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

Adriano Pisante; Eugenio Vecchi

Paper · Jun 21, 2024

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

Adriano Pisante, Eugenio Vecchi

math.APWider flowsarXiv:2406.15123v3

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Abstract

We consider the inverse mean curvature flow (IMCF) in the Heisenberg group $(\He^{n}, d_{ε})$ , where $d_{ε}$ is distance associated to either $∣ \cdot ∣_{ε}$ , $ε > 0$ , the natural family of left-invariant Riemannian metrics, or with their sub-Riemannian counterparts for $ε = 0$ . For $Ω \subseteq \He^{n}$ an open set with smooth boundary $Σ_{0} = \partial Ω$ satisfying a uniform exterior gauge-ball condition and bounded complement we show existence of a global weak IMCF of generalized hypersurfaces ${Σ_{s}^{ε}}_{s \geq 0} \subseteq H^{n}$ which are level sets of a proper globally Lipschitz function with logarithmic growth at infinity. Here, both in the Riemannian and in the sub-Riemannian setting, we adopt the weak formulation introduced by Huisken and Ilmanen in, following the approach in due to Moser and based on the the link between IMCF and $p$ -harmonic functions.

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Submitted: Jun 21, 2024
Last revised: Dec 23, 2025 (v3)
Subjects: math.AP

Comments: A new section on the relevant Bochner inequality has been added. A relevant error in the computation of the Ricci curvature has been fixed, together with the proofs of Theorem 1.2 and 1.4

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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12C.9
After PerelmanBook 12C · Extinction and the Proof
Singular flows, the Smale conjecture, and Bamler’s theory.
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.

[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.

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

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