Anisotropic mean curvature type flow and capillary Alexandrov-Fenchel inequalities
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| Format: | Preprint |
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2024
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| author | Ding, Shanwei Gao, Jinyu Li, Guanghan |
| author_facet | Ding, Shanwei Gao, Jinyu Li, Guanghan |
| contents | In this paper, an anisotropic volume-preserving mean curvature type flow for star-shaped anisotropic $ω_0$-capillary hypersurfaces in the half-space is studied, and the long-time existence and smooth convergence to a capillary Wulff shape are obtained. If the initial hypersurface is strictly convex, the solution of this flow remains to be strictly convex for all $t>0$ by adopting a new approach applicable to anisotropic capillary setting. In analogy with closed hypersurfaces, if the $ω_0$-capillary Wulff shape is a $θ$-capillary hypersurface with constant contact angle $θ$, the quermassintegrals for anisotropic capillary hypersurfaces match the mixed volume of two $θ$-capillary convex bodies. Thus, generalized quermassintegrals for anisotropic capillary hypersurfaces with general Wulff shapes (i.e., the $ω_0$-capillary Wulff shape has a variable contact angle) can be defined, which satisfy certain monotonicity properties along the flow. As applications, we establish an anisotropic capillary isoperimetric inequality for star-shaped anisotropic capillary hypersurfaces and a family of new Alexandrov-Fenchel inequalities for strictly convex anisotropic capillary hypersurfaces. In particular, we provide a flow's method to derive the Alexandrov-Fenchel inequalities for two $θ$-capillary hypersurfaces, demonstrated in [30] (arXiv:2408.13655) from the view of point in convex geometry. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_10740 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Anisotropic mean curvature type flow and capillary Alexandrov-Fenchel inequalities Ding, Shanwei Gao, Jinyu Li, Guanghan Differential Geometry 53E10, 35K93, 53C21 In this paper, an anisotropic volume-preserving mean curvature type flow for star-shaped anisotropic $ω_0$-capillary hypersurfaces in the half-space is studied, and the long-time existence and smooth convergence to a capillary Wulff shape are obtained. If the initial hypersurface is strictly convex, the solution of this flow remains to be strictly convex for all $t>0$ by adopting a new approach applicable to anisotropic capillary setting. In analogy with closed hypersurfaces, if the $ω_0$-capillary Wulff shape is a $θ$-capillary hypersurface with constant contact angle $θ$, the quermassintegrals for anisotropic capillary hypersurfaces match the mixed volume of two $θ$-capillary convex bodies. Thus, generalized quermassintegrals for anisotropic capillary hypersurfaces with general Wulff shapes (i.e., the $ω_0$-capillary Wulff shape has a variable contact angle) can be defined, which satisfy certain monotonicity properties along the flow. As applications, we establish an anisotropic capillary isoperimetric inequality for star-shaped anisotropic capillary hypersurfaces and a family of new Alexandrov-Fenchel inequalities for strictly convex anisotropic capillary hypersurfaces. In particular, we provide a flow's method to derive the Alexandrov-Fenchel inequalities for two $θ$-capillary hypersurfaces, demonstrated in [30] (arXiv:2408.13655) from the view of point in convex geometry. |
| title | Anisotropic mean curvature type flow and capillary Alexandrov-Fenchel inequalities |
| topic | Differential Geometry 53E10, 35K93, 53C21 |
| url | https://arxiv.org/abs/2408.10740 |