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

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 = 𝑓(𝑥, 𝑦) is a solution of the equation of minimal surfaces (1) given over the domain 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) = 1, 2, then 𝐾(𝑥, 𝑦) → 0 for (𝑥, 𝑦) tending to infinity along any path lying in the domain 𝐷.