Co-Axial Metrics on the Sphere and Algebraic Numbers
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 r...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2024 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2024
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/212155 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Co-Axial Metrics on the Sphere and Algebraic Numbers. Zhijie Chen, Chang-Shou Lin and Yifan Yang. SIGMA 20 (2024), 040, 30 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862684623392210944 |
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| author | Chen, Zhijie Lin, Chang-Shou Yang, Yifan |
| author_facet | Chen, Zhijie Lin, Chang-Shou Yang, Yifan |
| citation_txt | Co-Axial Metrics on the Sphere and Algebraic Numbers. Zhijie Chen, Chang-Shou Lin and Yifan Yang. SIGMA 20 (2024), 040, 30 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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.
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| first_indexed | 2026-03-17T10:05:30Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-212155 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-17T10:05:30Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
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| spelling | Chen, Zhijie Lin, Chang-Shou Yang, Yifan 2026-01-30T08:12:37Z 2024 Co-Axial Metrics on the Sphere and Algebraic Numbers. Zhijie Chen, Chang-Shou Lin and Yifan Yang. SIGMA 20 (2024), 040, 30 pages 1815-0659 2020 Mathematics Subject Classification: 57M50 arXiv:2205.13912 https://nasplib.isofts.kiev.ua/handle/123456789/212155 https://doi.org/10.3842/SIGMA.2024.040 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. The authors thank the referees very much for their careful reading and valuable comments. The research of the first author was supported by the National Key R&D Program of China (Grant 2022ZD0117000) and NSFC (No. 12222109, 12071240). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Co-Axial Metrics on the Sphere and Algebraic Numbers Article published earlier |
| spellingShingle | Co-Axial Metrics on the Sphere and Algebraic Numbers Chen, Zhijie Lin, Chang-Shou Yang, Yifan |
| title | Co-Axial Metrics on the Sphere and Algebraic Numbers |
| title_full | Co-Axial Metrics on the Sphere and Algebraic Numbers |
| title_fullStr | Co-Axial Metrics on the Sphere and Algebraic Numbers |
| title_full_unstemmed | Co-Axial Metrics on the Sphere and Algebraic Numbers |
| title_short | Co-Axial Metrics on the Sphere and Algebraic Numbers |
| title_sort | co-axial metrics on the sphere and algebraic numbers |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212155 |
| work_keys_str_mv | AT chenzhijie coaxialmetricsonthesphereandalgebraicnumbers AT linchangshou coaxialmetricsonthesphereandalgebraicnumbers AT yangyifan coaxialmetricsonthesphereandalgebraicnumbers |