球面多边形的新型四顶点型定理

Projective geometry of polygons and discrete 4-vertex and 6-vertex theorems

The paper concerns discrete versions of the three well-known results of projective differential geometry: the four vertex theorem, the six affine vertex theorem and the Ghys theorem on four zeroes of the Schwarzian derivative. We study geometry of closed polygonal lines in $\bbRP^d$ and prove that polygons satisfying a certain convexity condition have at least d+1 flattenings. This result provides a new approach to the above mentioned classical theorems.

Polygons of the Lorentzian plane and spherical simplexes

It is known that the space of convex polygons in the Euclidean plane with fixed normals, up to homotheties and translations, endowed with the area form, is isometric to a hyperbolic polyhedron. In this note we show a class of convex polygons in the Lorentzian plane such that their moduli space, if the normals are fixed and endowed with a suitable area, is isometric to a spherical polyhedron. These polygons have an infinite number of vertices, are space-like, contained in the future cone of the origin, and setwise invariant under the action of a linear isometry.

球面多边形面积公式的预量化解法

Area formula for spherical polygons via prequantization

本文提出了一种基于预量化解法计算球面多边形面积的新公式，替代传统高斯-博内定理方法，适用于更广泛的退化情况。

On the symplectic volume of the moduli space of Spherical and Euclidean polygons

In this paper, we study the symplectic volume of the moduli space of polygons by using Witten's formula. We propose to use this volume as a measure for the flexibility of a polygon with fixed side-lengths. The main result of our is that among all the Spherical and Euclidean polygons with fixed perimeter the regular one is the most flexible and that among all the spherical polygons the regular one with side-length $π/2$ is the most flexible.

H^{2,2}中的多项式极大曲面与球面多边形

Polynomial Maximal Surfaces in H^{2,2} and Spherical Polygons

研究伪双曲空间中具有类光多边形渐近边界的完备极大曲面模空间，引入转动不变量分类其连通分支，通过多项式给出模空间的几何参数化，关联Hitchin分支的相关研究。

Geometric decompositions of surfaces with spherical metric and conical singularities

We prove that any compact surface with constant positive curvature and conical singularities can be decomposed into irreducible components of standard shape, glued along geodesic arcs connecting conical singularities. This is a spherical analog of the geometric triangulations for flat surfaces with conical singularities. The irreducible components include not only spherical triangles but also other interesting spherical polygons. In particular, we present the class of \textit{half-spherical concave polygons} that are spherical polygons without diagonals and that can be arbitrarily complicated.…