The uniqueness of the helicoid
The uniqueness of the helicoid
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DOI：
10.4007/annals.2005.161.727
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发表时间：
2005-03
影响因子：
4.9
通讯作者：
W. Meeks;H. Rosenberg
中科院分区：
文献类型：
--
作者：
W. Meeks;H. Rosenberg
在本文中，我们将讨论在R3中适当嵌入最小曲面M的有限拓扑几何。有限拓扑的M意味着M同纯于紧曲面M³（k属和空边界）减去有限个数的点p1，…， pj∈M³，称为穿刺。M中一个穿孔的封闭邻域E称为M的一个端点。我们将选择足够小的端点，使它们在拓扑上是S1 ×[0,1]，因此是环状的。我们注意到M是可定向的，因为M适当地嵌入在R3中。最简单的例子（1776年由Meusnier发现）是螺旋面和链状面（当然还有一个平面）。直到1982年才发现了另一个例子。在他在英帕的论文中，Celso Costa写下了一个完整的最小曲面的Weierstrass表示，该曲面以3个穿孔环面为模型。他观察到该表面的三个末端被嵌入：一个上链型end1，一个下链型end1和一个中平面型end2[8]。随后，Hoffman和Meeks[15]证明了这个例子是嵌入的，他们构造了对于每一个有限正的k属嵌入的例子，k属和三个端点。1993年，Hoffman、Karcher和Wei[14]发现了1属和1环端完全极小曲面的Weierstrass数据。计算机生成的图像表明，该表面是嵌入的，其末端渐近于螺旋体的末端。霍夫曼、韦伯和沃尔夫现在已经证明了存在这样一个嵌入表面。此外，计算机证据表明，人们可以在螺旋面上添加任意有限数目的k个柄，以获得一个适当嵌入的k类最小曲面渐近于螺旋面。多年来，除了平面和螺旋面之外，人们一直在寻找单连通的例子。我们将证明没有这样的例子。
In this paper we will discuss the geometry of finite topology properly embedded minimal surfaces M in R3. M of finite topology means M is homeomorphic to a compact surface M̂ (of genus k and empty boundary) minus a finite number of points p1, ..., pj ∈ M̂ , called the punctures. A closed neighborhood E of a puncture in M is called an end of M . We will choose the ends sufficiently small so they are topologically S1 × [0, 1) and hence, annular. We remark that M̂ is orientable since M is properly embedded in R3. The simplest examples (discovered by Meusnier in 1776) are the helicoid and catenoid (and a plane of course). It was only in 1982 that another example was discovered. In his thesis at Impa, Celso Costa wrote down the Weierstrass representation of a complete minimal surface modelled on a 3-punctured torus. He observed the three ends of this surface were embedded: one top catenoidtype end1, one bottom catenoid-type end, and a middle planar-type end2 [8]. Subsequently, Hoffman and Meeks [15] proved this example is embedded and they constructed for every finite positive genus k embedded examples of genus k and three ends. In 1993, Hoffman, Karcher and Wei [14] discovered the Weierstrass data of a complete minimal surface of genus one and one annular end. Computer generated pictures suggested this surface is embedded and the end is asymptotic to an end of a helicoid. Hoffman, Weber and Wolf [17] have now given a proof that there is such an embedded surface. Moreover, computer evidence suggests that one can add an arbitrary finite number k of handles to a helicoid to obtain a properly embedded genus k minimal surface asymptotic to a helicoid. For many years, the search went on for simply connected examples other than the plane and helicoid. We shall prove that there are no such examples.