Total curvature of convex hypersurfaces in Cartan-Hadamard manifolds
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912833553301504 |
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| author | Ghomi, Mohammad Stavroulakis, John Ioannis |
| author_facet | Ghomi, Mohammad Stavroulakis, John Ioannis |
| contents | We show that if the curvature of a Cartan-Hadamard $n$-manifold is constant near a convex hypersurface $Γ$, then the total Gauss-Kronecker curvature $\mathcal{G}(Γ)$ is not less than that of any convex hypersurface nested inside $Γ$. This extends Borbély's monotonicity theorem in hyperbolic space. It follows that $\mathcal{G}(Γ)$ is bounded below by the volume of the unit sphere in Euclidean space $\mathbf{R}^n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_13280 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Total curvature of convex hypersurfaces in Cartan-Hadamard manifolds Ghomi, Mohammad Stavroulakis, John Ioannis Differential Geometry Primary: 53C20, 53C42, Secondary: 53C45, 53C65 We show that if the curvature of a Cartan-Hadamard $n$-manifold is constant near a convex hypersurface $Γ$, then the total Gauss-Kronecker curvature $\mathcal{G}(Γ)$ is not less than that of any convex hypersurface nested inside $Γ$. This extends Borbély's monotonicity theorem in hyperbolic space. It follows that $\mathcal{G}(Γ)$ is bounded below by the volume of the unit sphere in Euclidean space $\mathbf{R}^n$. |
| title | Total curvature of convex hypersurfaces in Cartan-Hadamard manifolds |
| topic | Differential Geometry Primary: 53C20, 53C42, Secondary: 53C45, 53C65 |
| url | https://arxiv.org/abs/2601.13280 |