On metrics of curvature 1 with four conic singularities on tori and on the sphere. (English) Zbl 1366.30029

Eremenko, Alexandre; Gabrielov, Andrei

Ill. J. Math. 59, No. 4, 925-947 (2015).

The authors discuss conformal metrics of curvature \(1\) on tori and on the sphere, with four conic singularities whose angles are multiples of \(\pi\). Let \(\rho(z)|dz|\) be the length element of the metric in a local conformal coordinate. The behavior at a singularity is \(\rho(z) \sim c| z|^{\alpha-1}\) where \(\alpha>0\). The considered question is related to linear differential equations with regular singular points \(w''+\frac 12 R\), \( w=0\). Special cases of \(R\) are considered.

Reviewer: Vladimir Mityushev (Kraków)

MSC:

30F45	Conformal metrics (hyperbolic, Poincaré, distance functions)
34M03	Linear ordinary differential equations and systems in the complex domain
34M05	Entire and meromorphic solutions to ordinary differential equations in the complex domain
35J91	Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
33E05	Elliptic functions and integrals

Keywords:

conformal metrics with conic singularities; complex linear differential equations

References:

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