Horocyclic coordinates for Riemann surfaces and moduli spaces. I. Teichmüller and Riemann spaces of Kleinian groups

Kra, Irwin

doi:10.1090/S0894-0347-1990-1049503-1

Horocyclic coordinates for Riemann surfaces and moduli spaces. I. Teichmüller and Riemann spaces of Kleinian groups
HTML articles powered by AMS MathViewer

by Irwin Kra

J. Amer. Math. Soc. 3 (1990), 499-578

DOI: https://doi.org/10.1090/S0894-0347-1990-1049503-1

PDF | Request permission

References

Lars V. Ahlfors, The complex analytic structure of the space of closed Riemann surfaces. , Analytic functions, Princeton Univ. Press, Princeton, N.J., 1960, pp. 45–66. MR 0124486
L. V. Ahlfors, The modular function and geometric properties of quasiconformal mappings, Proc. Conf. Complex Analysis (Minneapolis, 1964) Springer, Berlin, 1965, pp. 296–300. MR 0176064
Lars Ahlfors and Lipman Bers, Riemann’s mapping theorem for variable metrics, Ann. of Math. (2) 72 (1960), 385–404. MR 115006, DOI 10.2307/1970141
Lipman Bers, Spaces of Riemann surfaces, Proc. Internat. Congress Math. 1958., Cambridge Univ. Press, New York, 1960, pp. 349–361. MR 0124484
Lipman Bers, Simultaneous uniformization, Bull. Amer. Math. Soc. 66 (1960), 94–97. MR 111834, DOI 10.1090/S0002-9904-1960-10413-2
Lipman Bers, Uniformization by Beltrami equations, Comm. Pure Appl. Math. 14 (1961), 215–228. MR 132175, DOI 10.1002/cpa.3160140304
Lipman Bers, A non-standard integral equation with applications to quasiconformal mappings, Acta Math. 116 (1966), 113–134. MR 192046, DOI 10.1007/BF02392814
Lipman Bers, Spaces of Kleinian groups, Several Complex Variables, I (Proc. Conf., Univ. of Maryland, College Park, Md., 1970) Springer, Berlin, 1970, pp. 9–34. MR 0271333
Lipman Bers, Fiber spaces over Teichmüller spaces, Acta Math. 130 (1973), 89–126. MR 430318, DOI 10.1007/BF02392263
Lipman Bers, On spaces of Riemann surfaces with nodes, Bull. Amer. Math. Soc. 80 (1974), 1219–1222. MR 361165, DOI 10.1090/S0002-9904-1974-13686-4
Lipman Bers, Spaces of degenerating Riemann surfaces, Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973) Ann. of Math. Studies, No. 79, Princeton Univ. Press, Princeton, N.J., 1974, pp. 43–55. MR 0361051
Lipman Bers, Deformations and moduli of Riemann surfaces with nodes and signatures, Math. Scand. 36 (1975), 12–16. MR 374496, DOI 10.7146/math.scand.a-11557
Lipman Bers, An extremal problem for quasiconformal mappings and a theorem by Thurston, Acta Math. 141 (1978), no. 1-2, 73–98. MR 477161, DOI 10.1007/BF02545743
Lipman Bers, Finite-dimensional Teichmüller spaces and generalizations, Bull. Amer. Math. Soc. (N.S.) 5 (1981), no. 2, 131–172. MR 621883, DOI 10.1090/S0273-0979-1981-14933-8
Lipman Bers and Leon Greenberg, Isomorphisms between Teichmüller spaces, Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969) Ann. of Math. Studies, No. 66, Princeton Univ. Press, Princeton, N.J., 1971, pp. 53–79. MR 0289763
Peter Buser, The collar theorem and examples, Manuscripta Math. 25 (1978), no. 4, 349–357. MR 509590, DOI 10.1007/BF01168048
P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109. MR 262240, DOI 10.1007/BF02684599
Clifford J. Earle, Some intrinsic coordinates on Teichmüller space, Proc. Amer. Math. Soc. 83 (1981), no. 3, 527–531. MR 627684, DOI 10.1090/S0002-9939-1981-0627684-5
Clifford J. Earle and Irwin Kra, On holomorphic mappings between Teichmüller spaces, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 107–124. MR 0430319
Clifford J. Earle and Irwin Kra, Half-canonical divisors on variable Riemann surfaces, J. Math. Kyoto Univ. 26 (1986), no. 1, 39–64. MR 827157, DOI 10.1215/kjm/1250520963

