Modified defect relations for the gauss map of minimal surfaces, III
Modified defect relations for the gauss map of minimal surfaces, III
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DOI：
10.1017/s0027763000003755
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发表时间：
1991-12
影响因子：
0.8
通讯作者：
H. Fujimoto
中科院分区：
文献类型：
--
作者：
H. Fujimoto
在文[5]中，作者证明了浸没在R3中的非平坦完备极小曲面的Gauss映射至多可以省略球面上的四个点，并在文[7]中揭示了这一结果与亚纯函数值分布的Nevanlinna理论中的亏量关系之间的一些关系。随后，Mo和Osserman在他们的文献[11]中得到了这些结果的改进，即如果浸入R3的非平坦完备极小曲面M的Gauss映射仅有限次取五个不同的值，则M有有限的全曲率。给出了具有完全协调度量的黎曼曲面到n维复射影空间Pn(C)的全纯映射的修正亏量关系，作为应用，证明了如果浸没在Rm中的完备极小曲面M的(广义)Gauss映射G是非退化的，即像G(M)不包含在Pm−1(C)中的任何超平面上，则它在一般位置([8])上至多可以省略m(m+1)/2个超平面.这里，数字m(m+1)/2对于任意奇数和一些小的偶数m是最佳可能的(见[6])。最近，Ru证明了上述结果的“非退化”假设可以去掉([13])。本文引入了修正缺陷的新定义，并证明了修正缺陷关系的精化。作为应用，我们将在[5]、[7]、[8]、[11]和[13]中改进上述结果。
In [5], the author proved that the Gauss map of a nonflat complete minimal surface immersed in R 3 can omit at most four points of the sphere, and in [7] he revealed some relations between this result and the defect relation in Nevanlinna theory on value distribution of meromorphic functions. Afterwards, Mo and Osserman obtained an improvement of these results in their paper [11], which asserts that if the Gauss map of a nonflat complete minimal surface M immersed in R 3 takes on five distinct values only a finite number of times, then M has finite total curvature. The author also gave modified defect relations for holomorphic maps of a Riemann surface with a complete conformai metric into the n-dimensional complex projective space Pn(C) and, as its application, he showed that, if the (generalized) Gauss map G of a complete minimal surface M immersed in Rm is nondegenerate, namely, the image G(M) is not contained in any hyperplane in P m − 1(C), then it can omit at most m(m + 1)/2 hyperplanes in general position ([8]). Here, the number m(m + 1)/2 is best-possible for arbitrary odd numbers and some small even numbers m (see [6]). Recently, Ru showed that the “nondegenerate” assumption of the above result can be dropped ([13]). In this paper, we shall introduce a new definition of modified defect and prove a refined Modified defect relation. As its application, we shall give some improvements of the above-mentioned results in [5], [7], [8], [11] and [13].