Construction of submanifold with constant mean curvature, and its applications

利用魏尔斯特拉斯表示公式，研究了三维欧氏空间中极小曲面的性质，并推广了它们。首先，我们给出了三维欧氏空间中极小曲面的通量反问题的一个肯定结果。此外，作为(复解析)流的推广，我们定义了双曲空间中常平均曲率曲面的一个新的同调不变量，也称为“流”。利用通量平衡公式证明了双曲空间中常平均曲率单曲面的一些不存在性结果.作为这一非存在性结果的继续，我们尝试对低总绝对曲率的双曲空间中的完全常平均曲率单曲面进行分类，得到了总绝对曲率小于或等于4π的曲面的完全分类.另一方面，作为高维欧氏空间极小曲面的魏尔斯特拉斯型表示公式的推广，我们在某些非紧型对称空间中定义了具有全纯右高斯映射的曲面的概念，并得到了魏尔斯特拉斯-布赖恩特型表示公式.作为该公式的一个应用，我们得到了关于全绝对曲率的一个Osserman型不等式。

We investigated properties of minimal surfaces in the three dimensional euclidean space using the Weierstrass representation formula, and generalizations of them. First, we gave an affirmative result for an inverse problem of flux for minimal surfaces in the three dimensional euclidean space. Moreover, as a generalization of (a complex analytic) flux, we defined a new homology invariant, which is also called as "flux", for surfaces of constant mean curvature one in the hyperbolic three space. Using the balancing formula of the flux, we proved some non-existence results for constant mean curvature one surface in hyperbolic space.As a continuation of this non-existence results, we tried to classify the complete constant mean curvature one surface in hyperbolic space with low total absolute curvature, and we obtained the complete classification for surfaces with total absolute curvature less than or equal to 4π.On the other hand, as a generalization of the Weierstrass-type representation formula for minimal surface with higher dimensional euclidean space, we defined a notion of surfaces with holomorphic right gauss map in some non-compact type symmetric space, and obtained the Weierstrass-Bryant type representation formula. As an application of this formula, we obtained an Osserman-type inequality for total absolute curvature.