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Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica, 2016, Issue 1(32), Pages 6–10 (Mi vvgum90)  

This article is cited in 1 scientific paper (total in 1 paper)

Mathematics

On limit value of the Gaussian curvature of the minimal surface at infinity

R. S. Akopyan

Volgograd State Agricultural University

Abstract: A lot of works on researching of solutions of equation of the minimal surfaces, which are given over unbounded domains (see, for example, [1; 2; 4–6]) in which various tasks of asymptotic behavior of the minimal surfaces were studied. In the present paper the object of the research is a study of limit behavior of Gaussian curvature of the minimal surface given at infinity. We use a traditional approach for the solution of a similar kind of tasks which is a construction of auxiliary conformal mapping which appropriate properties are studied.
Let $z=f(x,y)$ is a solution of the equation of minimal surfaces (1) given over the domain $D$ bounded by two curves $L_1$ and $L_2$, coming from the same point and going into infinity. We assume that $f(x,y) \in C^2(\overline{D})$.
For the Gaussian curvature of minimal surfaces $K(x,y)$ will be the following theorem.
Theorem. If the Gaussian curvature $K(x,y)$ of the minimal surface (1) on the curves $L_1$ and $L_2$ satisfies the conditions
$$ K(x,y) \to 0, \quad ((x,y) \to \infty, (x,y) \in L_n) \quad n=1,2, $$
then $K(x,y) \to 0$ for $(x,y)$ tending to infinity along any path lying in the domain $D$.

Keywords: equations of the minimal surfaces, Gaussian curvature, asymptotic behavior, holomorphic function, isothermal coordinates, holomorphic function in the metric of the surface.

DOI: https://doi.org/10.15688/jvolsu1.2016.1.1

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Document Type: Article
UDC: 517.95
BBK: 22.161

Citation: R. S. Akopyan, “On limit value of the Gaussian curvature of the minimal surface at infinity”, Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica, 2016, no. 1(32), 6–10

Citation in format AMSBIB
\Bibitem{Ako16}
\by R.~S.~Akopyan
\paper On limit value of the Gaussian curvature of the minimal surface at infinity
\jour Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica
\yr 2016
\issue 1(32)
\pages 6--10
\mathnet{http://mi.mathnet.ru/vvgum90}
\crossref{https://doi.org/10.15688/jvolsu1.2016.1.1}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. R. S. Akopyan, “Teoremy tipa Lindelefa dlya minimalnoi poverkhnosti na beskonechnosti”, Vestn. Volgogr. gos. un-ta. Ser. 1, Mat. Fiz., 2016, no. 5(36), 7–12  mathnet  crossref
  • Mathematical Physics and Computer Simulation
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