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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.

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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
\Bibitem{ZogTak87}
\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
\mathnet{http://mi.mathnet.ru/msb1856}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=889594}
\zmath{https://zbmath.org/?q=an:0663.32017|0642.32011}
\transl
\jour Math. USSR-Sb.
\yr 1988
\vol 60
\issue 2
\pages 297--313
\crossref{https://doi.org/10.1070/SM1988v060n02ABEH003170}


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    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
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  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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