The Topology, Geometry and Conformal Structure of Properly Embedded Minimal Surfaces
The Topology, Geometry and Conformal Structure of Properly Embedded Minimal Surfaces
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正确嵌入的最小曲面的拓扑、几何和共形结构
DOI：
10.4310/jdg/1102536205
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发表时间：
2004
影响因子：
2.5
通讯作者：
IIIHarold Rosenberg
中科院分区：
文献类型：
--
作者：
P. Collin;R. Kusner;W. Meeks;IIIHarold Rosenberg
本文开发了一种新的工具，用于理解适当地极小地嵌入到欧几里得三维空间中的具有多个一端(通常是无限拓扑)的曲面。在这样的表面上，两端的集合形成了紧凑的Hausdorff空间，自然地按空间中两端的相对高度排序。我们的主要结果之一是曲面的中端具有二次面积增长，因此不是极限端。例如，这意味着曲面最多可以有两个极限端(在排序的顶部和底部)，这是一个很强的拓扑限制。对这种曲面的渐近几何和共形结构也有限制：例如，我们证明了如果曲面恰好有两个极限端(就像经典的Riemann楼梯例子一样)，那么它是常返的(也就是说，几乎所有的布朗路径在曲面上是稠密的，特别是曲面上的任何正调和函数都是常量的)。这些结果在证明该理论的几个最新进展方面发挥了重要作用，包括螺旋面的唯一性、适当浸入极小曲面上的坐标函数的通量不变性以及适当嵌入极小曲面的拓扑分类。
This paper develops new tools for understanding surfaces with more than one end (and usually, of infinite topology) which properly minimally embed into Euclidean three-space. On such a surface, the set of ends forms a compact Hausdorff space, naturally ordered by the relative heights of the ends in space. One of our main results is that the middle ends of the surface have quadratic area growth, and are thus not limit ends. This implies, for instance, that the surface can have at most two limit ends (at the top and bottom of the ordering), which is a strong topological restriction. There are also restrictions on the asymptotic geometry and conformal structure of such a surface: for example, we prove that if the surface has exactly two limit ends (as do the classical Riemann Staircase examples), then it is recurrent (that is, almost all Brownian paths are dense in the surface, and in particular any positive harmonic function on the surface is constant). These results have played an important role in the proof of several recent advances in the theory, including the uniqueness of the helicoid, the invariance of flux for a coordinate function on a properly immersed minimal surface, and the topological classification of properly embedded minimal surfaces.