Paper · Feb 11, 2025

Generalized Harnack Inequality for Mean Curvature Flow and Ancient Solutions

Junyoung Park

math.DGWider flowsarXiv:2502.07210v2
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Abstract

The goal of this paper is to relax convexity assumption on some classical results in mean curvature flow. In the first half of the paper, we prove a generalized version of Hamilton's differential Harnack inequality which holds for mean convex solutions to mean curvature flow with a lower bound on where is the smallest principal curvature. Then, we use classical maximum principle to provide several characterizations of family of shrinking spheres for closed, mean convex ancient solution to mean curvature flow with a lower bound on for some , where are the principal curvatures.

Comments: 11 pages

Mean curvature flowAncient solutionsHarnack 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 2021, plus the canon.

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
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    Xiang-Dong Li and Qi YanWider flows PDF

    Topics Harnack inequalities · Shared terms “Hamilton”

  2. The Harnack inequality without convexity for curve shortening flow

    Arjun Sobnack and Peter M. ToppingWider flows PDF

    Topics Harnack inequalities · Shared terms “convex solutions”, “Hamilton”

  3. Monotonicity of Perelman -Entropy of Mean Curvature Flow

    Xiang-Dong Li and Qi Yan PDF

    Topics Harnack inequalities · Shared terms “Hamilton”

  4. Ancient mean curvature flow asymptotic to a minimal quadratic cone

    Junyoung ParkWider flows PDF

    Topics Ancient solutions · Same author Junyoung Park

  5. Lectures on curve shortening flow

    Robert HaslhoferWider flows PDF

    Topics Harnack inequalities · Shared terms “maximum principle”

  6. A Bernstein theorem of ancient solutions to mean curvature flow

    Xiangzhi CaoWider flows PDF

    Topics Ancient solutions

  7. Parabolic frequency monotonicity for two nonlinear equations under Ricci flow

    Chuanhuan Li, Yi Li, Kairui Xu et al. PDF

    Topics Harnack inequalities · Shared terms “lower bound”

  8. A direct approach to sharp Li-Yau Estimates on closed manifolds with negative Ricci lower bound

    Xingyu Song, Ling Wu and Meng Zhu PDF

    Topics Harnack inequalities · Shared terms “classical maximum”, “lower bound”, “maximum principle”

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. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) 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. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  2. 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.

  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.