Geometric inequalities and their stabilities for modified quermassintegrals in hyperbolic space
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911766080913408 |
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| author | Gao, Chaoqun Zhou, Rong |
| author_facet | Gao, Chaoqun Zhou, Rong |
| contents | In this paper, we first consider the curve case of Hu-Li-Wei's flow for shifted principal curvatures of h-convex hypersurfaces in $\mathbb{H}^{n+1}$ proposed in [10]. We prove that if the initial closed curve is smooth and strictly h-convex, then the solution exists for all time and preserves strict h-convexity along the flow. Moreover, the evolving curve converges smoothly and exponentially to a geodesic circle centered at the origin. The key ingredient in our proof is the Heintze-Karcher type inequality for h-convex curves proved recently in [14].
As an application, we then provide a new proof of geometric inequalities involving weighted curvature integrals and modified quermassintegrals for h-convex curves in $\mathbb{H}^2$. We finally discuss the stability of these inequalities as well as Alexandrov-Fenchel type inequalities for modified quermassintegrals for strictly h-convex domains in $\mathbb{H}^{n+1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_09788 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Geometric inequalities and their stabilities for modified quermassintegrals in hyperbolic space Gao, Chaoqun Zhou, Rong Differential Geometry 53C42, 53E10 In this paper, we first consider the curve case of Hu-Li-Wei's flow for shifted principal curvatures of h-convex hypersurfaces in $\mathbb{H}^{n+1}$ proposed in [10]. We prove that if the initial closed curve is smooth and strictly h-convex, then the solution exists for all time and preserves strict h-convexity along the flow. Moreover, the evolving curve converges smoothly and exponentially to a geodesic circle centered at the origin. The key ingredient in our proof is the Heintze-Karcher type inequality for h-convex curves proved recently in [14]. As an application, we then provide a new proof of geometric inequalities involving weighted curvature integrals and modified quermassintegrals for h-convex curves in $\mathbb{H}^2$. We finally discuss the stability of these inequalities as well as Alexandrov-Fenchel type inequalities for modified quermassintegrals for strictly h-convex domains in $\mathbb{H}^{n+1}$. |
| title | Geometric inequalities and their stabilities for modified quermassintegrals in hyperbolic space |
| topic | Differential Geometry 53C42, 53E10 |
| url | https://arxiv.org/abs/2401.09788 |