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 Mat. Sb. (N.S.), 1970, Volume 83(125), Number 2(10), Pages 313–324 (Mi msb3514)

Some properties of surfaces with slowly varying negative extrinsic curvature in a Riemannian space

I. S. Brandt

Abstract: We consider surfaces of negative extrinsic curvature in a Riemannian space with nonpositive curvature. We prove that the following inequality holds on a surface which is complete in the sense of the intrinsic metric:
$$\sup_F\{|\operatorname{grad}\frac1k|+\frac{\Lambda-\lambda}{2k^2}\}=q>\frac1{\sqrt3},$$
here $F$ is the surface being considered, $k=\sqrt{K_e}$ ($K_e$ is the extrinsic curvature of $F$) and $\Lambda$ and $\lambda$ are the maximum and minimum of the Riemannian curvature of the space $R$ at a given point.
This theorem generalizes a theorem of Efimov concerning $q$-metrics. We give an example of a surface for which $q=4,5$.
Bibliography: 8 titles.

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English version:
Mathematics of the USSR-Sbornik, 1970, 12:2, 313–324

Bibliographic databases:

UDC: 513.736.3
MSC: 14H55, 53C21, 14Jxx

Citation: I. S. Brandt, “Some properties of surfaces with slowly varying negative extrinsic curvature in a Riemannian space”, Mat. Sb. (N.S.), 83(125):2(10) (1970), 313–324; Math. USSR-Sb., 12:2 (1970), 313–324

Citation in format AMSBIB
\Bibitem{Bra70} \by I.~S.~Brandt \paper Some properties of surfaces with slowly varying negative extrinsic curvature in a~Riemannian space \jour Mat. Sb. (N.S.) \yr 1970 \vol 83(125) \issue 2(10) \pages 313--324 \mathnet{http://mi.mathnet.ru/msb3514} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=268824} \zmath{https://zbmath.org/?q=an:0203.24303} \transl \jour Math. USSR-Sb. \yr 1970 \vol 12 \issue 2 \pages 313--324 \crossref{https://doi.org/10.1070/SM1970v012n02ABEH000923} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. I. V. Gribkov, “The problem of the correctness of Schur's theorem”, Math. USSR-Sb., 44:4 (1983), 471–481
2. I. V. Gribkov, “The multidimensional problem of the correctness of Schur's theorem”, Math. USSR-Sb., 48:2 (1984), 423–436
3. Connell Ch., Ullman J., “Ends of Negatively Curved Surfaces in Euclidean Space”, Manuscr. Math., 131:3-4 (2010), 275–303
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