开平面C上的亚纯函数与完备极小曲面的Gauss映射

Meromorphic Functions on C and the Gauss Map of Complete Minimal Surfaces

设■:D→R^3确定了以等温参数表示的极小曲面M,其中D是全平面R^2的开子区域,那么极小曲面的Gauss映射g(z)是D上的亚纯函数.Xavier与Chao提出了一个尚未解决的问题:任意给定区域■上的亚纯函数g(z),它是否是某完备极小曲面的Gauss映射?本文证明了若开平面C上的亚纯函数g(z)的零点列或极点列的收敛指数小于1/2,则g(z)—定是某完备极小曲面的Gauss映射.

Let ■:D →R^3 be a minimal surface M with isothermal parameter, where D is a domain in R^2, then the Gauss map of minimal surface is a meromorphic function on D. Supposing g(z) to be an arbitrary meromorphic function on ■, whether there exists a complete minimal surface M, such that g(z) is the Gauss map of M,which is proposed by Xavier and Chao is still to be unsolved. In this paper, we prove that for the meromorphic function g(z) on C with the property that either the exponent of convergence of zeros or the exponent of convergence of poles is less than 1/2,g(z) must be the Gauss map of some complete minimal surface.