Convexity and rigidity of hypersurfaces in Cartan-Hadamard manifolds
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909847840096256 |
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| author | Ghomi, Mohammad |
| author_facet | Ghomi, Mohammad |
| contents | We show that in Cartan-Hadamard manifolds $M^n$, $n\geq 3$, closed infinitesimally convex hypersurfaces $Γ$ bound convex flat regions, if curvature of $M^n$ vanishes on tangent planes of $Γ$. This encompasses Chern-Lashof-Sacksteder characterization of compact convex hypersurfaces in Euclidean space, and some results of Greene-Wu-Gromov on rigidity of Cartan-Hadamard manifolds. It follows that closed simply connected surfaces in $M^3$ with minimal total absolute curvature bound Euclidean convex bodies, as stated by Gromov in 1985. The proofs employ the Gauss-Codazzi equations, a generalization of Schur comparison theorem to CAT($k$) spaces, and other techniques from Alexandrov geometry outlined by Petrunin. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_15454 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Convexity and rigidity of hypersurfaces in Cartan-Hadamard manifolds Ghomi, Mohammad Differential Geometry Analysis of PDEs Metric Geometry Primary: 53C20, 58J05, Secondary: 53C44, 52A15 We show that in Cartan-Hadamard manifolds $M^n$, $n\geq 3$, closed infinitesimally convex hypersurfaces $Γ$ bound convex flat regions, if curvature of $M^n$ vanishes on tangent planes of $Γ$. This encompasses Chern-Lashof-Sacksteder characterization of compact convex hypersurfaces in Euclidean space, and some results of Greene-Wu-Gromov on rigidity of Cartan-Hadamard manifolds. It follows that closed simply connected surfaces in $M^3$ with minimal total absolute curvature bound Euclidean convex bodies, as stated by Gromov in 1985. The proofs employ the Gauss-Codazzi equations, a generalization of Schur comparison theorem to CAT($k$) spaces, and other techniques from Alexandrov geometry outlined by Petrunin. |
| title | Convexity and rigidity of hypersurfaces in Cartan-Hadamard manifolds |
| topic | Differential Geometry Analysis of PDEs Metric Geometry Primary: 53C20, 58J05, Secondary: 53C44, 52A15 |
| url | https://arxiv.org/abs/2308.15454 |