Lindelef type theorems for the minimal surface at infinity

R. S. Akopyan

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 5 (2016), pp. 7-12

Résumé

A lot of works on researching of 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 Lindelef 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 $z=f(x,y)$ 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 $$ K(x,y) \to b_n, \qquad ((x,y)\to \infty, (x,y)\in L_n ) \qquad n=1,\ 2, $$ and, in addition, the gradient of the function $f(x,y)$ on the curves $L_1$ and $L_2$ has the equal limit values for $(x,y)\to \infty$, this is one of two possibilities: or $ K(x,y) $ not limited to $D$, or $b_1 = b_2 = b$ and $K(x,y) \to b$ for $ (x,y)$ tending to infinity along any path lying in the domain $D$.

Détail

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