×

Metrics with four conic singularities and spherical quadrilaterals. (English) Zbl 1343.30007

Summary: 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.

MSC:

30C20 Conformal mappings of special domains
34M03 Linear ordinary differential equations and systems in the complex domain
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ahlfors, Lars V., Lectures on quasiconformal mappings, University Lecture Series 38, viii+162 pp. (2006), American Mathematical Society, Providence, RI · Zbl 1103.30001 · doi:10.1090/ulect/038
[2] Ahlfors, Lars V., Conformal invariants, xii+162 pp. (2010), AMS Chelsea Publishing, Providence, RI · Zbl 1211.30002
[3] Biswas, Indranil, A criterion for the existence of a parabolic stable bundle of rank two over the projective line, Internat. J. Math., 9, 5, 523-533 (1998) · Zbl 0939.14015 · doi:10.1142/S0129167X98000233
[4] Bonk, Mario; Eremenko, Alexandre, Uniformly hyperbolic surfaces, Indiana Univ. Math. J., 49, 1, 61-80 (2000) · Zbl 0961.53021
[5] REU R. Buckman and N. Schmitt, Spherical polygons and unitarization, www.\allowbreak gang.\allowbreak umass.\allowbreak edu/\allowbreak reu/2002/gon.pdf.
[6] Chai, Ching-Li; Lin, Chang-Shou; Wang, Chin-Lung, Mean field equations, hyperelliptic curves and modular forms: I, Camb. J. Math., 3, 1-2, 127-274 (2015) · Zbl 1327.35116
[7] Chen, Chiun-Chuan; Lin, Chang-Shou, Mean field equation of Liouville type with singular data: topological degree, Comm. Pure Appl. Math., 68, 6, 887-947 (2015) · Zbl 1319.35283 · doi:10.1002/cpa.21532
[8] Chen, Qing; Wang, Wei; Wu, Yingyi; Xu, Bin, Conformal metrics with constant curvature one and finitely many conical singularities on compact Riemann surfaces, Pacific J. Math., 273, 1, 75-100 (2015) · Zbl 1308.30050 · doi:10.2140/pjm.2015.273.75
[9] Dorfmeister, J.; Schuster, M., Construction of planar CMC 4-noids of genus \(g=0\), JP J. Geom. Topol., 6, 3, 319-381 (2006) · Zbl 1125.53006
[10] Dorfmeister, J.; Eschenburg, J.-H., Real Fuchsian equations and constant mean curvature surfaces, Mat. Contemp., 35, 1-25 (2008) · Zbl 1194.53007
[11] Eremenko, A., Metrics of positive curvature with conic singularities on the sphere, Proc. Amer. Math. Soc., 132, 11, 3349-3355 (electronic) (2004) · Zbl 1053.53025 · doi:10.1090/S0002-9939-04-07439-8
[12] Eremenko, A.; Gabrielov, A., Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry, Ann. of Math. (2), 155, 1, 105-129 (2002) · Zbl 0997.14015 · doi:10.2307/3062151
[13] Eremenko, Alexandre; Gabrielov, Andrei, The Wronski map and Grassmannians of real codimension 2 subspaces, Comput. Methods Funct. Theory, 1, 1, 1-25 (2001) · Zbl 1052.14057 · doi:10.1007/BF03320973
[14] Eremenko, Alexandre; Gabrielov, Andrei, An elementary proof of the B. and M. Shapiro conjecture for rational functions. Notions of positivity and the geometry of polynomials, Trends Math., 167-178 (2011), Birkh\"auser/Springer Basel AG, Basel · Zbl 1246.14062 · doi:10.1007/978-3-0348-0142-3\_10
[15] Eremenko, A.; Gabrielov, A., Counterexamples to pole placement by static output feedback, Linear Algebra Appl., 351/352, 211-218 (2002) · Zbl 1006.93032 · doi:10.1016/S0024-3795(01)00443-8
[16] Eremenko, Alexandre; Gabrielov, Andrei; Tarasov, Vitaly, Metrics with conic singularities and spherical polygons, Illinois J. Math., 58, 3, 739-755 (2014) · Zbl 1405.30005
