论文标题
几何结构和可还原球形锥形指标的存在
Geometric structure and existence of reducible spherical conical metrics
论文作者
论文摘要
共形度量$ {\ rm d} s^{2} $具有有限的许多圆锥形奇点，恒定高斯曲率$ k = 1 $在紧凑的riemann表面上称为球形锥形度量。当$ {\ rm d} s^{2} $的相关单型组可对角度溶液时，我们将$ {\ rm d} s^{2} $称为可还原的球形锥形指标。可简化的球形圆锥指标的最简单情况是“足球”，它表示带有球形圆锥指标的2杆，该指标的距离$π$恰好是两个奇异性。这项研究深入研究了紧凑的Riemann表面上的内在几何结构和可还原球形锥形指标的存在。我们证明，任何这样的球形表面都可以通过沿一组合适的地理素切割来将其分为有限数量的零件，这些大地测量会连接锥形奇点和度量的某些光滑点。尤其是，每件作品都是通过沿着连接两个圆锥形奇异性的大地测量的足球而获得的一部分。作为应用，提出了存在这种度量的角度条件。最值得注意的是，我们的研究证明了可还原的球形圆锥度度量的存在，其中摩尔斯函数的所有鞍点都位于同一测地上。
A conformal metric ${\rm d}s^{2}$ with finitely many conical singularities of constant Gaussian curvature $K=1$ on a compact Riemann surface is referred to as a spherical conical metric. When the associated monodromy group of ${\rm d}s^{2}$ is diagonalizable, we refer to ${\rm d}s^{2}$ as a reducible spherical conical metric. The simplest case of a reducible spherical conical metric is a `football', which denotes a 2-sphere with a spherical conical metric that has precisely two singularities separated by a distance of $π$. This study delves into the intrinsic geometric structure and existence of reducible spherical conical metrics on compact Riemann surfaces. We demonstrate that any such spherical surface can be divided into a finite number of pieces by cutting along a set of suitable geodesics, which connect the conical singularities and some smooth points of the metric. Especially, each piece is isometric to a portion obtained by cutting a football along a geodesic that joins the two conical singularities. As an application, an angle condition for the existence of such a metric is presented. Most notably, our study demonstrates the existence of a reducible spherical conical metric where all the saddle points of a Morse function are located on the same geodesic.
