Paper · Oct 9, 2025

Asymptotic behaviour of the weak inverse anisotropic mean curvature flow

Chaoqun Gao, Yong Wei, Rong Zhou

math.DGWider flowsarXiv:2510.08168v2
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Abstract

We first establish a local gradient estimate for anisotropic -harmonic functions. A key feature of our estimate is that the constant remains bounded as ; consequently, in the limit , this estimate yields the local gradient estimate for weak solutions of the inverse anisotropic mean curvature flow (IAMCF). As an application, we show that the weak IAMCF is asymptotic to the expanding Wulff shape solution at the infinity, thereby extending the result of Huisken and Ilmanen in [8] to the anisotropic case.

Comments: 22 pages, minor revision, add references

Mean curvature flowAnisotropic & nonlocal flowsWeak & singular flows

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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
  1. Higher regularity of the inverse anisotropic mean curvature flow

    Chaoqun Gao, Yong Wei and Rong ZhouWider flows PDF

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

  2. Threshold dynamics approximation schemes for anisotropic mean curvature flows with a forcing term

    Bohdan Bulanyi and Berardo RuffiniWider flows PDF

    Topics Anisotropic & nonlocal flows · Shared terms “Wulff shape”, “extending”

  3. A variational approach to the volume-preserving anisotropic mean curvature flow in 2D

    Andrea Kubin, Domenico Angelo La Manna and Enrico PasqualettoWider flows PDF

    Topics Anisotropic & nonlocal flows, Weak & singular flows

  4. Weinstock inequalities for outward-minimizing domains

    Chaoqun Gao, Yong Wei and Rong ZhouWider flows PDF

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

  5. Anisotropic Gauss curvature flow of complete non-compact graphs

    Shujing Pan and Yong WeiWider flows PDF

    Topics Anisotropic & nonlocal flows · Same author Yong Wei · Shared terms “Wulff shape”

  6. Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow

    Tim Laux, Kerrek Stinson and Clemens UllrichWider flows PDF

    Topics Anisotropic & nonlocal flows, Weak & singular flows

  7. 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 Anisotropic & nonlocal flows, Weak & singular flows · Shared terms “Wulff shape”, “harmonic functions”, “Huisken”

  8. Area-Preserving Anisotropic Mean Curvature Flow in Two Dimensions

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    Topics Anisotropic & nonlocal flows · Shared terms “Wulff shape”

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) 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) 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. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.