[17] EGT3 A. Eremenko, A. Gabrielov, and V. Tarasov, Spherical quadrilaterals with three non-integer angles, arXiv:1504.02928. To appear in Journal of Mathematical Physics, Analysis and Geometry. · Zbl 1364.30048
[18] Eremenko, A.; Gabrielov, A.; Shapiro, M.; Vainshtein, A., Rational functions and real Schubert calculus, Proc. Amer. Math. Soc., 134, 4, 949-957 (electronic) (2006) · Zbl 1110.14052 · doi:10.1090/S0002-9939-05-08048-2
[19] Fujimori, Shoichi; Kawakami, Yu; Kokubu, Masatoshi; Rossman, Wayne; Umehara, Masaaki; Yamada, Kotaro, CMC-1 trinoids in hyperbolic 3-space and metrics of constant curvature one with conical singularities on the 2-sphere, Proc. Japan Acad. Ser. A Math. Sci., 87, 8, 144-149 (2011) · Zbl 1242.53070
[20] Gantmacher, F. P.; Krein, M. G., Oscillation matrices and kernels and small vibrations of mechanical systems, viii+310 pp. (2002), AMS Chelsea Publishing, Providence, RI · Zbl 1002.74002
[21] Goldberg, Lisa R., Catalan numbers and branched coverings by the Riemann sphere, Adv. Math., 85, 2, 129-144 (1991) · Zbl 0732.14013 · doi:10.1016/0001-8708(91)90052-9
[22] Heins, Maurice, On a class of conformal metrics, Nagoya Math. J., 21, 1-60 (1962) · Zbl 0113.05603
[23] Hilb1 E. Hilb, \"Uber Kleinsche Theoreme in der Theorie der linearen Differentialgleichingen, Ann. Math., 66 (1909) 215-257. · JFM 39.0380.04
[24] Hilb2 E. Hilb, \"Uber Kleinsche Theoreme in der Theorie der linearen Differentialgleichungen (2 Mitteilung), Ann. Math., 68 (1910) 24-71. · JFM 40.0373.05
[25] Hurwitz, A., \"Uber die Nullstellen der hypergeometrischen Funktion, Math. Ann., 64, 4, 517-560 (1907) · JFM 38.0471.01 · doi:10.1007/BF01450062
[26] I W. Ihlenburg, \"Uber die geometrischen Eigenschaften der Kreisbogenvierecke, Nova Acta Leopoldina, 91 (1909) 1-79 and 5 pages of tables. · JFM 40.0532.01
[27] I2 W. Ihlenburg, Ueber die gestaltlichen Verg\`“altnisse der Kreisbogenvierecke, G\'”ottingen Nachrichten, (1908) 225-230. · JFM 39.0551.02
[28] Klein, Felix, Ueber die Nullstellen der hypergeometrischen Reihe, Math. Ann., 37, 4, 573-590 (1890) · JFM 22.0444.02 · doi:10.1007/BF01724773
[29] Klein, F., Bemerkungen zur Theorie der linearen Differentialgleichungen zweiter Ordnung, Math. Ann., 64, 2, 175-196 (1907) · JFM 38.0360.02 · doi:10.1007/BF01449891
[30] Luo, Feng; Tian, Gang, Liouville equation and spherical convex polytopes, Proc. Amer. Math. Soc., 116, 4, 1119-1129 (1992) · Zbl 0806.53012 · doi:10.2307/2159498
[31] Pn G. Mondello and D. Panov, Spherical metrics with conical singularities on a \(2\)-sphere: angle constraints, arXiv:1505.01994. · Zbl 1446.53027
[32] Mukhin, Evgeny; Tarasov, Vitaly; Varchenko, Alexander, The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz, Ann. of Math. (2), 170, 2, 863-881 (2009) · Zbl 1213.14101 · doi:10.4007/annals.2009.170.863
[33] Mukhin, E.; Tarasov, V.; Varchenko, A., On reality property of Wronski maps, Confluentes Math., 1, 2, 225-247 (2009) · Zbl 1181.81065 · doi:10.1142/S1793744209000092
[34] P1 E. Picard, De l’equation \(\Delta u=ke^u\) sur une surface de Riemann ferm\'ee, J. Math. Pures Appl 9 (1893) 273-292. · JFM 25.0683.05
[35] P2 E. Picard, De l’equation \(\Delta u=e^u\), J. Math Pures Appl., 4 (1898) 313-316. · JFM 29.0312.02
