Akopyan R.S. On Limit Value of the Gaussian Curvature of the Minimal Surface at Infinity


Ripsime Sergoevna Akopyan
Candidate of Physical and Mathematical Sciences, Associate Professor,
Department of Higher Mathematics,
Volgograd State Agrarian University
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Prosp. Universitetsky, 26, 400002 Volgograd, Russian Federation

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 L1 and L2, coming from the same point and going into infinity. We assume that f(x,y) ∈ C2(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 L1 and L2 satisfies the conditions

K(x,y) → 0, ((x,y)→ ∞, (x,y) ∈ Ln) n = 1, 2, 

then K(x,y) → 0 for (x,y) tending to infinity along any path lying in the domain D.

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

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On Limit Value of the Gaussian Curvature of the Minimal Surface at Infinity by Akopyan R.S. is licensed under a Creative Commons Attribution 4.0 International License.

Citation in EnglishScience Journal of Volgograd State University. Mathematics. Physics. No. 1 (32) 2016 pp. 6-10

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