General information
Latest issue
Forthcoming papers
Impact factor
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

Latest issue
Current issues
Archive issues
What is RSS

Mat. Sb.:

Personal entry:
Save password
Forgotten password?

Mat. Sb. (N.S.), 1987, Volume 132(174), Number 3, Pages 304–321 (Mi msb1856)  

This article is cited in 44 scientific papers (total in 44 papers)

On uniformization of Riemann surfaces and the Weil-Petersson metric on Teichmüller and Schottky spaces

P. G. Zograf, L. A. Takhtadzhyan

Abstract: A potential is constructed for the Weil–Petersson metric on the Teichmüller space $T_g$ of marked Riemann surfaces of genus $g>1$ in terms of the density of the Poincaré metric on the region of discontinuity of the corresponding normalized marked Schottky group. It is proved that the difference between the projective connections corresponding to the Fuchsian uniformization and the Schottky uniformization for a marked Riemann surface of genus $g>1$ is the $\partial$-derivative of this potential, and the Weil–Petersson symplectic form on Teichmüller space is the $\overline\partial$-derivative of the Fuchsian projective connection. The results establish how the accessory parameters of the Fuchsian uniformization and the Schottky uniformization of a Riemann surface are connected with the geometries of Teichmüller space and Schottky space.
Bibliography: 31 titles.

Full text: PDF file (998 kB)
References: PDF file   HTML file

English version:
Mathematics of the USSR-Sbornik, 1988, 60:2, 297–313

Bibliographic databases:

UDC: 517.9+512.7
MSC: Primary 30F10, 32G15; Secondary 11F67, 30F35
Received: 01.04.1986

Citation: P. G. Zograf, L. A. Takhtadzhyan, “On uniformization of Riemann surfaces and the Weil-Petersson metric on Teichmüller and Schottky spaces”, Mat. Sb. (N.S.), 132(174):3 (1987), 304–321; Math. USSR-Sb., 60:2 (1988), 297–313

Citation in format AMSBIB
\by P.~G.~Zograf, L.~A.~Takhtadzhyan
\paper On~uniformization of~Riemann surfaces and the Weil-Petersson metric on~Teichm\"uller and Schottky spaces
\jour Mat. Sb. (N.S.)
\yr 1987
\vol 132(174)
\issue 3
\pages 304--321
\jour Math. USSR-Sb.
\yr 1988
\vol 60
\issue 2
\pages 297--313

Linking options:

