Illinois Journal of Mathematics

On metrics of curvature $1$ with four conic singularities on tori and on the sphere

Alexandre Eremenko and Andrei Gabrielov

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We discuss conformal metrics of curvature $1$ on tori and on the sphere, with four conic singularities whose angles are multiples of $\pi$. Besides some general results we study in detail the family of such symmetric metrics on the sphere, with angles $(\pi,3\pi,\pi,3\pi)$. As a consequence we find new Heun’s equations whose general solution is algebraic.

Article information

Illinois J. Math., Volume 59, Number 4 (2015), 925-947.

Received: 10 January 2016
First available in Project Euclid: 27 February 2017

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Zentralblatt MATH identifier

Primary: 34M03: Linear equations and systems 34M05: Entire and meromorphic solutions 30C20: Conformal mappings of special domains 35J91: Semilinear elliptic equations with Laplacian, bi-Laplacian or poly- Laplacian 33E05: Elliptic functions and integrals


Eremenko, Alexandre; Gabrielov, Andrei. On metrics of curvature $1$ with four conic singularities on tori and on the sphere. Illinois J. Math. 59 (2015), no. 4, 925--947. doi:10.1215/ijm/1488186015.

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