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

Mathematics

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

R. S. Akopyan

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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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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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
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