Geometric structure and existence of reducible spherical conical metrics

TL;DR

This paper investigates the geometric structure and existence conditions of reducible spherical conical metrics on compact Riemann surfaces, revealing their decomposition into football-shaped pieces and establishing an angle criterion for their existence.

Contribution

It provides a detailed geometric decomposition of reducible spherical conical metrics and introduces an angle condition for their existence, including an example with Morse function saddle points.

Findings

Any reducible spherical conical metric can be decomposed into football-shaped pieces.

An explicit angle condition for the existence of such metrics is established.

Existence of a reducible metric with all Morse saddle points on a single geodesic is demonstrated.

Abstract

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 $\pi$. 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.