Fuchsian Equations with Three Non-Apparent Singularities
Fuchsian Equations with Three Non-Apparent Singularities
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DOI：
10.3842/sigma.2018.058
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发表时间：
2018-01
期刊：
影响因子：
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通讯作者：
A. Eremenko;V. Tarasov
中科院分区：
文献类型：
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作者：
A. Eremenko;V. Tarasov
我们证明了对于每一个有n个奇点且n-3明显的Fuchsian线性微分方程E，存在一个超几何方程H和一个带多项式系数的线性微分算子，它将H的解的空间映射到E的解的空间。这证明了Klein（1905）的一个说法是正确的。我们还计算了具有指定奇点和指数的这类方程的数目。我们将这些结果应用于具有二次奇异点的穿孔球上曲率1的共形度量的描述，除了三个点外，其余的点都具有整数角。
We show that for every Fuchsian linear differential equation E with n singularities of which n-3 are apparent there exists a hypergeometric equation H and a linear differential operator with polynomial coefficients which maps the space of solutions of H into the space of solutions of E. For generic parameters this map is surjective. This justifies one statement of Klein (1905). We also count the number of such equations with prescribed singularities and exponents. We apply these results to the description of conformal metrics of curvature 1 on the punctured sphere with conic singularities, all but three of them having integer angles.