Lindel"of-type theorems for the minimal surface at infinity

A lot of works on researching the solutions of equation of the minimal surfaces, which are given over unbounded domains (see, for example, [1-3; 5-7]) in which various tasks of asymptotic behavior of the minimal surfaces were studied. The obtained theorems of Lindel¨of type about the limiting value of the gradient of the solution of the equation of minimal surfaces and Gaussian curvature of the considered surface at infinity. Let = 𝑓(𝑥, 𝑦) be 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) → 𝑏𝑛, ((𝑥, 𝑦) → ∞, (𝑥, 𝑦) ∈ 𝐿𝑛) = 1, 2, and, in addition, the gradient of the function 𝑓(𝑥, 𝑦) on the curves 𝐿1 and 𝐿2 has the equal limit values for (𝑥, 𝑦) → ∞, this is one of two possibilities: or 𝐾(𝑥, 𝑦) not limited to 𝐷, or 𝑏1 = 𝑏2 = and 𝐾(𝑥, 𝑦) → for (𝑥, 𝑦) tending to infinity along any path lying in the domain 𝐷.