Complete surfaces of constant mean curvature-1 in the hyperbolic 3-space
Complete surfaces of constant mean curvature-1 in the hyperbolic 3-space
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DOI：
10.2307/2946533
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发表时间：
1993-05
影响因子：
4.9
通讯作者：
M. Umehara;Kotaro Yamada
中科院分区：
文献类型：
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作者：
M. Umehara;Kotaro Yamada
在欧几里得 3 空间中的极小曲面的研究中，Weierstrass 表示起着重要作用。 Bryant [Br] 表明，Weierstrass 表示公式的类似物适用于双曲 3 空间 X3 中平均曲率 -i 恒定的表面。在本文中，我们将术语“恒定平均曲率-i”缩写为 CMC-1。与欧几里得空间中的最小曲面一样，CMC-1 曲面的双曲高斯图被定义为 C U {oo} 的全纯图。然而，与欧几里德情况相反，即使总高斯曲率是有限的，CMC-1 表面的双曲高斯图也可能不会延伸到两端。我们将一个完整的 CMC-1 曲面（其高斯图可以扩展到其所有端点）称为规则端点的 CMC-1 曲面。在本文中，我们制作了一个显式工具来构造规则端部的 CMC-1 表面。在第 2 节中，我们展示了这样的曲面是通过求解一些具有正则奇点的常微分方程来构造的。在我们的 CMC-1 类别中，奥瑟曼不等式不是预期的，科恩-沃森不等式是最好的不等式。我们在第 4 节中表明，Cohn-Vossen 不等式的等式对于 XH3 中的完整 CMC-1 曲面永远不成立。在第 5 节中，我们给出了嵌入 CMC-1 表面的规则端的充分必要条件。在第 6 节中，我们对具有两个规则末端的属 0 的完整 CMC-1 表面进行分类。我们的分类包含新的例子。此外，在第 7 节中，我们构造了几个具有规则嵌入端的新 CMC-1 表面。这些例子中的每一个都有一个不平凡的变形，这在第 3 节中提到。应该指出的是，我们的构造不适用于具有不规则端部的表面。但还有另一种构造：通过扰动欧几里得 3 空间中的最小曲面，作者构造了 CMC-1 曲面，其所有端点都是不规则的（参见 [UY1]）。
In the study of minimal surfaces in the euclidean 3-space, the Weierstrass representation plays an important role. Bryant [Br] showed that an analogue of the Weierstrass-representation formula holds for surfaces of constant mean curvature-i in the hyperbolic 3-space X3. In this article we abbreviate the term "constant mean curvature-i" as CMC-1. Like minimal surfaces in the euclidean space, the hyperbolic Gauss map of CMC-1 surfaces is defined as a holomorphic map to C U {oo}. However, in contrast to the euclidean case, the hyperbolic Gauss map of a CMC-1 surface may not be extended across the ends even if the total Gaussian curvature is finite. We call a complete CMC-1 surface, whose Gauss map can be extended across all of its ends, a CMC-1 surface of regular ends. In this article we produce an explicit tool to construct CMC-1 surfaces of regular ends. In Section 2 we show that such surfaces are constructed by solving some ordinary differential equations with regular singularity. In our CMC-1 category, Ossermann's inequality is not expected and the Cohn-Vossen inequality is the best possible one. We show in Section 4 that the equality of the Cohn-Vossen inequality never holds for complete CMC-1 surfaces in XH3. In Section 5 we give a necessary and sufficient condition that a regular end of a CMC-1 surface be embedded. In Section 6 we classify complete CMC-1 surfaces of genus 0 with two regular ends. Our classification contains new examples. Furthermore, in Section 7, we construct several new CMC-1 surfaces with regular embedded ends. Each of these examples has a nontrivial deformation, which is mentioned in Section 3. It should be remarked that our construction does not work for surfaces with irregular ends. But there is another construction: By perturbing minimal surfaces in the euclidean 3-space, the authors constructed CMC-1 surfaces, all of whose ends are irregular (see [UY1]).