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An analog of Efimov’s theorem in variable curvature. (Un analogue du théorème d’Efimov en courbure variable.) (French) Zbl 0937.53024

Séminaire de théorie spectrale et géométrie. Année 1994-1995. St. Martin d’Hères: Univ. de Grenoble I, Institut Fourier, Sémin. Théor. Spectrale Géom., Chambéry-Grenoble. 13, 67-79 (1995).
That there is no regular isometric immersion of the hyperbolic plane \(H^2\) into Euclidean 3-space \(\mathbb{R}^3\) was already known a century ago from a work of D. Hilbert [Trans. Am. Math. Soc. 2, 87-99 (1901; JFM 32.0608.01)]. This result was generalized by N. Efimov who proved that there exists no complete surface with uniformly negative curvature that is isometrically \(C^2\)-immersible into \(\mathbb{R}^3.\) In the work under review, the author gives an analogous theorem in the case of an ambient space of variable curvature with some extra conditions on the sectional curvature. Namely, let \(M\) be a 3-dimensional manifold whose sectional curvature is bounded by \(K_2\) and \(K_3\), and let \(\Sigma\) be a complete surface with curvature less than \(K_1 < 0.\) If the gradients of the sectional curvatures of \(M\) and \(\Sigma\) are bounded and the numbers \(K_1, K_2, K_3\) satisfy a certain inequality (e.g., when \(K_3 \geq 0, \) \((K_3 - K_2)^2 < 16 |K_1|(K_2 - K_1)\)) then the impossibility of an isometric \(C^3\)-immersion of \(\Sigma\) into \(M\) follows.
For the entire collection see [Zbl 0833.00019].

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
35L70 Second-order nonlinear hyperbolic equations
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature

Citations:

JFM 32.0608.01
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