Co-Axial Metrics on the Sphere and Algebraic Numbers

Abstract

In this paper, we consider the following curvature equation Δ𝑢 + eᵘ = 4π ((𝜃₀ − 1)δ₀ + (𝜃₁−1)δ₁ + ∑ⁿ⁺ᵐⱼ₌₁(𝜃′ⱼ − 1)δₜⱼ)in ℝ², 𝑢(𝑥) = −2(1+𝜃∞)ln|𝑥| + O(1) as |𝑥| → ∞, where 𝜃₀, 𝜃₁, 𝜃∞, and 𝜃′ⱼ are positive non-integers for 1 ≤ j ≤ 𝑛, while 𝜃′ⱼ ∈ ℕ≥₂ are integers for 𝑛 + 1 ≤ j ≤ 𝑛 + 𝑚. Geometrically, a solution 𝑢 gives rise to a conical metric ds² =1/2eᵘ|d𝑥|² of curvature 1 on the sphere, with conical singularities at 0, 1, ∞ and tⱼ, 1 ≤ j ≤ 𝑛 + 𝑚, with angles 2π𝜃₀, 2π𝜃₁, 2π𝜃∞, and 2π𝜃′ⱼ at 0, 1, ∞ and tⱼ, respectively. The metric ds² 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 𝘵₁,…, 𝘵ₙ₊ₘ. Let 𝐴 ⊂ ℂⁿ⁺ᵐ be the set of those (𝘵₁,…, 𝘵ₙ₊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 on 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 ≥ 2, then 𝐴 is an algebraic set of dimension ≤ 𝑚 − 1.