Toda systems and hypergeometric equations

Lin, Chang-Shou; Nie, Zhaohu; Wei, Juncheng

doi:10.1090/tran/7577

Toda systems and hypergeometric equations
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by Chang-Shou Lin, Zhaohu Nie and Juncheng Wei PDF

Trans. Amer. Math. Soc. 370 (2018), 7605-7626 Request permission

Abstract:

This paper establishes certain existence and classification results for solutions to $\mathrm {SU}(n)$ Toda systems with three singular sources at 0, 1, and $\infty$. First, we determine the necessary conditions for such an $\mathrm {SU}(n)$ Toda system to be related to an $n$th order hypergeometric equation. Then, we construct solutions for $\mathrm {SU}(n)$ Toda systems that satisfy the necessary conditions and also the interlacing conditions from Beukers and Heckman. Finally, for $\mathrm {SU}(3)$ Toda systems satisfying the necessary conditions, we classify, under a natural reality assumption, that all the solutions are related to hypergeometric equations. This proof uses the Pohozaev identity.

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