Paper · Jan 28, 2019

Li-Yau gradient estimates for curvature flows in positively curved manifolds

Paul Bryan, Heiko Kröner, Julian Scheuer

math.DGWider flowsarXiv:1901.09763
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Abstract

We prove differential Harnack inequalities for flows of strictly convex hypersurfaces by powers , , of the mean curvature in Einstein manifolds with a positive lower bound on the sectional curvature. We assume that this lower bound is sufficiently large compared to the derivatives of the curvature tensor of the ambient space and that the mean curvature of the initial hypersurface is sufficiently large compared to the ambient geometry. We also obtain some new Harnack inequalities for more general curvature flows in the sphere, as well as a monotonicity estimate for the mean curvature flow in non-negatively curved, locally symmetric spaces.

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Submitted
Jan 28, 2019
Subjects
math.DG, math.AP
Journal
Methods Appl. Anal. 27, no. 4, (2021), p. 341-358
DOI
10.4310/MAA.2020.v27.n4.a2

Comments: 16 pages

Mean curvature flowOther curvature flowsEinstein metricsHarnack inequalities

Its neighbourhood

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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 Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature (Heiko Kröner and Julian Scheuer, 2017)2Kröner 2017 Harnack inequalities for curvature flows in Riemannian and Lorentzian manifolds (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2017)1Bryan 2017 Negatively Curved Three-Manifolds, Hyperbolic Metrics, Isometric Embeddings In Minkowski Space And The Cross Curvature Flow (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2019)4Bryan 2019 Differential Harnack inequalities via Concavity of the arrival time (Theodora Bourni and Mat Langford, 2019)7Bourni 2019 Parabolic approaches to curvature equations (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2020)8Bryan 2020 Generalized Harnack Inequality for Mean Curvature Flow and Ancient Solutions (Junyoung Park, 2025)6Park 2025 Harnack Inequality for f-Mean Curvature Flow (Xiang-Dong Li and Qi Yan, 2025)3Li 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Harnack inequalities for curvature flows in Riemannian and Lorentzian manifolds

    Paul Bryan, Mohammad N. Ivaki 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. Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature

    Heiko Kröner and Julian ScheuerWider flows

    Topics Other curvature flows · Same authors Heiko Kröner, Julian Scheuer · Shared terms “initial hypersurface”, “strictly convex”

  3. Harnack Inequality for -Mean Curvature Flow

    Xiang-Dong Li and Qi YanWider flows

    Topics Harnack inequalities · Shared terms “Yau”

  4. Negatively Curved Three-Manifolds, Hyperbolic Metrics, Isometric Embeddings In Minkowski Space And The Cross Curvature Flow

    Paul Bryan, Mohammad N. Ivaki and Julian Scheuer

    Topics Other curvature flows · Same authors Paul Bryan, Julian Scheuer · Shared terms “negatively curved”

  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, Julian Scheuer

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

    Junyoung ParkWider flows

    Topics Harnack inequalities · Shared terms “differential Harnack”, “lower bound”

  7. Differential Harnack inequalities via Concavity of the arrival time

    Theodora Bourni and Mat LangfordWider flows

    Topics Harnack inequalities · Shared terms “differential Harnack”

  8. Parabolic approaches to curvature equations

    Paul Bryan, Mohammad N. Ivaki and Julian ScheuerWider flows

    Topics Other curvature flows · Same authors Paul Bryan, Julian Scheuer · Shared terms “strictly convex”

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.