Holomorphic Representation of Constant Mean Curvature Surfaces in Minkowski Space: Consequences of Non-Compactness in Loop Group Methods
Holomorphic Representation of Constant Mean Curvature Surfaces in Minkowski Space: Consequences of Non-Compactness in Loop Group Methods
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DOI：
10.1016/j.aim.2009.09.006
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发表时间：
2008-04
影响因子：
1.7
通讯作者：
D. Brander;W. Rossman;N. Schmitt
中科院分区：
文献类型：
--
作者：
D. Brander;W. Rossman;N. Schmitt
在三维Minkowski空间R2，1中，给出了类空常平均曲率曲面的无穷维广义Weierstrass表示.该公式类似于Dorfmeister，Pedit和Wu在欧几里得空间中对CMC曲面给出的公式，用SU 1，1代替SU 2。然而，后一组的非紧性意味着用于构造曲面的循环组的岩泽分解不是全局的。我们证明了它是定义在一个开的稠密子集，后加倍的大小的真实的形式SU 1，1，并证明了几个结果有关的行为表面的边界，这个开集遇到。然后，我们使用广义Weierstrass表示创建和分类的新的例子，类空CMC表面在R2，1。特别是，我们分类表面的革命和表面的螺旋运动对称性，以及研究另一类表面的度量旋转不变。
We give an infinite dimensional generalized Weierstrass representation for spacelike constant mean curvature (CMC) surfaces in Minkowski 3-space R2,1. The formulation is analogous to that given by Dorfmeister, Pedit and Wu for CMC surfaces in Euclidean space, replacing the group SU2with SU1,1. The non-compactness of the latter group, however, means that the Iwasawa decomposition of the loop group, used to construct the surfaces, is not global. We prove that it is defined on an open dense subset, after doubling the size of the real form SU1,1, and prove several results concerning the behavior of the surface as the boundary of this open set is encountered. We then use the generalized Weierstrass representation to create and classify new examples of spacelike CMC surfaces in R2,1. In particular, we classify surfaces of revolution and surfaces with screw motion symmetry, as well as studying another class of surfaces for which the metric is rotationally invariant.