Metrics of positive curvature with conic singularities on the sphere
HTML articles powered by AMS MathViewer
- by A. Eremenko PDF
- Proc. Amer. Math. Soc. 132 (2004), 3349-3355 Request permission
Abstract:
A simple proof is given of the necessary and sufficient condition on a triple of positive numbers $\theta _1,\theta _2,\theta _3$ for the existence of a conformal metric of constant positive curvature on the sphere, with three conic singularities of total angles $2\pi \theta _1,2\pi \theta _2, 2\pi \theta _3$. The same condition is necessary and sufficient for the triple $\pi \theta _1,\pi \theta _2,\pi \theta _3$ to be interior angles of a spherical triangular membrane.References
- V. I. Arnol′d and A. L. Krylov, Uniform distribution of points on a sphere and certain ergodic properties of solutions of linear ordinary differential equations in a complex domain, Dokl. Akad. Nauk SSSR 148 (1963), 9–12 (Russian). MR 0150374
- A. Eremenko and A. Gabrielov, Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry, Ann. of Math. (2) 155 (2002), no. 1, 105–129. MR 1888795, DOI 10.2307/3062151
- A. Eremenko and A. Gabrielov, Counterexamples to pole placement by static output feedback, Linear Algebra Appl. 351/352 (2002), 211–218. Fourth special issue on linear systems and control. MR 1917479, DOI 10.1016/S0024-3795(01)00443-8
- Adel Bilal and Jean-Loup Gervais, Construction of constant curvature punctured Riemann surfaces with particle-scattering interpretation, J. Geom. Phys. 5 (1988), no. 2, 277–304. MR 1029430, DOI 10.1016/0393-0440(88)90007-1
- Lisa R. Goldberg, Catalan numbers and branched coverings by the Riemann sphere, Adv. Math. 85 (1991), no. 2, 129–144. MR 1093002, DOI 10.1016/0001-8708(91)90052-9
- M. Furuta and Y. Hattori, $2$-dimensional singular spherical space forms, manuscript.
- Maurice Heins, On a class of conformal metrics, Nagoya Math. J. 21 (1962), 1–60. MR 143901
- Albert Eagle, Series for all the roots of the equation $(z-a)^m=k(z-b)^n$, Amer. Math. Monthly 46 (1939), 425–428. MR 6, DOI 10.2307/2303037
- Felix Klein, Vorlesungen über die hypergeometrische Funktion, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 39, Springer-Verlag, Berlin-New York, 1981 (German). Reprint of the 1933 original. MR 668700
- J. Liouville, Sur l’équation aux dérivées partielles $\partial ^2\log \lambda /\partial u\partial v\pm 2\lambda a^2=0$, J. de Math., 18 (1853) 71-72.
- Feng Luo and Gang Tian, Liouville equation and spherical convex polytopes, Proc. Amer. Math. Soc. 116 (1992), no. 4, 1119–1129. MR 1137227, DOI 10.1090/S0002-9939-1992-1137227-5
- Robert C. McOwen, Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc. 103 (1988), no. 1, 222–224. MR 938672, DOI 10.1090/S0002-9939-1988-0938672-X
- Émile Picard, Traité d’analyse. Tome I, Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics], Éditions Jacques Gabay, Sceaux, 1991 (French). Intégrales simples et multiples. L’équation de Laplace et ses applications. Développements en séries. Applications géométriques du calcul infinitésimal. [Simple and multiple integrals. The Laplace equation and its applications. Series expansions. Geometric applications of differential and integral calculus]; With a preface by Gaston Julia; Reprint of the fourth (1942) edition. MR 1188642
- É. Picard, De l’intégration de l’équation $\Delta u=e^u$ sur une surface de Riemann fermée, J. reine angew. Math., 130 (1905) 243–258.
- H. Poincaré, Fonctions fuchsiennes et l’équation $\Delta u=e^u$, J. de math. pures et appl., 5 (4) (1898) 137–230.
- B. Riemann, Beitrage zur Theorie der durch Gauss’sche Reihe $F(\alpha ,\beta ,\gamma ,x)$ darstellbaren Funktionen, Ges. Math. Werke, 67–83; Vorlesungen über die hypergeometrische Reihe, Nachträge, III, 69–94. US edition: Dover, NY, 1953.
- Marc Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991), no. 2, 793–821. MR 1005085, DOI 10.1090/S0002-9947-1991-1005085-9
- Marc Troyanov, Les surfaces euclidiennes à singularités coniques, Enseign. Math. (2) 32 (1986), no. 1-2, 79–94 (French). MR 850552
- Marc Troyanov, Metrics of constant curvature on a sphere with two conical singularities, Differential geometry (Peñíscola, 1988) Lecture Notes in Math., vol. 1410, Springer, Berlin, 1989, pp. 296–306. MR 1034288, DOI 10.1007/BFb0086431
- Masaaki Umehara and Kotaro Yamada, Metrics of constant curvature $1$ with three conical singularities on the $2$-sphere, Illinois J. Math. 44 (2000), no. 1, 72–94. MR 1731382
- V. S. Varadarajan, Meromorphic differential equations, Exposition. Math. 9 (1991), no. 2, 97–188. MR 1101951
Additional Information
- A. Eremenko
- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
- MR Author ID: 63860
- Email: eremenko@math.purdue.edu
- Received by editor(s): May 16, 2003
- Received by editor(s) in revised form: July 22, 2003
- Published electronically: April 21, 2004
- Additional Notes: The author was supported by NSF grant DMS 0100512 and by the Humboldt Foundation
- Communicated by: Juha M. Heinonen
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 3349-3355
- MSC (2000): Primary 53C45, 33C05
- DOI: https://doi.org/10.1090/S0002-9939-04-07439-8
- MathSciNet review: 2073312