Teichmuller theory and handle addition for minimal surfaces
Teichmuller theory and handle addition for minimal surfaces
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Teichmuller 理论和最小曲面的手柄添加
DOI：
10.2307/3597281
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发表时间：
1998
影响因子：
4.9
通讯作者：
Michael Wolf
中科院分区：
文献类型：
--
作者：
Matthias J. Weber;Michael Wolf
我们开发了 Teichmuller 理论方法，通过向 $\BE^3$ 中现有的最小曲面添加手柄和平面端来构造 $\BE^3$ 中的新最小曲面。我们在一类有趣的最小曲面上展示了这种方法，这些曲面可能被嵌入，并且它们的属具有低度高\ss图；这些表面的（Weierstrass 数据）周期问题是任意维度的。
特别是，我们在欧几里得三空间 $\BE^3$ 中展示了一个完整最小曲面的二参数族，它推广了 C. Costa 的突破性最小曲面；这些新的表面（至少）嵌入在一个紧凑的集合之外，并通过它们具有的末端数量及其属数（大致）进行索引。尽管它们属于任意大的属，但它们最多具有八个自对称性，并且由于多种原因而令人感兴趣。此外，我们的方法还扩展到证明一些自然候选曲面类别不能实现为 $\BE^3$ 中的最小曲面。作为这项工作的两个方面的结果，我们获得了将一系列曲面分类为可实现或不可实现的最小曲面。
We develop Teichmuller theoretical methods to construct new minimal surfaces in $\BE^3$ by adding handles and planar ends to existing minimal surfaces in $\BE^3$. We exhibit this method on an interesting class of minimal surfaces which are likely to be embedded, and have a low degree Gau\ss map for their genus; the (Weierstrass data) period problem for these surfaces is of arbitrary dimension.
In particular, we exhibit a two-parameter family of complete minimal surfaces in the Euclidean three-space $\BE^3$ which generalize the breakthrough minimal surface of C. Costa; these new surfaces are embedded (at least) outside a compact set, and are indexed (roughly) by the number of ends they have and their genus. They have at most eight self-symmetries despite being of arbitrarily large genus, and are interesting for a number of reasons. Moreover, our methods also extend to prove that some natural candidate classes of surfaces cannot be realized as minimal surfaces in $\BE^3$. As a result of both aspects of this work, we obtain a classification of a family of surfaces as either realizable or unrealizable as minimal surfaces.