Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Metrics of positive curvature with conic singularities on the sphere

Author: A. Eremenko
Journal: Proc. Amer. Math. Soc. 132 (2004), 3349-3355
MSC (2000): Primary 53C45, 33C05
Published electronically: April 21, 2004
MathSciNet review: 2073312
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. V. I. Arnol'd and A. L. Krylov, Uniform distribution of points on the sphere and some ergodic properties of solutions of linear ordinary differential equations in the complex domain, Dokl. Akad. Nauk SSSR, 148 (1963) 9-12; English transl., Soviet Math. Dokl., 4 (1963) 1-5. MR 27:375
  • 2. 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., 155 (2002) 105-129. MR 2003c:58028
  • 3. A. Eremenko and A. Gabrielov, Counterexamples to pole placement by static output feedback, Linear Algebra and Appl., 351/352 (2002) 211-218. MR 2003f:93045
  • 4. A. Bilal and J-L. Gervais, Construction of constant curvature punctured Riemann surfaces with particle-scattering interpretation, J. Geom. Phys., 5 (1988) 277-304. MR 91e:81088
  • 5. L. Goldberg, Catalan numbers and branched coverings of the sphere, Adv. Math., 85 (1991) 129-144. MR 92b:14014
  • 6. M. Furuta and Y. Hattori, $2$-dimensional singular spherical space forms, manuscript.
  • 7. M. Heins, On a class of conformal metrics, Nagoya Math. J., 21 (1962) 1-60. MR 26:1451
  • 8. E. Ince, Ordinary differential equations, Dover, NY, 1956. MR 6,65f
  • 9. F. Klein, Vorlesungen über die hypergeometrische Funktionen, Springer, Berlin, 1933. MR 84b:01060
  • 10. J. Liouville, Sur l'équation aux dérivées partielles $\partial^2\log\lambda/\partial u\partial v\pm2\lambda a^2=0$, J. de Math., 18 (1853) 71-72.
  • 11. F. Luo and G. Tian, Liouville equation and spherical convex polytopes, Proc. AMS, 116 (1992) 1119-1129. MR 93b:53034
  • 12. R. McOwen, Point singularities and conformal metrics on Riemann surfaces, Proc. AMS, 103 (1988) 222-224. MR 89m:30089
  • 13. É. Picard, Traité d'Analyse, t. III, Gauthier-Villars, Paris, 1896. MR 93i:01032c (reprint of 1928 ed.)
  • 14. 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.
  • 15. H. Poincaré, Fonctions fuchsiennes et l'équation $\Delta u=e^u$, J. de math. pures et appl., 5 (4) (1898) 137-230.
  • 16. 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.
  • 17. M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. AMS, 324 (1991) 793-821. MR 91h:53059
  • 18. M. Troyanov, Surfaces euclidiennes à singularités coniques, Enseign. Math., 32 (1986) 79-94. MR 87i:30079
  • 19. M. Troyanov, Metrics of constant curvature on a sphere with two conical singularities, Lect. Notes Math., 1410, Springer, NY, 1989, 296-308. MR 90m:53057
  • 20. M. Umehara and K. Yamada, Metrics of constant curvature $1$ with three conical singularities on $2$-sphere, Illinois J. Math., 44 (2000) 72-94. MR 2001f:53072
  • 21. V. Varadarajan, Meromorphic differential equations. Expos. Math. 9 (1991) 97-188. MR 92i:32024

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 53C45, 33C05

Retrieve articles in all journals with MSC (2000): 53C45, 33C05

Additional Information

A. Eremenko
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

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
Article copyright: © Copyright 2004 American Mathematical Society