The space of complete embedded maximal surfaces with isolated singularities in the 3-dimensional Lorentz-Minkowski space
The space of complete embedded maximal surfaces with isolated singularities in the 3-dimensional Lorentz-Minkowski space
复制标题
3维Lorentz-Minkowski空间中具有孤立奇点的完全嵌入极大曲面空间
DOI：
10.1007/s00208-005-0642-6
复制
发表时间：
2003
影响因子：
1.4
通讯作者：
Rabah Souam
中科院分区：
文献类型：
--
作者：
I. Fernández;F. J. López;Rabah Souam
证明了在=(∈3,dx12+dx22-dx32)中具有有限个奇点的完全内嵌极大曲面是在任意类空间平面上具有圆锥奇点的完整极大图，因此它是渐近于类空间平面或半链面。证明了{x3=0}上n+1≥2个奇点和无穷远处垂直极限法向量的整个极大图的模空间是一个3n+4维可微流形。其中的收敛性是指共形结构和Weierstrass数据的收敛性，等价于图在{x3=0}的紧子集上的一致收敛性。此外，奇异点的位置和无穷远处的对数增长可以作为具有相同底层拓扑结构的全局解析坐标。我们还引入了包含+1个奇点的标记图的模空间（图中的标记是其奇点的排序），它是的（n+1）层覆盖。我们证明了识别由平移、沿垂直轴的旋转、沿水平面的同质或对称不同的标记图，相应的商空间是一个维数为3n−1的解析流形。该流形可以用与图的Weierstrass数据的模空间相关联的旋束来标识。
We prove that a complete embedded maximal surface in = (ℝ3,dx12+dx22-dx32) with a finite number of singularities is an entire maximal graph with conelike singularities over any spacelike plane, and so, it is asymptotic to a spacelike plane or a half catenoid. We show that the moduli space of entire maximal graphs over {x3=0} in withn+1≥2 singular points and vertical limit normal vector at infinity is a 3n+4-dimensional differentiable manifold. The convergence in means the one of conformal structures and Weierstrass data, and it is equivalent to the uniform convergence of graphs on compact subsets of {x3=0}. Moreover, the position of the singular points in ℝ3and the logarithmic growth at infinity can be used as global analytical coordinates with the same underlying topology. We also introduce the moduli space ofmarkedgraphs withn+1 singular points (a mark in a graph is an ordering of its singularities), which is a (n+1)-sheeted covering of . We prove that identifying marked graphs differing by translations, rotations about a vertical axis, homotheties or symmetries about a horizontal plane, the corresponding quotient space is an analytic manifold of dimension 3n−1. This manifold can be identified with a spinorial bundle associated to the moduli space of Weierstrass data of graphs in .