到H^n中等距极小浸入的注记(英文)

A NOTE ON ISOMETRIC MINIMAL IMMERSIONS INTO H^n

设M是一个m维流形,H^n是曲率为-1的标准双曲空间.本文研究了等距极小浸入h=(x_1, x_2,…,x_n):M→H^n的坐标函数,得到:如下结论:如果h=(x_1,x_2,…,x_n):M→H^u是一个等距极小浸入,则对k=1,2,…,n. △xk=-(m/xk)〈(E_n)~n,(E_k)_N〉, 这里是常向量场.由此可以准出如下事实:h同上,则只要m≥2,x_n就是关于h~*(,)的上调和函数,而只要m≥1,x_n就是关于h~*〈,〉的上调和函数.限制在m=2的情形,并借助于黎曼面理论,得到下述的重要结果:设M是一个抛物型黎曼面,则不存在M到H^n中的等距极小浸入。

Let M be a m-dimensional Riemannian manifold, H^n be the standardhyperbolic n-space with curvature -1. Working primarily within the isometric minimalimmersion (?): M→H^n, the author explores the coordinate functions and obtains thefollowing results: If (h)=(x,…,x_n):M→H^n is an isometric minimal immersion, thenfor k=1,…, n △x_k=-(m/x_n)<(E_n)~N, (E_k)~N>,where E_k=(0.…1,…,0) is a constant vector field. From this it easily follows thatIf (h)=(x_1…,x_n): M→H^n is isometric and minimal, then x_n is a superharmonic function to ■(,) if m≥2 and to h*<,> if m≥1. Restricted to the case m=2, using thetheory of Riemann surfaces, author gets the following conclusion: Let M be a Rieman-nian surface of parabolic type, then there is noisometric minimal immersion of M intoH^n.