Minimal Surfaces of Higher Genus with Finite Total Curvature
Minimal Surfaces of Higher Genus with Finite Total Curvature
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总曲率有限的高维极小曲面
DOI：
10.1007/s002050050021
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发表时间：
1997
影响因子：
2.5
通讯作者：
M. Wohlgemuth
中科院分区：
文献类型：
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作者：
M. Wohlgemuth
在极小曲面的研究中，有限全曲率的完全嵌入是一个重要的研究课题。这是一个长期存在的猜想，在13的唯一例子是平面和悬链。由于奥斯曼[奥斯]表明，完整的极小曲面的有限总曲率可以表示的亚纯数据紧凑黎曼曲面，有一个新的兴趣应用技术的复杂分析，在这里的连接几何和分析是通过维尔斯特拉斯表示。更一般地说，如果极小曲面可以用紧致Riemann曲面上的亚纯数据来描述，则我们将其分类为代数曲面。近十年来，代数极小曲面的研究取得了很大的进展。GACKSTATTER [Ga]和JORGE & MEEKS [Jo/Me]讨论了这种曲面的端点的行为; 1984年，COSTA [Co]构造了一个亏格为1的完全极小曲面，其总曲率为12，有三个嵌入端点。霍夫曼和米克斯[何/梅尔]在1985年表明，这个表面确实是嵌入式的。此外，通过增加对称性，他们建设，为每属p 1，一个嵌入的极小曲面与三个结束和总曲率<$4 2p 1。Callahan、HOFFMAN和MEEKS [Ca/Ho/Me]描述了具有四个端点的有限总曲率的极小曲面，并以数值方式考虑了周期问题。在目前的文件中，我们提供了一个分析证明，这些表面的周期问题有一个解决方案。对于极小曲面的构造，对称性假设起着重要的作用。更高的对称数使得处理周期问题和调整剩余的自由参数以获得所需的最小曲面容易得多。由于这些原因，我们试图在所有的结构中实现最大的对称性。从悬链面开始，我们提出了一个封闭的公式，用于向先前构建的曲面添加手柄和嵌入端点。这一程序有许多优点。首先，它是构造性的，这意味着“复杂”的极小曲面来自“简单”的极小曲面，这使得更容易理解它们的拓扑和形态。第二、
In the study of minimal surfaces, complete embeddings of finite total curvature are of major interest. It was a long-standing conjecture that the only examples in 13 are the plane and the catenoid. Since OSSERMAN [Oss] showed that complete minimal surfaces of finite total curvature can be represented by meromorphic data on compact Riemann surfaces, there has been a renewed interest in the application of techniques of complex analysis, where here the connection between geometry and analysis is given via the Weierstrass representation. More generally, we classify a minimal surface as algebraic if it can be described in terms of meromorphic data on a compact Riemann surface. In the last ten years there has been much progress in the study of algebraic minimal surfaces. GACKSTATTER [Ga] and JORGE & MEEKS [Jo/Me] have discussed the behavior at the ends of such surfaces; in 1984 COSTA [Co] constructed a complete minimal surface of genus 1 with total curvature ÿ12 with three embedded ends. HOFFMAN & MEEKS [Ho/Mel] showed in 1985 that this surface is indeed embedded. Furthermore by increasing the symmetry they constructed, for every genus p 1, an embedded minimal surface with three ends and total curvature ÿ4 2p 1. CALLAHAN, HOFFMAN & MEEKS [Ca/Ho/Me] described minimal surfaces of finite total curvature with four ends, considering the period problem numerically. In the current paper we provide an analytic proof that the period problem for these surfaces has a solution. For the constructions of minimal surfaces, symmetry assumptions play an important role. A higher number of symmetries makes it much easier to handle the period problem and to adjust the remaining free parameters to obtain the desired minimal surface. For these reasons we try to achieve a maximum of symmetry in all our constructions. Starting with the catenoid, we present a closed formula for adding handles and embedded ends to previously constructed surfaces. This procedure has a number of advantages. First, it is constructive, which means that ‘‘complicated’’minimal surfaces arise from ‘‘simpler’’minimal surfaces, making it easier to understand their topology and morphology. Secondly,