Illinois Journal of Mathematics

On metrics of curvature $1$ with four conic singularities on tori and on the sphere

Alexandre Eremenko and Andrei Gabrielov

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We discuss conformal metrics of curvature $1$ on tori and on the sphere, with four conic singularities whose angles are multiples of $\pi$. Besides some general results we study in detail the family of such symmetric metrics on the sphere, with angles $(\pi,3\pi,\pi,3\pi)$. As a consequence we find new Heun’s equations whose general solution is algebraic.

Article information

Source
Illinois J. Math., Volume 59, Number 4 (2015), 925-947.

Dates
Received: 10 January 2016
First available in Project Euclid: 27 February 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1488186015

Digital Object Identifier
doi:10.1215/ijm/1488186015

Mathematical Reviews number (MathSciNet)
MR3628295

Zentralblatt MATH identifier
1366.30029

Subjects
Primary: 34M03: Linear equations and systems 34M05: Entire and meromorphic solutions 30C20: Conformal mappings of special domains 35J91: Semilinear elliptic equations with Laplacian, bi-Laplacian or poly- Laplacian 33E05: Elliptic functions and integrals

Citation

Eremenko, Alexandre; Gabrielov, Andrei. On metrics of curvature $1$ with four conic singularities on tori and on the sphere. Illinois J. Math. 59 (2015), no. 4, 925--947. doi:10.1215/ijm/1488186015. https://projecteuclid.org/euclid.ijm/1488186015


Export citation

References

  • N. I. Akhiezer, Elements of the theory of elliptic functions, AMS, Providence, RI, 1990.
  • F. Baldassari and B. Dwork, On second order differential equations with algebraic solutions, Amer. J. Math. 101 (1979), 42–76.
  • W. Bergweiler and A. Eremenko, Green's function and anti-holomorphic dynamics on a torus, Proc. Amer. Math. Soc. 144 (2016), no. 7, 2911–2922.
  • C.-L. Chai, C.-S. Lin and C.-L. Wang, Mean field equations, hyperelliptic curves and modular forms: I, Cambridge J. Math. 3 (2015), 127–274.
  • C.-L. Chai, C.-S. Lin and C.-L. Wang, Mean field equations, hyperelliptic curves and modular forms: II. Available at \arxivurlarXiv1502.03295.
  • G. Darboux, Sur une équation linéaire, C. R. Acad. Sci. Paris 94 (1882), 1645–1648.
  • A. Eremenko, Metrics of positive curvature with conic singularities on the sphere, Proc. Amer. Math. Soc. 132 (2004), 3349–3355.
  • A. Eremenko and A. Gabrielov, Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry, Ann. Math. 155 (2002), 105–129.
  • A. Eremenko and A. Gabrielov, Spherical rectangles, Arnold Math. J. 2 (2016), no. 4, 463–486.
  • A. Eremenko, A. Gabrielov, M. Shapiro and A. Vainshtein, Rational functions and real Schubert calculus, Proc. Amer. Math. Soc. 134 (2006), no. 4, 949–957.
  • A. Eremenko, A. Gabrielov and V. Tarasov, Metrics with conic singularities and spherical polygons, Illinois J. Math. 58 (2014), no. 3, 739–755.
  • A. Eremenko, A. Gabrielov and V. Tarasov, Metrics with four conic singularities and spherical quadrilaterals, Conform. Geom. Dyn. 20 (2016), 128–175.
  • A. Eremenko, A. Gabrielov and V. Tarasov, Spherical quadrilaterals with three non-integer angles, Math. Phys. Anal. Geom. 12 (2016), no. 2, 134–167.
  • S. Finch, Mathematical constants, Cambridge UP, Cambridge, 2003.
  • S. Fujimori, Y. Kawakami, M. Kokubu, W. Rossman, M. Umehara and 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.
  • F. Gesztesy and Weikard, On Picard potentials, Differential Integral Equations 8 (1995), no. 6, 1453–1476.
  • G.-H. Halphen, Traité des fonctions elliptiques et de leurs applications, première partie, Gauthier-Villars, Paris, 1886.
  • C. Hermite, Sur l'équation de lamé, extrait de feuilles authographiées du course d'Analyse de l'École polytechnique, $1^{\mathrm{re}}$ division, 1872-73, $32^e$ leçon, Oeuvres, t. III, Gauthier-Villars, Paris, 1912, pp. 118–122.
  • F. Klein, Vorlesungen über die hypergeometrische funktion, reprint of the 1933 original, Springer, Berlin-New York, 1981.
  • F. Klein, Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade, Birkhäuser Verlag, Basel, 1993.
  • C.-S. Lin and C.-L. Wang, Elliptic functions Green functions and the mean field equations on tori, Ann. Math. 172 (2010), 911–954.
  • F. Luo and G. Tian, Liouville equation and spherical convex polytopes, Proc. Amer. Math. Soc. 116 (1992), no. 4, 1119–1129.
  • G. Mondello and D. Panov, Spherical metrics with conical singularities on a 2-sphere: angle constraints, Int. Math. Res. Not. 16 (2016), 4937–4995.
  • I. Scherbak, Rational functions with prescribed critical points, Geom. Funct. Anal. 12 (2002), no. 6, 1365–1380.
  • L. Schneps, Dessins d'enfants on the Riemann sphere, The Grothendieck theory of dessins d'enfants (Luminy, 1993), Cambridge Univ. Press, Cambridge, 1994, pp. 47–77.
  • G. Tarantello, Analytical, geometrical and topological aspects of a class of mean field equations on surfaces, Discrete Contin. Dyn. Syst. 28 (2010), no. 3, 931–973.
  • M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991), 793–821.
  • E. Van Vleck, A determination of the number of real and imaginary roots of the hypergeometric series, Trans. Amer. Math. Soc. 3 (1902), 110–131.
  • A. Veselov, On Darboux–Treibich–Verdier potentials, Lett. Math. Phys. 96 (2011), 209–216.
  • R. Vidunas, Degenerate and dihedral Heun functions with parameters, Hokkaido Math. J. 1 (2016), 93–108.
  • R. Vidunas and G. Filipuk, Parametric transformations between the Heun and Gauss hypergeometric functions, Funkcial. Ekvac. 56 (2013), no. 2, 271–321.
  • R. Vidunas and G. Filipuk, A classification of coverings yielding Heun-to-hypergeometric reductions, Osaka J. Math. 51 (2014), no. 4, 867–903.
  • W. von Koppenfels and F. Stallmann, Praxis der konformen Abbildung, Springer, Berlin, 1959.