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Conformal Geometry and Dynamics

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Metrics with four conic singularities and spherical quadrilaterals


Authors: Alexandre Eremenko, Andrei Gabrielov and Vitaly Tarasov
Journal: Conform. Geom. Dyn. 20 (2016), 128-175
MSC (2010): Primary 30C20, 34M03
DOI: https://doi.org/10.1090/ecgd/295
Published electronically: May 16, 2016
MathSciNet review: 3500744
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Abstract: A spherical quadrilateral is a bordered surface homeomorphic to a closed disk, with four distinguished boundary points called corners, equipped with a Riemannian metric of constant curvature $ 1$, except at the corners, and such that the boundary arcs between the corners are geodesic. We discuss the problem of classification of these quadrilaterals and perform the classification up to isometry in the case that two angles at the corners are multiples of $ \pi $. The problem is equivalent to classification of Heun's equations with real parameters and unitary monodromy.


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Additional Information

Alexandre Eremenko
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-2067

Andrei Gabrielov
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-2067

Vitaly Tarasov
Affiliation: Department of Mathematics, IUPUI, Indianapolis, Indiana 46202-3216 — and — St. Petersburg branch of Steklov Mathematical Institute, 27, Fontanka, 191023 St. Petersburg, Russia

DOI: https://doi.org/10.1090/ecgd/295
Keywords: Surfaces of positive curvature, conic singularities, Heun equation, Schwarz equation, accessory parameters, conformal mapping, circular polygons
Received by editor(s): July 7, 2015
Received by editor(s) in revised form: March 11, 2016, and March 12, 2016
Published electronically: May 16, 2016
Additional Notes: The first author was supported by NSF grant DMS-1361836.
The second author was supported by NSF grant DMS-1161629.
Article copyright: © Copyright 2016 American Mathematical Society