Fuchsian Equations with Three Non-Apparent Singularities

Symmetry Integrability and Geometry-Methods and Applications

article

Alexandre Eremenko¹  Vitaly Tarasov² 

[1] Purdue University;Indiana University – Purdue University Indianapolis

关键词: Fuchsian equations;    hypergeometric equation;    difference equations;    apparent singularities;    bispectral duality;    positive curvature;    conic singularities;   

DOI  :  10.3842/SIGMA.2018.058

来源: National Academy of Science of Ukraine

【 摘 要 】

We show that for every second order 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 $E$. This map is surjective for generic parameters. This justifies one statement of Klein (1905). We also count the number of such equations $E$ 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.

【 授权许可】

Unknown