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The Selberg zeta function and a new Kähler metric on the moduli space of punctured Riemann surfaces. (English) Zbl 0739.30032

Summary: A local index theorem for families of \(\overline\partial\)-operators on Riemann surfaces with punctures is proved. A new Kähler metric on the moduli space of puntured surfaces is described in terms of the Eisenstein-Maass series.

MSC:

30F30 Differentials on Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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[1] Quillen, D., Determinants of Cauchy-Riemann operators over a Riemann surface, Funk. Anal. i Prilozen, 19, 37-41 (1985), (in Russian)
[2] Belavin, A.; Knizhnik, V., Complex geometry and the theory of quantum strings, JETP, 91, 364-390 (1986), (in Russian) · Zbl 0693.58043
[3] Zograf, P.; Takhtajan, L., A potential of the Weil-Peterson metric on the Torelli space, Zap. Nauch. Sem. LOMI, 160, 110-120 (1987), (in Russian) · Zbl 0631.32020
[4] Zograf, P.; Takhtajan, L., Local index theorem for families of ∂̄-operator on Riemann surfaces, Uspehi Mat. Nauk, 42, n. 6, 133-150 (1987), (in Russian) · Zbl 0659.58043
[5] Gelfand, I. M.; Graev, M. I.; Piatetski-Shapiro, I. I., Representation theory and automorphic functions (1969), W.B. Saunders: W.B. Saunders Philadelphia · Zbl 0177.18003
[6] Lehner, J., Discontinuous groups and automorphic functions (1964), Providence · Zbl 0178.42902
[7] Fay, J., Fourier coefficients of the resolvent for a Fuchsian group, J. Reine Angew. Math., 293/294, 9, 143-203 (1977) · Zbl 0352.30012
[8] Ahlfors, L., Some remarks on Teichmüller’s space of Riemann surfaces, Ann. Math., 74, 171-191 (1961) · Zbl 0146.30602
[9] Wolpert, S., Chern forms and the Riemann tensor for the moduli space of curves, Inv. Math., 85, 119-145 (1986) · Zbl 0595.32031
[10] Bers, L., Holomorphic differentials as functions of moduli, Bull. Amer. Math. Soc., 67, 206-210 (1961) · Zbl 0102.06702
[11] Rauch, H., A transcendental view of the space of algebraic Riemann surfaces, Bull. Amer. Math. Soc., 71, 1-39 (1965) · Zbl 0154.33002
[12] Venkov, A., Spectral theory of automorphic functions, Trudy Mat. Inst. Steklov, 153 (1981), (in Russian) · Zbl 0483.10029
[13] Mumford, D., Stability of projective varieties, L’Ens. Math., 24, 39-110 (1977) · Zbl 0363.14003
[14] Harer, J., The second homology group of the mapping class group of an orientable surface, Inv. Math., 72, 221-239 (1983) · Zbl 0533.57003
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