Co-Axial Metrics on the Sphere and Algebraic Numbers

• Опубліковано в: : Symmetry, Integrability and Geometry: Methods and Applications
• Дата: 2024
• Автори: Chen, Zhijie, Lin, Chang-Shou, Yang, Yifan
• Формат: Стаття
• Мова: Англійська
• Опубліковано: Інститут математики НАН України 2024
• Онлайн доступ: https://nasplib.isofts.kiev.ua/handle/123456789/212155
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Цитувати: Co-Axial Metrics on the Sphere and Algebraic Numbers. Zhijie Chen, Chang-Shou Lin and Yifan Yang. SIGMA 20 (2024), 040, 30 pages

In this paper, we consider the following curvature equation Δ + eᵘ = 4π ((₀ − 1)δ₀ + (₁−1)δ₁ + ∑ⁿ⁺ᵐⱼ₌₁(′ⱼ − 1)δₜⱼ)in ℝ², () = −2(1+∞)ln|| + O(1) as || → ∞, where ₀, ₁, ∞, and ′ⱼ are positive non-integers for 1 ≤ j ≤ , while ′ⱼ ∈ ℕ≥₂ are integers for + 1 ≤ j ≤ + . Geometrically, a solution gives rise to a conical metric ds² =1/2eᵘ|d|² of curvature 1 on the sphere, with conical singularities at 0, 1, ∞ and tⱼ, 1 ≤ j ≤ + , with angles 2π₀, 2π₁, 2π∞, and 2π′ⱼ at 0, 1, ∞ and tⱼ, respectively. The metric ds² or the solution is called co-axial, which was introduced by Mondello and Panov, if there is a developing map () of 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 ₁,…, ₙ₊ₘ. Let ⊂ ℂⁿ⁺ᵐ be the set of those (₁,…, ₙ₊ₘ)'s such that a co-axial metric exists. Among other things, we prove that (i) If = 1, i.e., there is only one integer ′ₙ₊₁ among ′ⱼ, then is a finite set. Moreover, for the case = 0, we obtain a sharp bound on the cardinality of the set . We apply a result due to Eremenko, Gabrielov, and Tarasov (2016) and the monodromy of hypergeometric equations to obtain such a bound. (ii) If ≥ 2, then is an algebraic set of dimension ≤ − 1.