Representation formulas of Weierstrass type in submanifold theory

研究了Weierstrass表示的一个整体性质。首先，我们研究了一个与全球问题相关的基本问题。特别地，由于欧氏几何中极小曲面的单值问题被认为是一类全纯形式的积分的周期问题，因此双曲空间中CMC-1曲面的单值问题可以被看作是黎曼曲面上常微分方程的单值问题.在本文中，单值问题是SL（2，C）-单值群化为酉群的条件。要找到这样一个条件的标准是困难的。然而，当一个问题满足某些对称性质时，它可以显式求解。利用这一事实，我们已经构建了大量的CMC-1曲面的例子。与此相关，我们研究了具有圆锥奇点的常正曲率度量，得到了一个分类结果;与CMC-1曲面的分类相关，我们定义了CMC-1曲面上的一个同调不变量，称为通量，并利用这个不变量证明了一些不存在性定理。利用三维Minkowski空间中极大曲面的Weierstrass表示，对具有锥状奇点的曲面进行了分类。

We investigated a global properties of Weierstrass representation. First of all, we studied a fundamental problem related to global problems. In particular, as the monodromy problem for minimal surfaces in Euclidean geometry is considered as a period problem of certain integral of holomorphic forms, that of CMC-1 surfaces in hyperbolic space can be considerd as a monodromy problem of a ordinary differential equation on Riemann surfaces. In this context, the monodromy problem is the condition for SL(2, C)-monodromy group to be reduced to the unitary group. To find a criterion of such a condition is difficult. However, when a problem satisfies some symmetric properties, it can be solved explicitly. Using this fact, we have constructed a large amount of examples of CMC-1 surfaces. Related to this problem, we investigated metrics of constant positive curvature with conical singularities, and obtained a classification result.Related to classification of CMC-1 surfaces, we defined a homology invariant on CMC-1 surface, which is called flux, and proved some non-existence theorem using this invariant. Moreover, using Weierstrass representation for maximal surfaces in Minkowski 3-space, we have classified of surfaces with cone-like singularities with some finiteness.