Weil-Petersson metric on the universal Teichmüller space

Memoirs of the American Mathematical Society

Volumn 183, Issue 861, 2006, Pages 1-119

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

^a NONE

Author keywords

Bers coordinates; Bers embedding; Fredholm determinant; Grunsky operators; Hilbert manifold structure; K hler potential; Riemann curvature tensor; Universell Teichm ller space; Variation of the hyperbolic metric; Weil Petersson metric

Indexed keywords

EID: 33749259771     PISSN: 00659266     EISSN: None     Source Type: Journal    
DOI: 10.1090/memo/0861     Document Type: Article

References (53)

1. [AB60] Lars Ahlfors and Lipman Bers, Riemann's mapping theorem for variable metrics, Ann. of Math. (2) 72 (1960), 385-404.
2. [Ahl61] Lars V. Ahlfors, Some remarks on Teichmüller's space of Riemann surfaces, Ann. of Math. (2) 74 (1961), 171-191.
3. [Ahl62] _, Curvature properties of Teichmüller's space, J. Analyse Math. 9 (1961/1962), 161-176.
4. [Ahl87] _, Lectures on guasiconformal mappings, Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1987, With the assistance of Clifford J. Earle, Jr., Reprint of the 1966 original.
5. [Ber65] L Bers, Automorphic forms and general Teichmüller spaces, Proc. Conf. Complex Analysis (Minneapolis, 1964), Springer, Berlin, 1965, pp. 109-113.
6. [Ber66] Lipman Bers, A non-standard integral equation with applications to quasiconformal mappings, Acta Math. 116 (1966), 113-134.
7. [Ber72] _, Uniformization, moduli and Kleinian groups, Bull. London. Math. Soc. 4 (1972), 257-300.
8. [Ber73] _, Fiber spaces over Teichmüller spaces, Acta. Math. 130 (1973), 89-126.
9. [BFK92] D. Burghelea, L. Friedlander, and T. Kappeler, Meyer- Vietoris type formula for determinants of elliptic differential operators, J. Funct. Anal. 107 (1992), no. 1, 34-65.
10. [Bou67] N. Bourbaki, Éléments de mathématique. Fasc. XXXIII. Variétés différentielles et analytiques. Fascicule de résultats (Paragraphes 1 à 7), Actualités Scientifiques et Industrielles, No. 1333, Hermann, Paris, 1967.
11. [BP78] Jochen Becker and Christian Pommerenke, Über die quasikonforme Fortsetzung schlichter Funktionen, Math. Z. 161 (1978), no. 1, 69-80.
12. 1, Nuclear Phys. B 293 (1987), no. 2, 348-384.
13. [BR87b] _, String theory as the Kähler geometry of loop space, Phys. Rev. Lett. 58 (1987), no. 6, 535-538.
14. [Bra65] J. E. Brazilevič, On dispersion of coefficients of univalent functions, Mat. Sb. (N.S.) 68 (110) (1965), 549-560.
15. [DE86] Adrien Douady and Clifford J. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math. 157 (1986), no. 1-2, 23-48.
16. [Dur83] P. L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983.
17. [EN88] Clifford J. Earle and Subhashis Nag, Conformally natural reflections in Jordan curves with applications to Teichmüller spaces, Holomorphic functions and moduli, Vol. II (Berkeley, CA, 1986), Math. Sci. Res. Inst. Publ., vol. 11, Springer, New York, 1988, pp. 179-194.
18. [GK69] I. C. Gohberg and M. G. Kreǐn, Introduction to the theory of linear nonselfadjoint operators, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969.
19. [GS92] Frederick P. Gardiner and Dennis P. Sullivan, Symmetric structures on a closed curve, Amer. J. Math. 114 (1992), no. 4, 683-736.
20. [Ham02] D. H. Hamilton, Conformal welding, Handbook of complex analysis: geometric function theory, Vol. 1, North-Holland, Amsterdam, 2002, pp. 137-146.
21. [Hej76] Dennis A. Hejhal, The Selberg trace formula for PSL(2, A). Vol. I, Springer-Verlag, Berlin, 1976, Lecture Notes in Mathematics, Vol. 548.
22. [Hum72] J. A. Hummel, Inequalities of Grunsky type for Aharonov pairs, J. Analyse Math. 25 (1972), 217-257.
23. [HZ99] Andrew Hassell and Steve Zelditch, Determinants of Laplacians in exterior domains, Internat. Math. Res. Notices (1999), no. 18, 971-1004.
