The theory of minimal surfaces in $M \times \mathbb{R}$
The theory of minimal surfaces in $M \times \mathbb{R}$
复制标题
$M imes mathbb{R}$ 中的最小曲面理论
DOI：
10.4171/cmh/36
复制
发表时间：
2005
影响因子：
0.9
通讯作者：
H. Rosenberg
中科院分区：
文献类型：
--
作者：
W. Meeks;H. Rosenberg
在本文中，我们发展了$M \乘以$ mathbb{R}$中适当嵌入极小曲面的理论，其中$M$是一个封闭的可定向黎曼曲面。我们构造了许多不同拓扑和几何的例子。我们建立了几个全球性的结果。第一个定理说明有界曲率的例子具有线性面积增长，因此是准周期的。然后，我们将该定理应用于稳定例子的研究和分类。我们证明了拓扑结果，即每个例子都有有限个数的端点。我们应用Colding和Minicozzi的最新理论证明了有限拓扑的例子具有有界曲率。并证明了其中一些曲面嵌入的拓扑唯一性
In this paper, we develop the theory of properly embedded minimal surfaces in $M \times \mathbb{R}$, where $M$ is a closed orientable Riemannian surface. We construct many examples of different topology and geometry. We establish several global results. The first of these theorems states that examples of bounded curvature have linear area growth, and so, are quasiperiodic. We then apply this theorem to study and classify the stable examples. We prove the topological result that every example has a finite number of ends. We apply the recent theory of Colding and Minicozzi to prove that examples of finite topology have bounded curvature. Also we prove the topological unicity of the embedding of some of these surfaces