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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2405.03161 |
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| _version_ | 1866909190882066432 |
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| author | Mu, Jingyu Shi, Yiqian Xu, Bin |
| author_facet | Mu, Jingyu Shi, Yiqian Xu, Bin |
| contents | On a compact Riemann surface (X) with finite punctures (P_1, \ldots, P_k), we define toric curves as multi-valued, totally unramified holomorphic maps to (\mathbb{P}^n) with monodromy in a maximal torus of ({\rm PSU}(n+1)). \textit{Toric solutions} for the ({\rm SU}(n+1)) system on $X\setminus\{P_1,\ldots, P_k\}$ are recognized by their associated {\it toric} curves in (\mathbb{P}^n). We introduce a character n-ensemble as an (n)-tuple of meromorphic one-forms with simple poles and purely imaginary periods, generating toric curves on (X) minus finitely many points. We establish on $X$ a correspondence between character $n$-ensembles and toric solutions to the ({\rm SU}(n+1)) system with finitely many cone singularities. Our approach not only broadens seminal solutions for up to two cone singularities on the Riemann sphere, as classified by Jost-Wang (Int. Math. Res. Not., (6):277-290, 2002) and Lin-Wei-Ye (Invent. Math., 190(1):169-207, 2012), but also advances beyond the limits of Lin-Yang-Zhong's existence theorems (J. Differential Geom., 114(2):337-391, 2020) by introducing a new solution class. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_03161 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Solutions to ${\rm SU}(n+1)$ Toda system with cone singularities via toric curves on compact Riemann surfaces Mu, Jingyu Shi, Yiqian Xu, Bin Differential Geometry Mathematical Physics Analysis of PDEs Complex Variables Primary 37K10, Secondary 35J47 On a compact Riemann surface (X) with finite punctures (P_1, \ldots, P_k), we define toric curves as multi-valued, totally unramified holomorphic maps to (\mathbb{P}^n) with monodromy in a maximal torus of ({\rm PSU}(n+1)). \textit{Toric solutions} for the ({\rm SU}(n+1)) system on $X\setminus\{P_1,\ldots, P_k\}$ are recognized by their associated {\it toric} curves in (\mathbb{P}^n). We introduce a character n-ensemble as an (n)-tuple of meromorphic one-forms with simple poles and purely imaginary periods, generating toric curves on (X) minus finitely many points. We establish on $X$ a correspondence between character $n$-ensembles and toric solutions to the ({\rm SU}(n+1)) system with finitely many cone singularities. Our approach not only broadens seminal solutions for up to two cone singularities on the Riemann sphere, as classified by Jost-Wang (Int. Math. Res. Not., (6):277-290, 2002) and Lin-Wei-Ye (Invent. Math., 190(1):169-207, 2012), but also advances beyond the limits of Lin-Yang-Zhong's existence theorems (J. Differential Geom., 114(2):337-391, 2020) by introducing a new solution class. |
| title | Solutions to ${\rm SU}(n+1)$ Toda system with cone singularities via toric curves on compact Riemann surfaces |
| topic | Differential Geometry Mathematical Physics Analysis of PDEs Complex Variables Primary 37K10, Secondary 35J47 |
| url | https://arxiv.org/abs/2405.03161 |