Paper · Jul 7, 2026

Infinity-harmonic functions and inverse mean curvature flow clusters

As on arXiv

Abstract

An $\infty$ -harmonic function is a viscosity solution of $\nabla^{2} u (\nabla u, \nabla u) = 0$ , or equivalently, an absolute minimizer of $∥\nabla u ∥_{L^{\infty}}$ . We prove a variety of new structural and regularity results in two dimensions, including: 1. $\infty$ -harmonic functions in domains of $R^{2}$ are $C^{1, 1/3}$ . 2. Critical points are isolated, and at each critical point, the solution has a unique quasiradial blow-up. 3. Entire solutions with polynomial growth have unique quasiradial blow-downs, and are determined by their Fourier modes at infinity. These results are consequences of a new theory relating $\infty$ -harmonic functions to inverse mean curvature flow (IMCF) clusters – which are piecewise weak solutions of IMCF with common obstacle-type boundary conditions on the interfaces (a simple example is an embedded family of cuspidal curves evolving by inverse curvature). This connection arises as the $p \to \infty$ limit of the classical duality between $p$ -harmonic and $q$ -harmonic functions in $R^{2}$ , where $\frac{1}{p} + \frac{1}{q} = 1$ .

Comments: 151 pages, 42 figures. Comments are welcome!

Mean curvature flow Other curvature flows Inverse mean curvature flow Singularities Weak & singular flows Boundaries & non-compact

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.

Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other

Inverse mean curvature flow with outer obstacle
Kai Xu2024Wider flows PDF

Topics Inverse mean curvature flow, Weak & singular flows, Boundaries & non-compact · Same author Kai Xu · Shared terms “obstacle”, “boundary condition”
The streamlines of $\infty$ -harmonic functions obey the inverse mean curvature flow
Roger Moser2021Wider flows PDF

Topics Inverse mean curvature flow, Weak & singular flows, Other curvature flows · Shared terms “harmonic functions”
Global weak solutions for the inverse mean curvature flow in the Heisenberg group
Adriano Pisante and Eugenio Vecchi2024Wider flows PDF

Topics Inverse mean curvature flow, Weak & singular flows, Other curvature flows · Shared terms “harmonic functions”, “IMCF”
Isoperimetry and the properness of weak inverse mean curvature flow
Kai Xu2023Wider flows PDF

Topics Inverse mean curvature flow, Other curvature flows · Same author Kai Xu
3-Manifolds with positive scalar curvature and bounded geometry
Otis Chodosh, Yi Lai and Kai Xu2025Wider flows PDF

Topics Inverse mean curvature flow, Weak & singular flows, Other curvature flows · Same author Kai Xu
A topological gap theorem for the $π_{2}$ -systole of positive scalar curvature 3-manifolds
Kai Xu2023Wider flows PDF

Topics Inverse mean curvature flow, Other curvature flows · Same author Kai Xu
On weak inverse mean curvature flow and Minkowski-type inequalities in hyperbolic space
Brian Harvie2024Wider flows PDF

Topics Inverse mean curvature flow, Weak & singular flows, Boundaries & non-compact · Shared terms “IMCF”
On the evolution of hypersurfaces along their inverse spacetime mean curvature
Gerhard Huisken and Markus Wolff2022Wider flows PDF

Topics Inverse mean curvature flow, Weak & singular flows, Boundaries & non-compact

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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993)

In the guidebook

12C.9
After PerelmanBook 12C · Extinction and the Proof
Singular flows, the Smale conjecture, and Bamler’s theory.
6A.13
Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives
Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.
6C.8
Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods
Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.

Infinity-harmonic functions and inverse mean curvature flow clusters

Abstract

Connected papers

Background reading

In the guidebook

Keyboard shortcuts

Anywhere

Lists of papers

Guidebook chapters