The Structure of Constant Mean Curvature Embeddings in Euclidean Three Space
The Structure of Constant Mean Curvature Embeddings in Euclidean Three Space
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欧氏三空间中常平均曲率嵌入的结构
DOI：
10.1090/pspum/054.1/1216589
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
Nicholas J. Korevaar
中科院分区：
文献类型：
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作者：
Nicholas J. Korevaar
在这些笔记中，我们将概略地描述具有非零常平均曲率和有限拓扑的适当嵌入曲面的已知必要条件。近年来，通过H. Wente [13]，N. Kapouleas [4，5]，H. Karcher [6]，U.平考尔和我。Sterling [12].圆环面和圆柱体的浸入已经具有非常复杂的行为[12，13]，因此一般浸入的常平均曲率曲面在结构上必须非常不同。如果需要适当地嵌入，则其结构可以相当好地表征。这种结构不仅本身很有趣，而且还可以帮助分析涉及多种材料或相的物理系统，例如Cahn-Hilliard方程[2，3]所描述的那些，其中认为缓慢演变的界面可能具有随时间变化的近似恒定的平均曲率。设λ具有平均曲率向量H，|H|定义“外”法线ν为−H/H，并将这样的一个曲面称为M CH曲面。由两个边界的区域被称为外部和内部相应。我们可以缩放空间，使得H = 1。半径为2的球，通常的外法线为ν，是一个MC 1曲面，其内部是球。轴对称MC 1曲面的单参数族称为Delaunay曲面。它们沿轴呈沿着周期性，通过积分一阶O.D.E.对于初始半径ρ ≤ 2，初始导数为零的轮廓曲线，它又是二阶O.D.E.对于常数平均曲率嵌入的Delaunay曲面在半径为1的圆柱体和半径为2的球体链之间插值。下面我们列出有限拓扑MC 1曲面的已知必要条件：
In these notes we will sketch the known necessary conditions for properly embedded surfaces Σ ⊂ 3 which have non-zero constant mean curvature and finite topology. Many examples of constant mean curvature immersions and embeddings have been constructed in recent years through the work of H. Wente [13], N. Kapouleas [4, 5], H. Karcher [6], U. Pinkall and I. Sterling [12]. Immersions of tori and cylinders can already have extremely complicated behavior [12, 13], so general immersed constant mean curvature surfaces must be quite varied in structure. If Σ is required to be properly embedded, it turns out that its structure can be characterized fairly well. Not only is this structure interesting in its own right, but it could also aid in the analysis of physical systems involving several materials or phases, such as those described by the Cahn-Hilliard equation [2, 3], where it is believed that slowly evolving interfaces may have time-varying, approximately constant mean curvature. Let Σ have mean curvature vector H with |H| ≡ H > 0, define the “exterior” normal ν to be −H/H, and call such a Σ an MCH surface. The two regions bounded by Σ are called the exterior and interior accordingly. We may scale space so that H = 1. The sphere of radius 2, with ν the usual exterior normal, is an MC1 surface, whose interior is the ball. The one-parameter family of axially symmetric MC1 surfaces are known as Delaunay surfaces. They are periodic along the axis and are obtained by integrating a first order O.D.E. for the profile curve (say with initial radius ρ ≤ 2 and initial derivative zero) which in turn is the first integral of the second order O.D.E. for constant mean curvature. The embedded Delaunay surfaces interpolate between a cylinder of radius 1 and a chain of radius 2 spheres. Below we list the known necessary conditions on finite topology MC1 surfaces Σ