The value distribution of the hyperbolic Gauss map
The value distribution of the hyperbolic Gauss map
复制标题
双曲高斯图的值分布
DOI：
10.1090/s0002-9939-97-03937-3
复制
发表时间：
1997
期刊：
影响因子：
--
通讯作者：
Zuhuan Yu
中科院分区：
文献类型：
--
作者：
Zuhuan Yu
在极小曲面理论中，高斯映射的值分布问题一直是人们研究的热点。一个早期的结果是经典的伯恩斯坦定理：R 3中的任何完全极小图都是平面。用一些几何条件代替图，R。Osserman [3]证明了如果完备极小曲面是非平坦的，则Gauss映射不能省略一组正对数容量，这回答了Nirenberg的一个猜想.后来，F. Xavier [8]指出，这种曲面的高斯映射最多可以省略六个点。1989年，H. Fujimoto [2]最后证明了R3中非平坦完备极小曲面的Gauss映射最多可省略4个点的最佳结果。考虑H f中极小曲面的相似性质是很自然的。一个有趣的特征是H”中存在一族绝对面积极小化超曲面，并且其中只有一个是全测地的（见[7]）.问题似乎是如何在双曲空间中提出一个充分的伯恩斯坦问题。Y教授L.辛向我指出，R. Bryant [1]为双曲空间中的伯恩斯坦问题提供了一个求解框架。本文研究了常平均曲率双曲空间中的曲面。我们用CMC-1表示“常平均曲率1”。这些曲面与R”中的极小曲面有许多共同的性质。它们具有用全纯数据表示的Weierstrass表示。这个公式是由R. Bryant [1].在M. Umhara和K. Yamada（[4]，[5]）.在这里，我们试图研究高斯映射的双曲模拟。双曲高斯映射的值如何分布是一个很自然的问题。使用布莱恩特的代表性公式，我们能够回答这个问题如下。
In minimal surface theory, the value distribution of the Gauss map has been studied for a long time. An early result is the classical Bernstein Theorem: Any complete minimal graph in R 3 is a plane. Replacing the graph by some geometric conditions, R. Osserman [3] showed that if the complete minimal surface is nonflat, then the Gauss map cannot omit a set of positive logarithmic capacity, which answered a conjecture of Nirenberg. Afterwards, it was generalized by F. Xavier [8] that the Gauss map of such a surface can omit at most six points. In 1989, H. Fujimoto [2] proved the best result finally that the Gauss map of the nonflat complete minimal surface in R 3 can omit at most four points. It is natural to consider the similar properties of minimal surfaces in H f. An interesting feature is that there exists a family of absolutely area-minimizing hypersurfaces in H" and only one of them is totally geodesic (see [7]). The question seems to be how to raise an adequate Bernstein problem in hyperbolic space. Professor Y. L. Xin pointed out to me that the striking work done by R. Bryant [1] supplies a framework to solve the Bernstein problem in hyperbolic space. In this paper, we shall be concerned with the surfaces in hyperbolic space of constant mean curvature one. We abbreviate "constant mean curvature one" by CMC-1. These surfaces share many properties with minimal surfaces in R". They possess the "Weierstrass representation" in terms of holomorphic data. This formula was discovered by R. Bryant [1]. Many other properties may be found in papers by M. Umhara and K. Yamada ([4], [5]). Here we try to investigate the hyperbolic analogue of the Gauss map. It is a natural question how the values of the hyperbolic Gauss map distribute. Using Bryant's representation formula we are able to answer this question as follows.
影响因子：
0.8
作者：
H. Fujimoto
通讯作者：
H. Fujimoto
影响因子：
4.9
作者：
M. Umehara;Kotaro Yamada
通讯作者：
M. Umehara;Kotaro Yamada