    SHARE: FaceBook Twitter Livejournal

    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. P. G. Zograf, L. A. Takhtadzhyan, “A local index theorem for families of $\bar \partial$-operators on Riemann surfaces”, Russian Math. Surveys, 42:6 (1987), 169–190  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. P. G. Zograf, L. A. Takhtadzhyan, “On the geometry of moduli spaces of vector bundles over a Riemann surface”, Math. USSR-Izv., 35:1 (1990), 83–100  mathnet  crossref  mathscinet  zmath
    3. Kra I., “Accessory Parameters for Punctured Spheres”, Trans. Am. Math. Soc., 313:2 (1989), 589–617  crossref  mathscinet  zmath  isi
    4. S. Chakravarty, M. Ablowitz, P. Clarkson, “Reductions of self-dual Yang-Mills fields and classical systems”, Phys. Rev. Lett, 65:9 (1990), 1085  crossref
    5. Zheng-Mao Sheng, J Phys A Math Gen, 27:10 (1994), L339  crossref
    6. E. Aldrovandi, L. Bonora, “Liouville and Toda field theories on Riemann surfaces”, Journal of Geometry and Physics, 14:1 (1994), 65  crossref
    7. S.A. Apikyan, “Liouville field theory on a hyperelliptic surface”, Physics Letters B, 388:3 (1996), 557  crossref
    8. P. Winternitz, A. Yu. Orlov, “$P_\infty$ algebra of KP, free fermions and 2-cocycle in the Lie algebra of pseudodifferential operators”, Theoret. and Math. Phys., 113:2 (1997), 1393–1417  mathnet  crossref  crossref  mathscinet  isi
    9. Indranil Biswas, “Schottky uniformization and the symplectic structure of the cotangent bundle of a Teichmüller space”, Journal of Geometry and Physics, 35:1 (2000), 57  crossref
    10. Kirill Krasnov, Class Quantum Grav, 18:7 (2001), 1291  crossref  mathscinet  zmath  adsnasa
    11. Luigi Cantini, Pietro Menotti, Domenico Seminara, “Proof of Polyakov conjecture for general elliptic singularities”, Physics Letters B, 517:1-2 (2001), 203  crossref
    12. Kirill Krasnov, “Analytic continuation for asymptotically AdS 3D gravity”, Class Quantum Grav, 19:9 (2002), 2399  crossref  mathscinet  zmath  adsnasa  elib
    13. Kirill Krasnov, “$\Lambda$ < 0 quantum gravity in 2 $plus$ 1 dimensions: II. Black-hole creation by point particles”, Class Quantum Grav, 19:15 (2002), 3999  crossref  mathscinet  zmath  adsnasa  elib
    14. Luigi Cantini, Pietro Menotti, Domenico Seminara, “Liouville theory, accessory parameters and (2+1)-dimensional gravity”, Nuclear Physics B, 638:3 (2002), 351  crossref
    15. Aldrovandi E., “Homological Algebra of Multivalued Action Functionals”, Lett. Math. Phys., 60:1 (2002), 47–58  crossref  mathscinet  zmath  isi
    16. Kirill Krasnov, “Black-hole thermodynamics and Riemann surfaces”, Class Quantum Grav, 20:11 (2003), 2235  crossref  mathscinet  zmath  adsnasa  elib
    17. Kirill Krasnov, “On holomorphic factorization in asymptotically AdS 3D gravity”, Class Quantum Grav, 20:18 (2003), 4015  crossref  mathscinet  zmath  adsnasa  elib
    18. Korotkin D., “Matrix Riemann–Hilbert Problems Related to Branched Coverings of Cp”, Factorization and Integrable Systems, Operator Theory : Advances and Applications, 141, eds. Gohberg I., Manojlovic N., DosSantos A., Birkhauser Verlag Ag, 2003, 103–129  mathscinet  zmath  isi
    19. V. V. Chueshev, “An Explicit Variational Formula for the Monodromy Group”, Siberian Adv. Math., 15:2 (2005), 1–32  mathnet  mathscinet  zmath  elib
    20. Leszek Hadasz, Zbigniew Jaskólski, “Classical Liouville action on the sphere with three hyperbolic singularities”, Nuclear Physics B, 694:3 (2004), 493  crossref
    21. J. TESCHNER, “ON THE RELATION BETWEEN QUANTUM Liouville THEORY AND THE QUANTIZED Teichmüller SPACES”, Int. J. Mod. Phys. A, 19:supp02 (2004), 459  crossref
    22. Aldrovandi E., “On Hermitian-Holomorphic Classes Related to Uniformization, the Dilogarithm, and the Liouville Action”, Commun. Math. Phys., 251:1 (2004), 27–64  crossref  mathscinet  zmath  adsnasa  isi