24. [Kir87] A. A. Kirillov, Kähler structure on the K-orbits of a group of diffeomorphisms of the circle, Funktsional. Anal. i Prilozhen. 21 (1987), no. 2, 42-45.
25. 1), Funktsional. Anal. i Prilozhen. 21 (1987), no. 4, 35-46.
26. [KY88] A. A. Kirillov and D. V. Yuriev, Representations of the Virasoro algebra by the orbit method, J. Geom. Phys. 5 (1988), no. 3, 351-363.
27. 2(R), Graduate Texts in Mathematics, vol. 105, Springer-Verlag, New York, 1985, Reprint of the 1975 edition.
28. [Lan95] _, Differential and Riemannian manifolds, third ed., Springer-Verlag, New York, 1995.
29. [Leh87] Olli Lehto, Univalent functions and Teichmüller spaces, Springer-Verlag, New York, 1987.
30. [Nag88] Subhashis Nag, The complex analytic theory of Teichmüller spaces, John Wiley & Sons Inc., New York, 1988, A Wiley-Interscience Publication.
31. [Nag92] _, A period mapping in universal Teichmüller space, Bull. Amer. Math. Soc. (N.S.) 26 (1992), no. 2, 280-287.
32. 1/2 space on the circle, Osaka J. Math. 32 (1995), no. 1, 1-34.
33. 1) and the Teichmüller spaces, Comm. Math. Phys. 130 (1990), no. 1, 123-138.
34. [Pat75] S. J. Patterson, A lattice-point problem in hyperbolic space, Mathematika 22 (1975), no. 1, 81-88.
35. [Pom92] Ch. Pommerenke, Boundary behaviour of conformal maps, Springer-Verlag, Berlin, 1992.
36. [PS86] Andrew Pressley and Graeme Segal, Loop groups, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1986.
37. [Roy75] H. L. Royden, Intrinsic metrics on Teichmüller space, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 2, Canad. Math. Congress, Montreal, Que., 1975, pp. 217-221.
38. [Sch57] M. Schiffer, The Fredholm eigen values of plane domains, Pacific J. Math. 7 (1957), 1187-1225.
39. [Sch59] _, Fredholm eigen values of multiply-connected domains, Pacific J. Math. 9 (1959), 211-269.
40. [Sch81] Menahem Schiffer, Fredholm eigenvalues and Grunsky matrices, Ann. Polon. Math. 39 (1981), 149-164.
41. [Seg81] Graeme Segal, Unitary representations of some infinite-dimensional groups, Comm. Math. Phys. 80 (1981), no. 3, 301-342.
42. [SH62] M. Schiffer and N. S. Hawley, Connections and conformal mapping, Acta Math. 107 (1962), 175-274.
43. [Sie64] Carl Ludwig Siegel, Symplectic geometry, Academic Press, New York, 1964.
44. [STZ99] Maria E. Schonbek, Andrey N. Todorov, and Jorge P. Zubelli, Geodesic flows on diffeomorphisms of the circle, Grassmannians, and the geometry of the periodic KdV equation, Adv. Theor. Math. Phys. 3 (1999), no. 4, 1027-1092.
45. [SW85] Graeme Segal and George Wilson, Loop groups and equations of KdV type, Inst. Hautes Études Sci. Publ. Math. (1985), no. 61, 5-65.
46. [Teo03] Lee-Peng Teo, Analytic functions and integrable hierarchies - characterization of tau functions, Lett. Math. Phys. 64 (2003), no. 1, 75-92.
47. [Teo04] _, The Veiling-Kirillov metric on the universal Teichmüller curve, J. Analyse Math. 93 (2004), 271-308.
48. [TT03] Leon A. Takhtajan and Lee-Peng Teo, Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography, Comm. Math. Phys. 239 (2003), no. 1-2, 183-240.
49. [TZ91] L. A. Takhtajan and P. G. Zograf, A local index theorem for families of ∂̄-operators on punctured Riemann surfaces and a new Kähler metric on their moduli spaces, Comm. Math. Phys. 137 (1991), no. 2, 399-426.
50. [Vel] John A. Veiling, A protectively natural metric on Teichmüller's spaces, unpublished manuscript.
51. [Wol86] Scott A. Wolpert, Chern forms and the Riemann tensor for the moduli space of curves, Invent. Math. 85 (1986), no. 1, 119-145.
52. [YB53] K. Yano and S. Bochner, Curvature and Betti numbers, Annals of Mathematics Studies, No. 32, Princeton University Press, Princeton, N. J., 1953.
53. [ZT87] P. G. Zograf and L. A. Takhtadzhyan, A local index theorem for families of ∂̄-operators on Riemann surfaces, Uspekhi Mat. Nauk 42 (1987), no. 6(258), 133-150.