Horizontal inverse mean curvature flow in the Heisenberg group
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866910730067902464 |
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| author | Cui, Jingshi Zhao, Peibiao |
| author_facet | Cui, Jingshi Zhao, Peibiao |
| contents | 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 $\mathbb{H}^{1}$, and provides a positive answer to an open problem: the Heintze-Karcher inequality in $\mathbb{H}^{1}$. Furthermore, we introduce a $\mathbb{H}$-perimeter preserving flow (1.8) in the first Heisenberg group $\mathbb{H}^{1}$, which is derived by applying the Heisenberg dilation to HIMCF. This rescaled flow is subsequently applied to establish a Minkowski-type formula in $\mathbb{H}^{1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_15469 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Horizontal inverse mean curvature flow in the Heisenberg group Cui, Jingshi Zhao, Peibiao Differential Geometry 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 $\mathbb{H}^{1}$, and provides a positive answer to an open problem: the Heintze-Karcher inequality in $\mathbb{H}^{1}$. Furthermore, we introduce a $\mathbb{H}$-perimeter preserving flow (1.8) in the first Heisenberg group $\mathbb{H}^{1}$, which is derived by applying the Heisenberg dilation to HIMCF. This rescaled flow is subsequently applied to establish a Minkowski-type formula in $\mathbb{H}^{1}$. |
| title | Horizontal inverse mean curvature flow in the Heisenberg group |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2306.15469 |