Two families of reducible spherical conical metrics

TL;DR

The paper analyzes reducible spherical conical metrics on the sphere by examining two explicit families: a heart-shaped metric with cone angles $4\\pi, 2\\pi\\\\beta, 2\\pi\\gamma$ where $\\beta+\\gamma=1$, and a three-football metric with angles $4\\pi,4\\pi,2\\pi\\beta,2\\pi(\\alpha+\\beta),2\\pi(\\alpha+\\gamma),2\\pi\\gamma$. Building on the structure theorem for reducible metrics, it constructs explicit Abelian differentials of the third kind $\\omega$ and develops the metrics via the developing map $F$, yielding closed-form expressions for $ds^2$ in terms of $F$ and the parameter $c$. The heart-shaped metric decomposes into two footballs along a common boundary, with geodesic lengths determined by $c$ through $L(0,1)=L(0,-\\frac{\\gamma}{\\beta})=2\\arctan(|F(0)|)$, illustrating a concrete geometric realization of the theory. For the three-football case, a 3-parameter family arises from the residues and pole positions of $\\omega$, encoded by $F(z)=c(z-P_{\\beta})^{-\\beta}(z-P_{\\alpha})^{\\alpha+\\beta}(z-P_{\\gamma})^{\\gamma}$, with geodesic lengths $L(a,b)=2|\\arctan|F(b)|-\\arctan|F(a)||$ and an additional angle parameter between geodesics; this provides explicit decompositions into three footballs and validates the broader framework of the interaction between spherical geometry and the complex-analytic structure of reducible conical metrics. The results connect to and extend prior structure theorems (e.g., Wei–Wu–Xu 2022; Tahar 2022) by giving tangible constructions and metric-geometric decompositions for these two families.

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.

Figures (6)

• Figure 2.1: A picture of $\mathbb{S}_{\{2,\beta,\gamma\}}^2$ with $\beta+\gamma=1$, obtained from gluing two footballs $\mathbb{S}^2_{\beta,\beta}$ and $\mathbb{S}^2_{\gamma, \gamma}$
• Figure 2.2:
• Figure 2.3: Geodesics from $1$ to $\infty$ and from $-\frac{\gamma}{\beta}$ to $\infty$. They both go through 0 and our calculation shows that the lengths are both $\pi$.
• Figure 3.1: One example of $\mathbb{S}^2_{2,2,\beta,\alpha+\beta, \alpha+\gamma,\gamma}$ obtained by gluing three footballs $\mathbb{S}^2_{\beta, \beta}, \mathbb{S}^2_{\alpha, \alpha}, \mathbb{S}^2_{\gamma, \gamma}$
• Figure 3.2: Another example of $\mathbb{S}^2_{2,2,\beta,\alpha+\beta,\alpha+\gamma, \gamma}$

Theorems & Definitions (8)

• Theorem 1.1
• Theorem 1.2
• Remark 1.1
• Theorem 1.3
• Remark 1.2
• Remark 2.1
• Theorem 2.1
• Theorem 3.1