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

Roger Moser

Paper · Aug 12, 2021

The streamlines of $\infty$ -harmonic functions obey the inverse mean curvature flow

Roger Moser

math.APWider flowsarXiv:2108.05807

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Abstract

Given an $\infty$ -harmonic function $u_{\infty}$ on a domain $Ω \subseteq R^{2}$ , consider the function $w = - lo g ∣\nabla u_{\infty} ∣$ . If $u_{\infty} \in C^{2} (Ω)$ with $\nabla u_{\infty} \neq = 0$ and $\nabla∣\nabla u_{\infty} ∣ \neq = 0$ , then it is easy to check that (1) the streamlines of $u_{\infty}$ are the level sets of $w$ and (2) $w$ solves the level set formulation of the inverse mean curvature flow. For less regular solutions, neither statement is true in general, but even so, $w$ is still a weak solution of the inverse mean curvature flow under far weaker assumptions. This is proved through an approximation of $u_{\infty}$ by $p$ -harmonic functions, the use of conjugate $p^{'}$ -harmonic functions, and the known connection of the latter with the inverse mean curvature flow. A statement about the regularity of $∣\nabla u_{\infty} ∣$ arises as a by-product.

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Submitted: Aug 12, 2021
Subjects: math.AP
Journal: Communications in Partial Differential Equations, 47 (2022), no. 11, 2124-2145
DOI: 10.1080/03605302.2022.2109487

Mean curvature flow Other curvature flows Inverse mean curvature flow Weak & singular flows

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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) 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)

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[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.

The streamlines of $\infty$ -harmonic functions obey the inverse mean curvature flow

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