[36] Picard, E., De l’int\'egration de l’\'equation \(\Delta u=e^u\) sur une surface de Riemann ferm\'ee, J. Reine Angew. Math., 130, 243-258 (1905) · JFM 37.0375.02 · doi:10.1515/crll.1905.130.243
[37] P3 E. Picard, Quelques applications analytiques de la th\'eorie des courbes et des surfaces alg\'ebriques, Gauthier-Villars, Paris, 1931. · Zbl 0003.02203
[38] Pontrjagin, L., Hermitian operators in spaces with indefinite metric, Bull. Acad. Sci. URSS. S\'er. Math. [Izvestia Akad. Nauk SSSR], 8, 243-280 (1944) · Zbl 0061.26004
[39] Hodgson, Craig D.; Rivin, Igor, A characterization of compact convex polyhedra in hyperbolic \(3\)-space, Invent. Math., 111, 1, 77-111 (1993) · Zbl 0784.52013 · doi:10.1007/BF01231281
[40] Heun’s differential equations, Oxford Science Publications, xxiv+354 pp. (1995), The Clarendon Press, Oxford University Press, New York · Zbl 0847.34006
[41] de Saint-Gervais, Henri Paul, Uniformisation des surfaces de Riemann, 544 pp. (2010), ENS \'Editions, Lyon · Zbl 1228.30001
[42] Scherbak, I., Rational functions with prescribed critical points, Geom. Funct. Anal., 12, 6, 1365-1380 (2002) · Zbl 1092.14065 · doi:10.1007/s00039-002-1365-4
[43] Sch{\"o}nflies, A., Ueber Kreisbogendreiecke und Kreisbogenvierecke, Math. Ann., 44, 1, 105-124 (1894) · JFM 25.0693.02 · doi:10.1007/BF01446976
[44] Sch{\"o}nflies, A., Ueber Kreisbogenpolygone, Math. Ann., 42, 3, 377-408 (1893) · JFM 25.0693.01 · doi:10.1007/BF01444164
[45] Sm V. I. Smirnov, The problem of inversion of a linear differential equation of second order with four singularities, Petrograd, 1918 (Russian). Reproduced in V. I Smirnov, Selected works, vol. 2, Analytic theory of ordinary differential equations, St. Peterburg University, St. Peterburg, 1996.
[46] Sm2 V. Smirnoff, Sur les \'equations diff\'erentialles lin\'eaires du second ordre et la th\'eorie des fonctions automorphes, Bull. Sci. Math., 45 (1921) 93-120, 126-135. · JFM 48.0510.02
[47] Tarantello, Gabriella, Analytical, geometrical and topological aspects of a class of mean field equations on surfaces, Discrete Contin. Dyn. Syst., 28, 3, 931-973 (2010) · Zbl 1207.35148 · doi:10.3934/dcds.2010.28.931
[48] Troyanov, Marc, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc., 324, 2, 793-821 (1991) · Zbl 0724.53023 · doi:10.2307/2001742
[49] Troyanov, Marc, Metrics of constant curvature on a sphere with two conical singularities. Differential geometry, Pe\~n\'\i scola, 1988, Lecture Notes in Math. 1410, 296-306 (1989), Springer, Berlin · Zbl 0697.53037 · doi:10.1007/BFb0086431
[50] Umehara, Masaaki; Yamada, Kotaro, Metrics of constant curvature \(1\) with three conical singularities on the \(2\)-sphere, Illinois J. Math., 44, 1, 72-94 (2000) · Zbl 0958.30029
[51] Van Vleck, Edward B., A determination of the number of real and imaginary roots of the hypergeometric series, Trans. Amer. Math. Soc., 3, 1, 110-131 (1902) · JFM 33.0460.04 · doi:10.2307/1986319
[52] Yoshida, Masaaki, A naive-topological study of the contiguity relations for hypergeometric functions. PDEs, submanifolds and affine differential geometry, Banach Center Publ. 69, 257-268 (2005), Polish Acad. Sci., Warsaw · Zbl 1107.33002 · doi:10.4064/bc69-0-20
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.