Paper · Jul 2, 2026

Higher regularity of the inverse anisotropic mean curvature flow

Chaoqun Gao, Yong Wei, Rong Zhou

math.DGWider flowsarXiv:2607.02067
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Abstract

We prove an anisotropic analogue of the higher regularity theorem of Huisken and Ilmanen for inverse mean curvature flow. For an arbitrary smooth Minkowski norm, we first prove a Huisken–Ilmanen type Harnack estimate for smooth closed strictly star-shaped solutions. We then construct global smooth solutions starting from strictly star-shaped hypersurfaces with bounded nonnegative weak anisotropic mean curvature. Combining this construction with the asymptotic theory for weak inverse anisotropic mean curvature flow, we show that weak solutions starting from bounded smooth initial sets become smooth outside a compact set.

Comments: 31pages

Mean curvature flowOther curvature flowsInverse mean curvature flowAnisotropic & nonlocal flowsHarnack inequalitiesWeak & singular flows

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    Topics Anisotropic & nonlocal flows, Weak & singular flows · Same authors Chaoqun Gao, Yong Wei, Rong Zhou · Shared terms “inverse anisotropic”, “weak inverse”, “Huisken”

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    Topics Inverse mean curvature flow, Other curvature flows · Same authors Chaoqun Gao, Yong Wei, Rong Zhou · Shared terms “weak inverse”, “Huisken”, “Ilmanen”

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    Haizhong Li, Ruixuan Li and Changwei XiongWider flows PDF

    Topics Anisotropic & nonlocal flows, Inverse mean curvature flow, Other curvature flows · Shared terms “star shaped”, “compact set”, “weak inverse”

  4. Weak solutions of Anisotropic (and crystalline) inverse mean curvature flow as limits of -capacitary potentials

    Esther Cabezas-Rivas, Salvador Moll and Marcos SoleraWider flows PDF

    Topics Harnack inequalities, Anisotropic & nonlocal flows, Inverse mean curvature flow · Shared terms “Huisken”, “Ilmanen”

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    Tianci Luo, Yong Wei, Ruijia Zhang et al.Wider flows PDF

    Topics Other curvature flows · Same authors Yong Wei, Rong Zhou · Shared terms “closed strictly”, “smooth closed”

  6. Contraction of hypersurfaces with positive sectional curvature in hyperbolic space

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    Topics Other curvature flows · Same authors Yong Wei, Rong Zhou

  7. Alexandrov–Fenchel inequalities for convex domains in the sphere

    Tianci Luo, Yong Wei and Rong ZhouWider flows PDF

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  8. Free boundary flows by powers of the Gauss curvature in the unit ball

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    Topics Other curvature flows · Same authors Yong Wei, Rong Zhou

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) 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) Inverse mean curvature flow Wider flows A hypersurface expands with normal speed 1/H, the reciprocal of its mean curvature. Huisken and Ilmanen's weak version keeps the Hawking mass… Start with: Flow of nonconvex hypersurfaces into spheres (Gerhardt, 1990) Anisotropic & nonlocal flows Wider flows When surface energy depends on direction, as in crystals, mean curvature flow becomes anisotropic, and the Wulff shape replaces the round sphere. Start with: Crystalline variational problems (Taylor, 1978)

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.