Research on the structure of ideal boundaries of piecewise Riemannian polyhedra from the viewpoint of total curvature

我们的研究领域是分段黎曼2-多面体，这是一个组合2-多面体。使得每个2-单形等距于某个黎曼2-流形上由三条光滑曲线围成的三角形。本研究项目的目的是从全曲率的角度研究非紧分片黎曼2-多面体的一些性质，主要结果如下：1. Y.Machigashira和F.Ohtsuka的“长度曲面上的全过剩“：在满足适当条件的长度空间上，其中可能包含一些离散奇点，我们推广了全曲率的概念（本文称之为全过剩），并证明了一些推广的定理.“非紧分片黎曼2-多面体的全曲率“J.Itoh和F.Ohtsuka：我们在分片黎曼2-多面体上引入了两种全曲率的定义（全曲率和w-全曲率），得到了它们容许全曲率的一些推广定理，并观察了全曲率和w-全曲率的区别.“Structure of flat piecewise Riemannian 2-polyhedra“by F.Ohtsuka：我们给出了分片Riemannian 2-polyhedra的平坦性的定义并刻画了它的特征，Hertsuka想研究常曲率2-polyhedra的结构以及多面体与其理想边界的关系等。

The field of our study is piecewise Riemannian 2-polyhedra, which is a combinatorial 2-polyhedra. such that each 2-simplex is isometric to a triangle bounded by three smooth curves on some Riemannian 2-manifold. The purpose in this research project is to investigate some properties of noncompact piecewise Riemannian 2-polyhedra from the viewpoint of total curvature.Main results in this project is the following.1."Total excess on length surfaces"by Y.Machigashira and F.Ohtsuka : On length spaces satisfying some suitable conditions, which may contain some discrete singular points, we generalize the notion of total curvature (we call it total excess in this paper) and prove some generalized theorems.2."Total curvature of noncompact piecewise Riemannian 2-polyhedra"by J.Itoh and F.Ohtsuka : We introduce two kind of definitions of total curvature (total curvature and w-total curvature) on piecewise Riemannian 2-polyhedra and obtain some generalized theorems for them admitting total curvature and observe the difference of total curvature and w-total curvature.3."Structure of flat piecewise Riemannian 2-polyhedra"by F.Ohtsuka : We give a definition of flatness of piecewise Riemannian 2-polyhedra and characterize them.Henceforth I would like to investigate the structure of 2-polyhedra of constant curvature and the relationship between polyhedra and its ideal boundaries and so on.