Paper · Nov 7, 2026

Local regularity and p-harmonic approximation for weak inverse mean curvature flow

Wei-Chun Chen, Chun-Kai Lien

math.APWider flowsarXiv:2611.10533
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Abstract

We prove local Lipschitz regularity for locally bounded solutions of the calibration equations for inverse mean curvature flow. We also approximate every prescribed variational weak flow locally by Moser's logarithmic transforms of positive -harmonic functions. On a fixed smooth domain, with the trace of a calibrated Lipschitz solution prescribed, both the uniform error and the absolute difference of total variations are as . In particular, the approximations converge strictly in .

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Submitted
Nov 7, 2026
Subjects
math.AP, math.DG

Comments: 19 pages

Mean curvature flowOther curvature flowsInverse mean curvature flowWeak & singular flows

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  1. The streamlines of -harmonic functions obey the inverse mean curvature flow

    Roger MoserWider flows

    Topics Inverse mean curvature flow, Weak & singular flows, Other curvature flows · Shared terms “harmonic functions”, “approximation”

  2. Global weak solutions for the inverse mean curvature flow in the Heisenberg group

    Adriano Pisante and Eugenio VecchiWider flows

    Topics Inverse mean curvature flow, Weak & singular flows, Other curvature flows · Shared terms “Moser”, “harmonic functions”, “Lipschitz”

  3. Infinity-harmonic functions and inverse mean curvature flow clusters

    Kai XuWider flows

    Topics Inverse mean curvature flow, Weak & singular flows, Other curvature flows · Shared terms “harmonic functions”, “absolute”

  4. Inverse Mean Curvature Flow with Singularities

    Beomjun Choi and Pei-Ken HungWider flows

    Topics Inverse mean curvature flow, Weak & singular flows, Other curvature flows · Shared terms “Lipschitz”

  5. Inverse mean curvature flow with outer obstacle

    Kai XuWider flows

    Topics Inverse mean curvature flow, Weak & singular flows, Other curvature flows

  6. Self similar solutions to the inverse mean curvature flow in

    Jui-En ChangWider flows

    Topics Inverse mean curvature flow, Other curvature flows

  7. Inverse Mean Curvature Flow over Non-Star-Shaped Surfaces

    Brian HarvieWider flows

    Topics Inverse mean curvature flow, Weak & singular flows, Other curvature flows · Shared terms “variational weak”

  8. Isoperimetry and the properness of weak inverse mean curvature flow

    Kai XuWider flows

    Topics Inverse mean curvature flow, Other curvature flows · Shared terms “weak inverse”

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) 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) 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.