Paper · May 11, 2026

Distance between minimal surfaces and flows

Tobias Holck Colding, William P. Minicozzi

math.DGWider flowsarXiv:2605.10589
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Abstract

We will show that the distance between two minimal hypersurfaces is a Lipschitz continuous supersolution, in the viscosity sense, of a natural elliptic partial differential equation. This not only recovers several well-known properties of minimal hypersurfaces, but also encodes substantially richer information. Moreover, if the reference hypersurface is allowed to evolve by mean curvature flow, one obtains comparably strong estimates for a corresponding parabolic PDE, leading in particular to local Harnack inequalities for the distance. There is even a fully parabolic extension in which both hypersurfaces evolve. The problem of tracking the distance between two evolving hypersurfaces arises naturally in a wide range of settings.

Mean curvature flowHarnack inequalitiesMinimal & CMC surfaces

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 2021, 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 Optimal growth bounds for eigenfunctions (Tobias Holck Colding and William P. Minicozzi, 2021)2Colding 2021 Singularities of Ricci flow and diffeomorphisms (Tobias Holck Colding and William P. Minicozzi, 2021)8Colding 2021 A strong Frankel Theorem for shrinkers (Tobias Holck Colding and William P. Minicozzi, 2023)5Colding 2023 Ancient mean curvature flows from minimal hypersurfaces (Yongheng Han, 2023)6Han 2023 Quantitative uniqueness for mean curvature flow (Tobias Holck Colding and William P. Minicozzi, 2025)4Colding 2025 When can minimal hypersurfaces be connected by mean curvature flow? (Jingwen Chen and Pedro Gaspar, 2025)1Chen 2025 Classification of Minimal Surfaces and Solitons to the Mean Curvature Flow in ℍ³ as Translation Surfaces (Tarcios Andrey Ferreira and João Paulo dos Santos, 2025)7Ferreira 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. When can minimal hypersurfaces be connected by mean curvature flow?

    Jingwen Chen and Pedro GasparWider flows PDF

    Topics Minimal & CMC surfaces · Shared terms “minimal hypersurface”

  2. Optimal growth bounds for eigenfunctions

    Tobias Holck Colding and William P. Minicozzi PDF

    Topics Mean curvature flow · Same authors Tobias Holck Colding, William P. Minicozzi · Shared terms “wide range”, “PDE”

  3. Harnack Inequality for -Mean Curvature Flow

    Xiang-Dong Li and Qi YanWider flows PDF

    Topics Harnack inequalities

  4. Quantitative uniqueness for mean curvature flow

    Tobias Holck Colding and William P. MinicozziWider flows PDF

    Topics Mean curvature flow · Same authors Tobias Holck Colding, William P. Minicozzi

  5. A strong Frankel Theorem for shrinkers

    Tobias Holck Colding and William P. Minicozzi PDF

    Topics Mean curvature flow · Same authors Tobias Holck Colding, William P. Minicozzi

  6. Ancient mean curvature flows from minimal hypersurfaces

    Yongheng HanWider flows PDF

    Topics Minimal & CMC surfaces · Shared terms “minimal hypersurface”

  7. Classification of Minimal Surfaces and Solitons to the Mean Curvature Flow in as Translation Surfaces

    Tarcios Andrey Ferreira and João Paulo dos SantosWider flows PDF

    Topics Minimal & CMC surfaces

  8. Singularities of Ricci flow and diffeomorphisms

    Tobias Holck Colding and William P. Minicozzi PDF

    Same authors Tobias Holck Colding, William P. Minicozzi · Shared terms “PDE”

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) 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) Minimal & CMC surfaces Minimal surfaces are critical points of area, with zero mean curvature; constant mean curvature (CMC) surfaces are critical for area at fixed volume. Start with: On the structure of manifolds with positive scalar curvature (Schoen et al., 1979)

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.