Co-Axial Metrics on the Sphere and Algebraic Numbers

Abstract

In this paper, we consider the following curvature equation $$Δu+{\rm e}^u=4π\biggl((θ_0-1)δ_0+(θ_1-1)δ_1 +\sum_{j=1}^{n+m}\bigl(θ_j'-1\bigr)δ_{t_j}\biggr)\qquad \text{in}\ \mathbb R^2,$$ $$u(x)=-2(1+θ_\infty)\ln|x|+O(1)\qquad \text{as} \ |x|\to\infty,$$ where $θ_0$, $θ_1$, $θ_\infty$, and $θ_{j}'$ are positive non-integers for $1\le j\le n$, while $θ_{j}'\in\mathbb{N}_{\geq 2}$ are integers for $n+1\le j\le n+m$. Geometrically, a solution $u$ gives rise to a conical metric ${\rm d}s^2=\frac12 {\rm e}^u|{\rm d}x|^2$ of curvature $1$ on the sphere, with conical singularities at $0$, $1$, $\infty$, and $t_j$, $1\le j\le n+m$, with angles $2πθ_0$, $2πθ_1$, $2πθ_\infty$, and $2πθ_{j}'$ at $0$, $1$, $\infty$, and $t_j$, respectively. The metric ${\rm d}s^2$ or the solution $u$ is called co-axial, which was introduced by Mondello and Panov, if there is a developing map $h(x)$ of $u$ such that the projective monodromy group is contained in the unit circle. The sufficient and necessary conditions in terms of angles for the existence of such metrics were obtained by Mondello-Panov (2016) and Eremenko (2020). In this paper, we fix the angles and study the locations of the singularities $t_1,\dots,t_{n+m}$. Let $A\subset\mathbb{C}^{n+m}$ be the set of those $(t_1,\dots,t_{n+m})$'s such that a co-axial metric exists, among other things we prove that (i) If $m=1$, i.e., there is only one integer $θ_{n+1}'$ among $θ_j'$, then $A$ is a finite set. Moreover, for the case $n=0$, we obtain a sharp bound of the cardinality of the set $A$. We apply a result due to Eremenko, Gabrielov, and Tarasov (2016) and the monodromy of hypergeometric equations to obtain such a bound. (ii) If $m\ge 2$, then $A$ is an algebraic set of dimension $\leq m-1$.