Paper · Mar 1, 2016

Smoothing and non-smoothing via a flow tangent to the Ricci flow

Matthias Erbar, Nicolas Juillet

math.DGarXiv:1603.00280
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Abstract

We study a transformation of metric measure spaces introduced by Gigli and Mantegazza consisting in replacing the original distance with the length distance induced by the transport distance between heat kernel measures. We study the smoothing effect of this procedure in two important examples. Firstly, we show that in the case of some Euclidean cones, a singularity persists at the apex. Secondly, we generalize the construction to a sub-Riemannian manifold, namely the Heisenberg group, and show that it regularizes the space instantaneously to a smooth Riemannian manifold.

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Submitted
Mar 1, 2016
Subjects
math.DG, math.MG

Comments: 33 pages, 0 figure

Ricci flowSingularitiesHeat kernelWeak & 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 weighted graphs (Matthias Erbar and Eva Kopfer, 2018)4Erbar 2018 Ricci limit flows and weak solutions (Beomjun Choi and Robert Haslhofer, 2021)8Choi 2021 Heat kernel on Ricci shrinkers (II) (Yu Li and Bing Wang, 2023)5Li 2023 Synthetic notions of Ricci flow for metric measure spaces (Matthias Erbar, Zhenhao Li and Timo Schultz, 2025)1Erbar 2025 Synthetic approaches to Ricci flows (Matthias Erbar, Marco Flaim, Eric Hupp et al., 2025)3Erbar 2025 Gromov-Hausdorff convergence of time-slices of singular Ricci flows in dimension three (Yu Li, 2026)7Li 2026 Second-Order Departure of the Gigli–Mantegazza Flow from Ricci Flow (Dongwoo Gang, 2026)2Gang 2026 Heat kernel on Ricci shrinker metric measure spaces (Bing Wang and Jie Wang, 2026)6Wang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Synthetic notions of Ricci flow for metric measure spaces

    Matthias Erbar, Zhenhao Li and Timo Schultz

    Topics Limit spaces & RCD, Weak & singular flows · Same author Matthias Erbar · Shared terms “metric measure”, “measure spaces”, “transport”

  2. Second-Order Departure of the Gigli–Mantegazza Flow from Ricci Flow

    Dongwoo Gang

    Topics Heat kernel, Limit spaces & RCD · Shared terms “Gigli”, “Mantegazza”, “Riemannian manifold”

  3. Synthetic approaches to Ricci flows

    Matthias Erbar, Marco Flaim, Eric Hupp et al.

    Topics Limit spaces & RCD, Weak & singular flows · Same author Matthias Erbar · Shared terms “metric measure”, “measure spaces”, “transport”

  4. Super Ricci flows for weighted graphs

    Matthias Erbar and Eva Kopfer

    Topics Limit spaces & RCD, Weak & singular flows, Singularities · Same author Matthias Erbar · Shared terms “transport”, “metric measure”, “measure spaces”

  5. Heat kernel on Ricci shrinkers (II)

    Yu Li and Bing Wang

    Topics Heat kernel, Limit spaces & RCD

  6. Heat kernel on Ricci shrinker metric measure spaces

    Bing Wang and Jie Wang

    Topics Heat kernel, Limit spaces & RCD · Shared terms “metric measure”, “measure spaces”

  7. Gromov-Hausdorff convergence of time-slices of singular Ricci flows in dimension three

    Yu Li

    Topics Heat kernel, Limit spaces & RCD, Weak & singular flows · Shared terms “kernel measures”, “Riemannian manifold”

  8. Ricci limit flows and weak solutions

    Beomjun Choi and Robert Haslhofer

    Topics Heat kernel, Limit spaces & RCD, 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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Heat kernel The heat kernel of the conjugate heat equation, coupled to Ricci flow, is the probe Perelman and Bamler use to measure geometry. 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)

In the guidebook

  1. 6A.3
    The Heat Equation on ℝⁿBook 6A · The Heat Equation and Its Relatives

    The heat kernel, smoothing, uniqueness, Brownian motion, and a Gaussian that runs to Perelman.

  2. 6A.4
    Fourier Methods and Scale SpaceBook 6A · The Heat Equation and Its Relatives

    Fourier's own method on the circle and the line, smoothing as the decay of high frequencies, and the heat equation as the mathematics of blurring images.

  3. 6A.12
    Entropy, Information and DiffusionBook 6A · The Heat Equation and Its Relatives

    Entropy along the heat flow, the log-Sobolev inequality and Li–Yau: the ancestors of Perelman’s entropy.