Horizontal inverse mean curvature flow in the Heisenberg group

2023/06/27 by Jingshi Cui, Cui, Jingshi, Peibiao Zhao +1 · 1 citation
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2306.15469

openalex publication_date 2023/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Huisken and Ilmanen in [37] created the theory of weak solutions for inverse mean curvature flows (IMCF) of hypersurfaces on Riemannian manifolds, and proved successfully a Riemannian version of the Penrose inequality. The present paper investigates and constructs a sub-Riemannian version of the theory of weak solutions for inverse mean curvature flows of hypersurfaces in the first Heisenberg group ℍ¹, and provides a positive answer to an open problem: the Heintze-Karcher inequality in ℍ¹. Furthermore, we introduce a ℍ-perimeter preserving flow (1.8) in the first Heisenberg group ℍ¹, which is derived by applying the Heisenberg dilation to HIMCF. This rescaled flow is subsequently applied to establish a Minkowski-type formula in ℍ¹.

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