Paper · Dec 13, 2019

Differential Harnack inequalities via Concavity of the arrival time

Theodora Bourni, Mat Langford

math.DGWider flowsarXiv:1912.06623v2
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Abstract

We present a simple connection between differential Harnack inequalities for hypersurface flows and natural concavity properties of their time-of-arrival functions. We prove these concavity properties directly for a large class of flows by applying a concavity maximum principle argument to the corresponding level set flow equations. In particular, this yields a short proof of Hamilton's differential Harnack inequality for mean curvature flow and, more generally, Andrews' differential Harnack inequalities for certain "-inverse-concave" flows.

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Submitted
Dec 13, 2019
Last revised
Mar 30, 2021 (v2)
Subjects
math.DG
Mean curvature flowHarnack inequalitiesWeak & singular flows

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 Harnack inequalities for Witten Laplacian on Riemannian manifolds with super Ricci flows (Songzi Li and Xiang-Dong Li, 2017)6Li 2017 Hamilton differential Harnack inequality and W-entropy for Witten Laplacian on Riemannian manifolds (Songzi Li and Xiang-Dong Li, 2017)2Li 2017 New differential Harnack inequalities for nonlinear heat equations (Jia-Yong Wu, 2018)5Wu 2018 Li-Yau gradient estimates for curvature flows in positively curved manifolds (Paul Bryan, Heiko Kröner and Julian Scheuer, 2019)8Bryan 2019 Convex ancient solutions to mean curvature flow (Theodora Bourni, Mat Langford and Giuseppe Tinaglia, 2019)1Bourni 2019 Generalized Harnack Inequality for Mean Curvature Flow and Ancient Solutions (Junyoung Park, 2025)3Park 2025 Harnack Inequality for f-Mean Curvature Flow (Xiang-Dong Li and Qi Yan, 2025)4Li 2025 The Harnack inequality without convexity for curve shortening flow (Arjun Sobnack and Peter M. Topping, 2026)7Sobnack 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Convex ancient solutions to mean curvature flow

    Theodora Bourni, Mat Langford and Giuseppe TinagliaWider flows

    Topics Harnack inequalities · Same authors Theodora Bourni, Mat Langford · Shared terms “differential Harnack”, “directly”

  2. Hamilton differential Harnack inequality and -entropy for Witten Laplacian on Riemannian manifolds

    Songzi Li and Xiang-Dong Li

    Topics Harnack inequalities, Weak & singular flows · Shared terms “differential Harnack”, “Hamilton differential”

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

    Junyoung ParkWider flows

    Topics Harnack inequalities · Shared terms “differential Harnack”, “Hamilton differential”, “maximum principle”

  4. Harnack Inequality for -Mean Curvature Flow

    Xiang-Dong Li and Qi YanWider flows

    Topics Harnack inequalities · Shared terms “Hamilton”

  5. New differential Harnack inequalities for nonlinear heat equations

    Jia-Yong Wu

    Topics Harnack inequalities · Shared terms “differential Harnack”

  6. On Harnack inequalities for Witten Laplacian on Riemannian manifolds with super Ricci flows

    Songzi Li and Xiang-Dong Li

    Topics Harnack inequalities, Weak & singular flows · Shared terms “Hamilton”

  7. The Harnack inequality without convexity for curve shortening flow

    Arjun Sobnack and Peter M. ToppingWider flows

    Topics Harnack inequalities · Shared terms “Hamilton”

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

    Paul Bryan, Heiko Kröner and Julian ScheuerWider flows

    Topics Harnack inequalities · Shared terms “differential Harnack”

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) 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) 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. 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.