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

P. G. Zograf   ; L. A. Takhtadzhyan

Sbornik. Mathematics, Tome 60 (1988) no. 2, pp. 297-313

Résumé

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.

Bibliographie

[1] Zograf P. G., Takhtadzhyan L. A., “Deistvie uravneniya Liuvillya – proizvodyaschaya funktsiya dlya aktsessornykh parametrov i potentsial metriki Veilya–Petersona na prostranstve Teikhmyullera”, Funkts. analiz i ego pril., 19:3 (1985), 67–68 | MR | Zbl

[2] Zograf P. G., Takhtadzhyan L. A., Ob uravnenii Liuvillya, aktsessornykh parametrakh i geometrii prostranstva Teikhmyullera I. Rimanovy poverkhnosti roda 0, Preprint LOMI. R-11-85, Leningrad, LOMI, 1985 | MR

[3] Zograf P. G., Takhtadzhyan L. A., “O svyaznosti Narasimkhana–Seshadri i kelerovoi strukture prostranstva modulei golomorfnykh vektornykh rassloenii nad rimanovymi poverkhnostyami”, Funkts. analiz i ego pril., 20:3 (1986), 84–85 | MR | Zbl

[4] Zograf P. G., Takhtadzhyan L. A., Ob uniformizatsii rimanovykh poverkhnostei i metrike Veilya–Petersona na prostranstvakh Teikhmyullera i Shottki, Preprint LOMI. R-12-85, Leningrad, LOMI, 1985 | MR

[5] Klein F., “Neue Beiträge zur Riemann'schen Funktiontheorie”, Math. Ann., 21 (1883), 141–218 | DOI | MR

[6] Poincaré H., “Sur les groupes des équations linéaires”, Acta Math., 4 (1884), 201–312 ; Puankare A., Izbrannye trudy, Nauka, M., 1974, t. 3 | DOI | MR

[7] Maskit B., “A characterization of Schottky groups”, J. Analyse Math., 19 (1967), 227–230 | DOI | MR | Zbl

[8] Chuckrow V., “On Schottky groups with application to kleinian groups”, Ann. Math., 88 (1968), 47–61 | DOI | MR | Zbl

[9] Koebe P., “Über die Uniformisierung der algebraischen Kurven IV”, Math. Ann., 75 (1914), 42–129 | DOI | MR | Zbl

[10] Ford L. R., Avtomorfnye funktsii, ONTI, M., 1936

[11] Hejhal D., “Monodromy groups and linearly polymorphic functions”, Acta Math., 135 (1975), 1–55 | DOI | MR | Zbl

[12] Hejhal D., “Monodromy groups and Poincare series”, Bull. AMS., 84:3 (1978), 339–376 | DOI | MR | Zbl

[13] Schottky F., “Über eine specielle Function, welche bei einer bestimmten linearen Transformation ihres Arguments unverandert bleibt”, J. Reine Angew. Math., 101 (1887), 227–272

[14] Schiffer M., Hawley N., “Half-order differentials on Riemann surfaces”, Acta Math., 115 (1966), 199–236 | DOI | MR | Zbl

[15] Tyurin A. N., “O periodakh kvadratichnykh differentsialov”, UMN, 33:6 (1978), 149–195 | MR | Zbl

[16] Hejhal D., “Sur les paramètres accessoires pour l'uniformisation de Schottky”, C. R. Acad. Sci. Paris Sér. A–B, 279:17 (1974), 695–697 ; ibid:18, 713–716 | MR | Zbl | MR

[17] Hejhal D., “Sur les paramètres accessoires pour l'uniformisation fuchsienne”, C. R. Acad. Sci. Paris Sér. A–B, 282:8 (1976), 403–406 | MR | Zbl

[18] Hejhal D., “The variational theory of linearly polymorphic functions”, J. Analyse Math., 30 (1976), 215–264 | DOI | MR | Zbl

[19] Picard E., “Mémoire sur la théorie des équations aux dérivées partielles et la méthode des approximations successives”, J. math, pures et appl. 4e sér., 6 (1890), 145–211

[20] Poincaré H., “Les functions fuchsiennes et l'equation $\Delta u=e^u$”, J. math, pures et appl. 5e sér., 4 (1898), 137–230 ; Puankare A., Izbrannye trudy, t. 3, Nauka, M., 1974, S. 235–309 | Zbl

[21] Alfors L., Lektsii po kvazikonformnym otobrazheniyam, Mir, M., 1969 | MR

[22] Bers L., Spaces of Riemann surfaces, Proc. ICM, 1958, Cambridge, I960

[23] Hejhal D., “On Schottky and Teichmüller spaces”, Adv. in Math., 15:2 (1975), 133–156 | DOI | MR | Zbl

[24] Weil A., “Sur les modules des surfaces de Riemann”, Sém. Bourbaki, 1958

[25] Ahlfors L., “Some remarks on Teichmüller's space of Reimann surfaces”, Ann. Math., 74 (1961), 171–191 | DOI | MR | Zbl

[26] Wolpert S., “The topology and geometry of the moduli space of Riemann surfaces”, Lect. Notes in Math., 1111, 1985, 431–451 | MR | Zbl

[27] Ahlfors L., Bers L., “Riemann's mapping theorem for variable metrics”, Ann. Math., 72 (1960), 385–404 | DOI | MR | Zbl

[28] Wolpert S., Chern forms and the Riemann tensor for the moduli space of curves, Preprint, University of Maryland. MD 84–20, College Park, 1984 | MR

[29] Bers L., “Fiber spaces over Teichmüller spaces”, Acta Math., 130 (1973), 89–126 | DOI | MR | Zbl

[30] Bers L., “Holomorphic differentials as functions of moduli”, Bull. Amer. Math. Soc., 67 (1961), 206–210 | DOI | MR | Zbl

[31] Wolpert S., Thurston's Riemannian metric for Teichmüller space, Preprint University of Maryland MD 85–6, College Park, 1985