Total mean curvatures of Riemannian hypersurfaces
We obtain a comparison formula for integrals of mean curvatures of Riemannian hypersurfaces via Reilly’s identities. As applications, we derive several geometric inequalities for a convex hypersurface Γ\Gamma in a Cartan-Hadamard manifold MM. In particular, we show that the first mean curvature int...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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De Gruyter
2023-01-01
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| Series: | Advanced Nonlinear Studies |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/ans-2022-0029 |
| _version_ | 1811171884476137472 |
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| author | Ghomi Mohammad Spruck Joel |
| author_facet | Ghomi Mohammad Spruck Joel |
| author_sort | Ghomi Mohammad |
| collection | DOAJ |
| description | We obtain a comparison formula for integrals of mean curvatures of Riemannian hypersurfaces via Reilly’s identities. As applications, we derive several geometric inequalities for a convex hypersurface Γ\Gamma in a Cartan-Hadamard manifold MM. In particular, we show that the first mean curvature integral of a convex hypersurface γ\gamma nested inside Γ\Gamma cannot exceed that of Γ\Gamma , which leads to a sharp lower bound for the total first mean curvature of Γ\Gamma in terms of the volume it bounds in MM in dimension 3. This monotonicity property is extended to all mean curvature integrals when γ\gamma is parallel to Γ\Gamma , or MM has constant curvature. We also characterize hyperbolic balls as minimizers of the mean curvature integrals among balls with equal radii in Cartan-Hadamard manifolds. |
| first_indexed | 2024-04-10T17:22:43Z |
| format | Article |
| id | doaj.art-69ed07647f1c45299276abf2897de22f |
| institution | Directory Open Access Journal |
| issn | 2169-0375 |
| language | English |
| last_indexed | 2024-04-10T17:22:43Z |
| publishDate | 2023-01-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Advanced Nonlinear Studies |
| spelling | doaj.art-69ed07647f1c45299276abf2897de22f2023-02-05T08:43:35ZengDe GruyterAdvanced Nonlinear Studies2169-03752023-01-0123132132510.1515/ans-2022-0029Total mean curvatures of Riemannian hypersurfacesGhomi Mohammad0Spruck Joel1School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, United StatesDepartment of Mathematics, Johns Hopkins University, Baltimore, MD 21218, United StatesWe obtain a comparison formula for integrals of mean curvatures of Riemannian hypersurfaces via Reilly’s identities. As applications, we derive several geometric inequalities for a convex hypersurface Γ\Gamma in a Cartan-Hadamard manifold MM. In particular, we show that the first mean curvature integral of a convex hypersurface γ\gamma nested inside Γ\Gamma cannot exceed that of Γ\Gamma , which leads to a sharp lower bound for the total first mean curvature of Γ\Gamma in terms of the volume it bounds in MM in dimension 3. This monotonicity property is extended to all mean curvature integrals when γ\gamma is parallel to Γ\Gamma , or MM has constant curvature. We also characterize hyperbolic balls as minimizers of the mean curvature integrals among balls with equal radii in Cartan-Hadamard manifolds.https://doi.org/10.1515/ans-2022-0029reilly’s formulasquermassintegralmixed volumegeneralized mean curvaturehyperbolic spacecartan-hadamard manifoldprimary: 53c2058j05secondary: 52a3849q15 |
| spellingShingle | Ghomi Mohammad Spruck Joel Total mean curvatures of Riemannian hypersurfaces Advanced Nonlinear Studies reilly’s formulas quermassintegral mixed volume generalized mean curvature hyperbolic space cartan-hadamard manifold primary: 53c20 58j05 secondary: 52a38 49q15 |
| title | Total mean curvatures of Riemannian hypersurfaces |
| title_full | Total mean curvatures of Riemannian hypersurfaces |
| title_fullStr | Total mean curvatures of Riemannian hypersurfaces |
| title_full_unstemmed | Total mean curvatures of Riemannian hypersurfaces |
| title_short | Total mean curvatures of Riemannian hypersurfaces |
| title_sort | total mean curvatures of riemannian hypersurfaces |
| topic | reilly’s formulas quermassintegral mixed volume generalized mean curvature hyperbolic space cartan-hadamard manifold primary: 53c20 58j05 secondary: 52a38 49q15 |
| url | https://doi.org/10.1515/ans-2022-0029 |
| work_keys_str_mv | AT ghomimohammad totalmeancurvaturesofriemannianhypersurfaces AT spruckjoel totalmeancurvaturesofriemannianhypersurfaces |