Surfaces with negative intrinsic curvature in hyperbolic space (Q2725497)
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scientific article; zbMATH DE number 1619327
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Surfaces with negative intrinsic curvature in hyperbolic space |
scientific article; zbMATH DE number 1619327 |
Statements
11 December 2001
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surface
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negative curvature
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isometric immersion
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complete Riemannian metric
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Efimov theorem
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Surfaces with negative intrinsic curvature in hyperbolic space (English)
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The author proves the following analogue of Efimov's theorem: Let \(\left(\Sigma,\sigma\right)\) be a complete Riemannian surface with curvature \(K\leq -1-\varepsilon\) (\(\varepsilon>0\)) such that \(\frac{\left|\left|\nabla K\right|\right|}{\left|K\right|^{3/2}}\) is bounded. Then \(\left(\Sigma,\sigma\right)\) does not admit any isometric immersion into \(H^3\). Similar results are proved for \(S^3\) and \(H^3_1\).
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