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

2024/06/21 by Adriano Pisante, Pisante, Adriano, Eugenio Vecchi +1 · 1 citation
Mathematics · #35B50 #35J92 #53E10 #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Primary 35R03 #Secondary 35B45 #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2406.15123

openalex publication_date 2024/06/21 · openalex created_date 2024/06/25 · openalex updated_date 2026/07/28

Abstract

We consider the inverse mean curvature flow (IMCF) in the Heisenberg group (\Heⁿ, d_ε), where d_ε is distance associated to either | ⋅ |_ε, ε>0, the natural family of left-invariant Riemannian metrics, or with their sub-Riemannian counterparts for ε=0. For Ω⊆ \Heⁿ an open set with smooth boundary Σ₀=∂ Ω satisfying a uniform exterior gauge-ball condition and bounded complement we show existence of a global weak IMCF of generalized hypersurfaces \Σ^ε_s\_{s ≥ 0} ⊆ ℍⁿ 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 \citeHuiskenIlmanen, following the approach in \citeMoser due to Moser and based on the the link between IMCF and p-harmonic functions.

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• Legendrian curve flow in Sasakian sub-Riemannian 3-manifolds

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