Geodesic
Spherical quadrilaterals with three non-integer angles
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 12 (2016) no. 2, pp. 134-167 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 one corner of a quadrilateral is integer (i.e., its angle is a multiple of $\pi$) while the angles at its other three corners are not multiples of $\pi$. The problem is equivalent to classification of Heun's equations with real parameters and unitary monodromy, with the trivial monodromy at one of its four singular point. 
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A. Eremenko; A. Gabrielov; V. Tarasov. Spherical quadrilaterals with three non-integer angles. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 12 (2016) no. 2, pp. 134-167. http://geodesic.mathdoc.fr/item/JMAG_2016_12_2_a2/

[1] L. Ahlfors, Conformal Invariants. Topics in Geometric Function Theory, Reprint of the 1973 original, AMS Chelsea Publishing, Providence, RI, 2010 | MR | Zbl

[2] C.-L. Chai, C.-S. Lin, C.-L. Wang, “Mean Field Equations, Hyperelliptic Curves and Modular Forms, I”, Camb. J. Math., 3:1, 2 (2015), 127–274 | DOI | MR | Zbl

[3] A. Eremenko, “Metrics of Positive Curvature with Conic Singularities on the Sphere”, Proc. Amer. Math. Soc., 132 (2004), 3349–3355 | DOI | MR | Zbl

[4] A. Eremenko, A. Gabrielov, V. Tarasov, “Metrics with Conic Singularities and Spherical Polygons”, Illinois J. Math., 58:3 (2014), 739–755 | MR | Zbl

[5] A. Eremenko, A. Gabrielov, V. Tarasov, Metrics with Four Conic Singularities and Spherical Quadrilaterals, arXiv: 1409.1529

[6] S. Fujimori, Y. Kawakami, M. Kokubu, W. Rossman, M. Umehara, K. Yamada, “CMC-1 Trinoids in Hyperbolic 3-Space and Metrics of Constant Curvature One with Conical Singularities on the 2-Sphere”, Proc. Japan Acad., 87 (2011), 144–149 | DOI | MR | Zbl

[7] F. Gantmakher, M. Krein, Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems, AMS, Chelsea–Providence, RI, 2011 | MR

[8] F. Klein, Mathematical seminar at Göttingen, winter semester 1905/6 under the direction of Professors Klein, Hilbert and Minkowski, talks by F. Klein, notes by O. Toeplitz, http://www.claymath.org/sites/default/files/klein1math.sem_._ws1905-06.pdf

[9] F. Klein, Forlesungen über die Hypergeometrische Funktion, Reprint of the 1933 original, Springer Verlag, Berlin–New York, 1981 | MR

[10] G. Mondello, D. Panov, Spherical Metrics with Conical Singularities on a $2$-Sphere: Angle Constraints, 1505.01994v1

[11] L. Pontrjagin, “Hermitian Operators in Spaces with Indefinite Metric”, Bull. Acad. Sci. URSS, Sér. Math., 8 (1944), 243–280 (in Russian) | MR

[12] F. Schilling, “Ueber die Theorie der symmetrischen s-Funktion mit einem einfachen Nebenpunkte”, Math. Ann., 51 (1899), 481–522 | DOI | MR | Zbl

[13] G. Tarantello, “Analytical, Geometrical and Topological Aspects of a Class of Mean Field Equations on Surfaces”, Discrete and Continuous Dynamical Systems, 29:3 (2010), 931–973 | DOI | MR

[14] M. Troyanov, “Metrics of Constant Curvaturer on a Sphere with Two Conical Singularities”, Lect. Notes Math., 1410, Springer, Berlin, 1989, 296–306 | DOI | MR