Fuchsian Equations with Three Non-Apparent Singularities

TL;DR

This paper demonstrates a method to relate certain second order Fuchsian equations with three non-apparent singularities to hypergeometric equations via polynomial coefficient operators, with applications to conformal metrics on punctured spheres.

Contribution

It establishes a surjective mapping from hypergeometric solutions to a class of Fuchsian equations with three non-apparent singularities, and counts such equations with fixed singularities and exponents.

Findings

Existence of a polynomial coefficient operator mapping hypergeometric solutions to Fuchsian solutions.

Surjectivity of this map for generic parameters.

Application to conformal metrics with conic singularities on punctured spheres.

Abstract

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.