曲面上奇异平坦结构的分类

How close are cone singularities on a random flat surface?

We study the shortest geodesics on flat cone spheres, i.e. flat metrics on the sphere with conical singularities. The length of the shortest geodesic between two singular points can be treated as a function on the moduli space of flat cone spheres with prescribed angle defects. We prove a recurrent relation on the distribution of this function with respect to Thurston's volume form on the moduli space.

Ideal Triangulation and Disk Unfolding of a Singular Flat Surface

An ideal triangulation of a singular flat surface is a geodesic triangulation such that its vertex set is equal to the set of singular points of the surface. Using the fact that each pair of points in a surface has a finite number of geodesics having length $\leq L$ connecting them, where $L$ is any positive number, we prove that each singular flat surface has an ideal triangulation provided that the surface has singular points when it has no boundary components, or each of its boundary components has a singular point. Also, we prove that such a surface contains a finite number of geodesics wh…

Classification of punctures on complete flat surfaces

We investigate the behavior of a complete flat metric on a surface near a puncture. We call a puncture on a flat surface regular if it has a neighborhood which is isometric to that of a point at infinity of a cone. We prove that there are punctures which are not regular if and only if the curvature at the puncture is $4π$.

Signature Calculation of Area Hermitian Form on Some Spaces of Polygons

This chapter is motivated by the paper by Thurston on triangulations of the sphere and singular flat metrics on the sphere. Thurston locally parametrized the moduli space of singular flat metrics on the sphere with prescribed positive curvature data by the complex hyperbolic space of appropriate dimension. This work can be considered as a generalization of signature calculation of the Hermitian form that he made in his paper. The moduli space of singular flat metrics having unit area on the sphere with prescribed curvature data can be locally parametrized by certain spaces of polygons. This ca…

Gromov-Hausdorff Convergence of Metric Quotients and Singular Conic-Flat Surfaces

Given metric quotients $S$ and $S_n$, $n \in \mathbb{N}$, of a metric space $X$, sufficient conditions are provided on the data defining them guaranteeing that $S$ is the Gromov-Hausdorff limit of $S_n$. These conditions are recognized within metric quotients of plane polygons determined by side-pairings known as plain paper-folding schemes. In particular, concrete examples are given of sequences of two-dimensional conic-flat spheres converging to spheres that are conic-flat except around certain singularities, some of them with unbounded curvature in the sense of comparative geometry.

平面的Wild奇点

Wild singularities of flat surfaces

我们考虑平面和他们的度量完成点，特别是奇点的平面结构的表面不延伸。奇点x附近的局部行为可以部分地由拓扑空间L（x）描述，该空间捕获了x可以“线性接近”的所有方式。L（x）的同胚类型是仿射不变量。当x不是锥点或无限角奇点时，我们说它是野的;在这种情况下，有必要向L（x）添加进一步的度量数据，以获得x附近曲面的定量描述。