Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography

Communications in Mathematical Physics

Volumn 239, Issue 1-2, 2003, Pages 183-240

  Takhtajan, Leon A ^a   Teo, Lee Peng ^b  

^a STONY BROOK UNIVERSITY   (United States)

^b NATIONAL CHIAO TUNG UNIVERSITY   (Taiwan)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0041975739     PISSN: 00103616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00220-003-0878-5     Document Type: Article

References (25)

1. [Ahl61] Ahlfors, L.V.: Some remarks on Teichmüller's space of Riemann surfaces. Ann. Math. 74(2), 171-191 (1961)
2. [Ahl87] Ahlfors, L.V.: Lectures on quasiconformal mappings. Monterey CA: Wadsworth & Brooks/Cole Advanced Books & Software, 1987, With the assistance of Clifford J. Earle Jr. Reprint of the 1966 original
3. n. Ph.D. thesis, Berkeley, 1998
4. [AT97] Aldrovandi, E., Takhtajan, L.A.: Generating functional in CFT and effective action for two-dimensional quantum gravity on higher genus Riemann surfaces. Commun. Math. Phys. 188(1), 29-67 (1997)
5. [Ber60] Bers, L.: Simultaneous uniformization. Bull. Am. Math. Soc. 66, 94-97 (1960)
6. [Ber70] Bers, L.: Spaces of Kleinian groups. In: Several Complex Variables, I (Proc. Conf. Univ. of Maryland, College Park, Md. 1970) Berlin: Springer, 1970, pp. 9-34
7. [Ber71] Bers, L.: Extremal quasiconformal mappings. In: Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y. 1969) Princeton NJ: Princeton, Univ. Press, 1971, pp. 27-52
8. [Dur83] Duren, P.L.: Univalent functions. New York: Springer-Verlag, 1983
9. [Eps84] Epstein, C.L.: Envelopes of horospheres and Weingarten surfaces in hyperbolic 3-space. Preprint, 1984
10. [Eps86] Epstein, C.L.: The hyperbolic Gauss map and quasiconformal reflections. J. Reine Angew. Math. 372, 96-135 (1986)
11. [Kra72a] Kra, I.: Automorphic forms and Kleinian groups. Reading, MA: W. A. Benjamin Inc., 1972, Mathematics Lecture Note Series
12. [Kra72b] Kra, I.: On spaces of Kleinian groups. Comment. Math. Helv. 47, 53-69 (1972)
13. [Kra00] Krasnov, K.: Holography and Riemann surfaces. Adv. Theor. Math. Phys. 4(4), 929-979 (2000)
14. [Kra01] Krasnov, K.: On holomorphic factorization in asymptotically Ads 3d gravity. e-preprint hep-th/0109198, 2001
15. [Mas88] Maskit, B.: Kleinian groups. Berlin: Springer-Verlag, 1988
16. [McM00] McMullen, C.T.: The moduli space of Riemann surfaces is Kähler hyperbolic. Ann. Math. (2) 151(1), 327-357 (2000)
17. [MM02] Manin, Y.I., Marcolli, M.: Holography principle and arithmetic of algebraic curves. Adv. Theor. Math. Phys. 5, 617-650 (2002)
18. [OPS88] Osgood, B., Phillips, R., Sarnak, P.: Extremals of determinants of Laplacians. J. Funct. Anal. 80(1), 148-211 (1988)
19. [Pol81] Polyakov, A.M.: Quantum geometry of bosonic strings. Phys. Lett. B 103(3), 207-210 (1981)
20. [Tak92] Takhtajan, L.: Semi-classical Liouville theory, complex geometry of moduli spaces, and uniformization of Riemann surfaces. In: New Symmetry Principles in Quantum Field Theory (Cargèse 1991), New York: Plenum, 1992, pp. 383-406
21. [Wit98] Witten, E.: Anti de Sitter space and holography. Adv. Theor. Math. Phys. 2(2), 253-291 (1998)
22. [Wol86] Wolpert, S.A.: Chern forms and the Riemann tensor for the moduli space of curves. Invent. Math. 85(1), 119-145 (1986)
23. [ZT85] Zograf P.G., Takhtadzhyan L.A.: Action of the Liouville equation as generating function for accessory parameters and the potential of the Weil-Petersson metric on Teichmüller space. Funkt. Anal. i Prilozhen. 19(3), 67-68 (1985) (Russian); English. transl. in: Funct. Anal. Appl. 19(3), 219-220 (1985)
24. [ZT87a] Zograf P.G., Takhtadzhyan L.A.: On the Liouville equation, accessory parameters and the geometry of Teichmüller space for Riemann surfaces of genus 0. Mat. Sb. (N.S.) 132(174)(2), 147-166 (1987) (Russian); English transl. in: Math. USSR Sb. 60(1), 143-161 (1988)
25. [ZT87b] Zograf P.G., Takhtadzhyan L.A.: On the uniformization of Riemann surfaces and on the Weil-Petersson metric on the Teichmüller and Schottky spaces. Mat. Sb. (N.S.) 132(174)(3), 297-313 (1987) (Russian); English transl. in: Math. USSR SB. 60(2), 297-313 (1988)