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

R. S. Akopyan

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 1 (2016), pp. 6-10

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; 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 $D$.

Détail

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