces (Matthias Erbar, Zhenhao Li and Timo Schultz, 2025)5Erbar 2025 Synthetic approaches to Ricci flows (Matthias Erbar, Marco Flaim, Eric Hupp et al., 2025)6Erbar 2025 Gaussian Volume Functional, Integral Scalar Curvature, and Minimal Super-Ricci Flows (Marco Flaim, Erik Hupp and Karl-Theodor Sturm, 2026)4Flaim 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Functional inequalities for the heat flow on time-dependent metric measure spaces

    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 “super Ricci”, “metric measure”, “measure spaces”

  3. Super Ricci flows for weighted graphs

    Matthias Erbar and Eva Kopfer

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

  4. 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 “super Ricci”, “time dependent”, “metric measure”

  5. 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”

  6. 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 “time dependent”, “heat flow”, “metric measure”

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

    Karl-Theodor Sturm

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

  8. Heat flow on time-dependent manifolds

    Beomjun Choi, Jianhui Gao, Robert Haslhofer et al.

    Topics Weak & singular flows, Stability & uniqueness · Shared terms “time dependent”, “heat flow”

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) 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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

In the guidebook

  1. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.