Paper · Mar 22, 2017

Harnack inequalities for curvature flows in Riemannian and Lorentzian manifolds

Paul Bryan, Mohammad N. Ivaki, Julian Scheuer

math.DGWider flowsarXiv:1703.07493
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Abstract

We obtain Harnack estimates for a class of curvature flows in Riemannian manifolds of constant non-negative sectional curvature as well as in the Lorentzian Minkowski and de Sitter spaces. Furthermore, we prove a Harnack estimate with a bonus term for mean curvature flow in locally symmetric Riemannian Einstein manifold of non-negative sectional curvature. Using a concept of "duality" for strictly convex hypersurfaces, we also obtain a new type of inequalities, so-called "pseudo"-Harnack inequalities, for expanding flows in the sphere and in the hyperbolic space.

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Submitted
Mar 22, 2017
Subjects
math.DG, math.AP
Journal
J. Reine Angew. Math. 764 (2020), 71-109
DOI
10.1515/crelle-2019-0006
Mean curvature flowOther curvature flowsEinstein metricsHarnack inequalitiesPhysics

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 the classification of ancient solutions to curvature flows on the sphere (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2016)5Bryan 2016 A unified flow approach to smooth, even Lₚ-Minkowski problems (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2016)7Bryan 2016 Li-Yau gradient estimates for curvature flows in positively curved manifolds (Paul Bryan, Heiko Kröner and Julian Scheuer, 2019)1Bryan 2019 The Minkowski inequality in de Sitter space (Julian Scheuer, 2019)4Scheuer 2019 Parabolic approaches to curvature equations (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2020)3Bryan 2020 Christoffel-Minkowski flows (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2020)6Bryan 2020 Generalized Harnack Inequality for Mean Curvature Flow and Ancient Solutions (Junyoung Park, 2025)8Park 2025 Harnack Inequality for f-Mean Curvature Flow (Xiang-Dong Li and Qi Yan, 2025)2Li 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Li-Yau gradient estimates for curvature flows in positively curved manifolds

    Paul Bryan, Heiko Kröner and Julian ScheuerWider flows

    Topics Harnack inequalities, Einstein metrics, Other curvature flows · Same authors Paul Bryan, Julian Scheuer · Shared terms “locally symmetric”, “sectional curvature”, “strictly convex”

  2. Harnack Inequality for -Mean Curvature Flow

    Xiang-Dong Li and Qi YanWider flows

    Topics Harnack inequalities · Shared terms “Harnack estimate”

  3. Parabolic approaches to curvature equations

    Paul Bryan, Mohammad N. Ivaki and Julian ScheuerWider flows

    Topics Physics, Other curvature flows · Same authors Paul Bryan, Mohammad N. Ivaki, Julian Scheuer · Shared terms “Sitter space”, “Lorentzian”, “strictly convex”

  4. The Minkowski inequality in de Sitter space

    Julian ScheuerWider flows

    Topics Physics, Other curvature flows · Same author Julian Scheuer · Shared terms “Sitter space”, “Lorentzian”, “Minkowski”

  5. On the classification of ancient solutions to curvature flows on the sphere

    Paul Bryan, Mohammad N. Ivaki and Julian ScheuerWider flows

    Topics Other curvature flows · Same authors Paul Bryan, Mohammad N. Ivaki, Julian Scheuer

  6. Christoffel-Minkowski flows

    Paul Bryan, Mohammad N. Ivaki and Julian ScheuerWider flows

    Topics Other curvature flows · Same authors Paul Bryan, Mohammad N. Ivaki, Julian Scheuer · Shared terms “Minkowski”

  7. A unified flow approach to smooth, even -Minkowski problems

    Paul Bryan, Mohammad N. Ivaki and Julian ScheuerWider flows

    Topics Other curvature flows · Same authors Paul Bryan, Mohammad N. Ivaki, Julian Scheuer · Shared terms “Minkowski”

  8. Generalized Harnack Inequality for Mean Curvature Flow and Ancient Solutions

    Junyoung ParkWider flows

    Topics Harnack inequalities

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.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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)

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.