Two families of reducible spherical conical metrics

TL;DR

This paper studies specific reducible spherical conical metrics, verifying their structure and linking them to Abelian differentials, with explicit geometric decompositions to illustrate their properties.

Contribution

It provides a detailed analysis of two families of reducible spherical conical metrics, confirming theoretical structure theorems and connecting geometry with complex analysis.

Findings

Verification of the structure theorem for these metrics

Explicit geometric decompositions and calculations

Connection between spherical geometry and Abelian differentials

Abstract

We analyze a 1-parameter family of heart shape and a 3-parameter family obtained by gluing three footballs, both of which are examples of reducible spherical conical metrics. For these examples we verify the structure theorem given in [15] and show that such metrics naturally arise from Abelian differentials of the third kind. We then obtain the geometric decomposition using explicit metric and geodesic calculations. This offers new evidence for the interaction between the synthetic spherical geometry and the complex analytic structure of reducible conical metrics.