    23. S Carlip, “Conformal field theory, (2 + 1)-dimensional gravity and the BTZ black hole”, Class Quantum Grav, 22:12 (2005), R85  crossref  mathscinet  zmath  adsnasa  isi
    24. Leszek Hadasz, Zbigniew Jaskólski, Marcin Pia̧tek, “Classical geometry from the quantum Liouville theory”, Nuclear Physics B, 724:3 (2005), 529  crossref
    25. Gaetano Bertoldi, Stefano Bolognesi, Gaston Giribet, Marco Matone, Yu Nakayama, “Zamolodchikov relations and Liouville hierarchy in WZNW model”, Nuclear Physics B, 709:3 (2005), 522  crossref
    26. Pietro Menotti, Gabriele Vajente, “Semiclassical and quantum Liouville theory on the sphere”, Nuclear Physics B, 709:3 (2005), 465  crossref
    27. Pietro Menotti, “Semiclassical and quantum Liouville theory”, J Phys Conf Ser, 33 (2006), 26  crossref  elib
    28. Leszek Hadasz, Zbigniew Jaskólski, “Semiclassical limit of the FZZT Liouville theory”, Nuclear Physics B, 757:3 (2006), 233  crossref
    29. Mcintyre A., Teo L.-P., “Holomorphic Factorization of Determinants of Laplacians Using Quasi-Fuchsian Uniformization”, Lett. Math. Phys., 83:1 (2008), 41–58  crossref  mathscinet  zmath  adsnasa  isi
    30. Kostas Skenderis, Balt C. Rees, “Holography and Wormholes in 2+1 Dimensions”, Commun. Math. Phys, 2010  crossref
    31. Colin Guillarmou, Sergiu Moroianu, Jinsung Park, “Eta invariant and Selberg zeta function of odd type over convex co-compact hyperbolic manifolds”, Advances in Mathematics, 225:5 (2010), 2464  crossref
    32. Guo R., Huang Zh., Wang B., “Quasi-Fuchsian 3-Manifolds and Metrics on Teichmüller Space”, Asian J. Math., 14:2 (2010), 243–256  mathscinet  zmath  isi
    33. Majid Heydarpour, “The Green’s functions of the boundaries at infinity of the hyperbolic 3-manifolds”, Journal of Geometry and Physics, 2011  crossref
    34. Franco Ferrari, Marcin Piatek, “Liouville theory, $ \mathcal{N} = 2 $ gauge theories and accessory parameters”, J. High Energ. Phys, 2012:5 (2012)  crossref
    35. A. G. Sergeev, “Lektsii ob universalnom prostranstve Teikhmyullera”, Lekts. kursy NOTs, 21, MIAN, M., 2013, 3–130  mathnet  crossref  zmath  elib
    36. A. Yu. Vasiliev, A. G. Sergeev, “Classical and quantum Teichmüller spaces”, Russian Math. Surveys, 68:3 (2013), 435–502  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    37. Taylor Barrella, Xi Dong, S.A.. Hartnoll, V.L.. Martin, “Holographic entanglement beyond classical gravity”, J. High Energ. Phys, 2013:9 (2013)  crossref
    38. Colin Guillarmou, Sergiu Moroianu, “Chern–Simons line bundle on Teichmüller space”, Geom. Topol, 18:1 (2014), 327  crossref
    39. Marcin Piatek, “Classical torus conformal block, $ \mathcal{N} $ = 2∗ twisted superpotential and the accessory parameter of Lamé equation”, J. High Energ. Phys, 2014:3 (2014)  crossref
    40. Bin Chen, Feng-yan Song, Jia-ju Zhang, “Holographic Rényi entropy in AdS3/LCFT2 correspondence”, J. High Energ. Phys, 2014:3 (2014)  crossref
    41. Andrew McIntyre, Jinsung Park, “Tau function and Chern–Simons invariant”, Advances in Mathematics, 262 (2014), 1  crossref
    42. Bin Chen, Jie-qiang Wu, “Single interval Rényi entropy at low temperature”, J. High Energ. Phys, 2014:8 (2014)  crossref
    43. Jan de Boer, Alejandra Castro, Eliot Hijano, J.I.. Jottar, Per Kraus, “Higher spin entanglement and W N
      $$ {\mathcal{W}}_{\mathrm{N}} $$
      conformal blocks”, J. High Energ. Phys, 2015:7 (2015)  crossref
    44. Seppi A., “Minimal discs in hyperbolic space bounded by a quasicircle at infinity”, Comment. Math. Helv., 91:4 (2016), 807–839  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
    Number of views:
    This page:985
    Full text:268

    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019