Convexity of moduli spaces

Geometric complex coordinates for Teichmüller space

Clifford J. Earle and Patricia L. Sipe, Families of Riemann surfaces over the punctured disk, Pacific J. Math. 150 (1991), no. 1, 79–96. MR 1120713, DOI 10.2140/pjm.1991.150.79
John D. Fay, Theta functions on Riemann surfaces, Lecture Notes in Mathematics, Vol. 352, Springer-Verlag, Berlin-New York, 1973. MR 0335789
Leon Greenberg, Maximal Fuchsian groups, Bull. Amer. Math. Soc. 69 (1963), 569–573. MR 148620, DOI 10.1090/S0002-9904-1963-11001-0
Dennis A. Hejhal, On Schottky and Teichmüller spaces, Advances in Math. 15 (1975), 133–156. MR 357862, DOI 10.1016/0001-8708(75)90128-0

Regular

-groups, degenerating Riemann surfaces, and spectral theory

Shoshichi Kobayashi, Hyperbolic manifolds and holomorphic mappings, Pure and Applied Mathematics, vol. 2, Marcel Dekker, Inc., New York, 1970. MR 0277770
Irwin Kra, Deformations of Fuchsian groups. II, Duke Math. J. 38 (1971), 499–508. MR 285734
Irwin Kra, On spaces of Kleinian groups, Comment. Math. Helv. 47 (1972), 53–69. MR 306485, DOI 10.1007/BF02566788
Irwin Kra, On Teichmüller’s theorem on the quasi-invariance of cross ratios, Israel J. Math. 30 (1978), no. 1-2, 152–158. MR 508260, DOI 10.1007/BF02760836
Irwin Kra, Canonical mappings between Teichmüller spaces, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 2, 143–179. MR 598682, DOI 10.1090/S0273-0979-1981-14870-9
Irwin Kra, On lifting Kleinian groups to $\textrm {SL}(2,\textbf {C})$, Differential geometry and complex analysis, Springer, Berlin, 1985, pp. 181–193. MR 780044

Non-variational global coordinates for Teichmüller spaces

Irwin Kra and Bernard Maskit, The deformation space of a Kleinian group, Amer. J. Math. 103 (1981), no. 5, 1065–1102. MR 630778, DOI 10.2307/2374258
Irwin Kra and Bernard Maskit, Bases for quadratic differentials, Comment. Math. Helv. 57 (1982), no. 4, 603–626. MR 694607, DOI 10.1007/BF02565877
A. Marden, Geometrically finite Kleinian groups and their deformation spaces, Discrete groups and automorphic functions (Proc. Conf., Cambridge, 1975) Academic Press, London, 1977, pp. 259–293. MR 0494117
A. Marden, Geometric relations between homeomorphic Riemann surfaces, Bull. Amer. Math. Soc. (N.S.) 3 (1980), no. 3, 1001–1017. MR 585180, DOI 10.1090/S0273-0979-1980-14836-3
Albert Marden, Geometric complex coordinates for Teichmüller space, Mathematical aspects of string theory (San Diego, Calif., 1986) Adv. Ser. Math. Phys., vol. 1, World Sci. Publishing, Singapore, 1987, pp. 341–354. MR 915828
Bernard Maskit, Self-maps on Kleinian groups, Amer. J. Math. 93 (1971), 840–856. MR 291453, DOI 10.2307/2373474
Bernard Maskit, Decomposition of certain Kleinian groups, Acta Math. 130 (1973), 243–263. MR 404614, DOI 10.1007/BF02392267
Bernard Maskit, Moduli of marked Riemann surfaces, Bull. Amer. Math. Soc. 80 (1974), 773–777. MR 346149, DOI 10.1090/S0002-9904-1974-13600-1
Bernard Maskit, Isomorphisms of function groups, J. Analyse Math. 32 (1977), 63–82. MR 463430, DOI 10.1007/BF02803575
Howard Masur, Extension of the Weil-Petersson metric to the boundary of Teichmuller space, Duke Math. J. 43 (1976), no. 3, 623–635. MR 417456
J. Peter Matelski, A compactness theorem for Fuchsian groups of the second kind, Duke Math. J. 43 (1976), no. 4, 829–840. MR 432921
H. E. Rauch, Variational methods in the problem of the moduli of Riemann surfaces. , Contributions to function theory (Internat. Colloq. Function Theory, Bombay, 1960) Tata Institute of Fundamental Research, Bombay, 1960, pp. 17–40. MR 0132173
H. L. Royden, Automorphisms and isometries of Teichmüller space, Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969) Ann. of Math. Studies, No. 66, Princeton Univ. Press, Princeton, N.J., 1971, pp. 369–383. MR 0288254
Scott Wolpert, On the homology of the moduli space of stable curves, Ann. of Math. (2) 118 (1983), no. 3, 491–523. MR 727702, DOI 10.2307/2006980

The hyperbolic metric and the geometry of the universal curve