New quermassintegral and Poincaré type inequalities for non-convex domains
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| Format: | Preprint |
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2024
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| author | Hu, Yingxiang Ivaki, Mohammad N. |
| author_facet | Hu, Yingxiang Ivaki, Mohammad N. |
| contents | In the first part of this paper, we study the following non-homogeneous, locally constrained inverse curvature flow in Euclidean space $\mathbb{R}^{n+1}$, \begin{align*} \dot{x}=\left(\frac{1}{\frac{E_k(\hatκ)}{E_{k-1}(\hatκ)}-α}-\langle x,ν\rangle\right)ν, \quad k=2,3,\ldots,n-1. \end{align*} Assuming that the initial hypersurface $\mathcal{M}_0 \subset \mathbb{R}^{n+1}$ is star-shaped and its shifted principal curvatures $\hatκ=κ+α(1,\ldots,1)$ lie in the convex set \begin{align*} Γ_{α,k}:=Γ_{k-1}\cap \{λ\in \mathbb{R}^n:\, E_k(λ)-αE_{k-1}(λ)>0\}, \end{align*} we show that the flow admits a smooth solution that exists for all positive times, and it converges smoothly to a round sphere. As a corollary, we obtain a new set of Alexandrov-Fenchel-type inequalities for non-convex domains.
In the second part, we derive a Poincaré type inequality for $k$-convex hypersurfaces which complements a more general version of the well-known Heintze-Karcher inequality. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_06057 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | New quermassintegral and Poincaré type inequalities for non-convex domains Hu, Yingxiang Ivaki, Mohammad N. Differential Geometry Analysis of PDEs In the first part of this paper, we study the following non-homogeneous, locally constrained inverse curvature flow in Euclidean space $\mathbb{R}^{n+1}$, \begin{align*} \dot{x}=\left(\frac{1}{\frac{E_k(\hatκ)}{E_{k-1}(\hatκ)}-α}-\langle x,ν\rangle\right)ν, \quad k=2,3,\ldots,n-1. \end{align*} Assuming that the initial hypersurface $\mathcal{M}_0 \subset \mathbb{R}^{n+1}$ is star-shaped and its shifted principal curvatures $\hatκ=κ+α(1,\ldots,1)$ lie in the convex set \begin{align*} Γ_{α,k}:=Γ_{k-1}\cap \{λ\in \mathbb{R}^n:\, E_k(λ)-αE_{k-1}(λ)>0\}, \end{align*} we show that the flow admits a smooth solution that exists for all positive times, and it converges smoothly to a round sphere. As a corollary, we obtain a new set of Alexandrov-Fenchel-type inequalities for non-convex domains. In the second part, we derive a Poincaré type inequality for $k$-convex hypersurfaces which complements a more general version of the well-known Heintze-Karcher inequality. |
| title | New quermassintegral and Poincaré type inequalities for non-convex domains |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2408.06057 |