Surfaces of constant mean curvaturec inH3(−c2) with prescribed hyperbolic Gauss map
Surfaces of constant mean curvaturec inH3(−c2) with prescribed hyperbolic Gauss map
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具有指定双曲高斯图的恒定平均曲率 c inH3(−c2) 的曲面
DOI：
10.1007/bf01446291
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发表时间：
1996
影响因子：
1.4
通讯作者：
Kotaro Yamada
中科院分区：
文献类型：
--
作者：
M. Umehara;Kotaro Yamada
我们用H3（-c2）表示具有常截面曲率-c2的双曲三维空间。曲面的常平均曲率c在H3（-c 2）有非常相似的性质，极小曲面在R 3和研究的几个作者。(See[CS]，[Di].）特别是Bryant [BI]建立了H3（-c 2）中这些曲面的表示公式，作为极小曲面理论中Weierstrass公式的类比。通过将李群SL（2，C）坍缩为C3 [U-Y2]，得到了经典的Weierstrass表示作为其极限.现在我们把”常平均曲率c”表示为“CMC-c”。“H3（-c2）中CMC-c曲面的双曲高斯映射也被定义为到PI（C）的全纯映射。虽然R3中有限全曲率完备极小曲面的Gauss映射在端点上是同胚扩张的，但一般的双曲Gauss映射不是这样。设M2是Riemann曲面，f：M2-，H3（-c2）是具有有限总曲率的完全CMC-c浸入.则存在一个紧的Riemann曲面~ 2和有限个点Pb…，P，在~ 2上，使得M2是~ 2\{p~..... pn}。如果双曲高斯映射在pj处最多有一个极点，则称浸入f的端点P2是圆的;如果浸入f的端点P2在pj处有一个本质奇点，则称浸入f的端点P2是不规则的。在[U-Y1]和[U-Y2]中，作者给出了许多具有规则或不规则端点的CMC-c曲面的例子。一个标准是否一个规则的结束是嵌入或不也是已知的[U-Y1]。
We denote by H3 (-c 2) the hyperbolic 3-space of constant sectional curvature-c 2. Surfaces of constant mean curvature c in H3 (-c 2) have quite similar properties to minimal surfaces in R 3 and investigated by several authors.(See [CS],[Di].) In particular, Bryant [BI] established a representation formula for these surfaces in H3 (-c 2) as an analogy of the Weierstrass formula in minimal surface theory. The classical Weierstrass representation is obtained as a limit of it by collapsing the Lie group SL (2, C) to C 3 [U-Y2]. Now we abbreviate the term" constant mean curvature c" as"'CMC-c." The hyperbolic Gauss maps of CMC-c surfaces in H3 (-c 2) are also defined as holomorphic maps to PI (C). Though the Gauss maps of complete minimal surfaces of finite total curvature in R 3 extend hoIomorphically across their ends, it is not true for the hyperbolic Gauss maps in general. Let M 2 be a Riemann surface and f: M 2---, H3 (-c 2) be a complete CMC-c immersion with finite total curvature. Then there exist a compact Riemann surface~ 2 and finite points Pb..., P, on~ 2 such that M 2 is biholomorphic to~ 2\{p~..... pn}. An end P2 of the immersion f is called reoular if the hyperbolic Gauss map has at most pole at pj, and called irregular if it has an essential singularity at pj. In [U-Y1] and [U-Y2], the authors have given many examples of CMC-c surfaces with regular or irregular ends. A criterion whether a regular end is embedded or not is also known [U-Y1].
影响因子：
2.5
作者：
H. Lawson, Jr.;R. Tribuzy
通讯作者：
H. Lawson, Jr.;R. Tribuzy
影响因子：
4.9
作者：
M. Umehara;Kotaro Yamada
通讯作者：
M. Umehara;Kotaro Yamada