Approximation of infinite-dimensional Teichmüller spaces

Gardiner, Frederick P.

doi:10.1090/S0002-9947-1984-0728718-7

Approximation of infinite-dimensional Teichmüller spaces
HTML articles powered by AMS MathViewer

by Frederick P. Gardiner

Trans. Amer. Math. Soc. 282 (1984), 367-383

DOI: https://doi.org/10.1090/S0002-9947-1984-0728718-7

PDF | Request permission

Abstract:

By means of an exhaustion process it is shown that Teichmüller’s metric and Kobayashi’s metric are equal for infinite dimensional Teichmüller spaces. By the same approximation method important estimates coming from the Reich-Strebel inequality are extended to the infinite dimensional cases. These estimates are used to show that Teichmüller’s metric is the integral of its infinitesimal form. They are also used to give a sufficient condition for a sequence to be an absolute maximal sequence for the Hamilton functional. Finally, they are used to give a new sufficient condition for a sequence of Beltrami coefficients to converge in the Teichmüller metric.

References

Lars V. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand Mathematical Studies, No. 10, D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966. Manuscript prepared with the assistance of Clifford J. Earle, Jr. MR 0200442
Lipman Bers, An approximation theorem, J. Analyse Math. 14 (1965), 1–4. MR 178287, DOI 10.1007/BF02806376
Lipman Bers, Automorphic forms and Poincaré series for infinitely generated Fuchsian groups, Amer. J. Math. 87 (1965), 196–214. MR 174737, DOI 10.2307/2373231
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, A new proof of a fundamental inequality for quasiconformal mappings, J. Analyse Math. 36 (1979), 15–30 (1980). MR 581797, DOI 10.1007/BF02798764
Clifford J. Earle and James Eells Jr., On the differential geometry of Teichmüller spaces, J. Analyse Math. 19 (1967), 35–52. MR 220923, DOI 10.1007/BF02788708
Clifford J. Earle and James Eells Jr., Foliations and fibrations, J. Differential Geometry 1 (1967), no. 1, 33–41. MR 215320
Fred Gardiner, An analysis of the group operation in universal Teichmüller space, Trans. Amer. Math. Soc. 132 (1968), 471–486. MR 224812, DOI 10.1090/S0002-9947-1968-0224812-9
Frederick P. Gardiner, On the variation of Teichmüller’s metric, Proc. Roy. Soc. Edinburgh Sect. A 85 (1980), no. 1-2, 143–152. MR 566072, DOI 10.1017/S0308210500011768
Richard S. Hamilton, Extremal quasiconformal mappings with prescribed boundary values, Trans. Amer. Math. Soc. 138 (1969), 399–406. MR 245787, DOI 10.1090/S0002-9947-1969-0245787-3
Irwin Kra, Automorphic forms and Kleinian groups, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Reading, Mass., 1972. MR 0357775
Irwin Kra, On spaces of Kleinian groups, Comment. Math. Helv. 47 (1972), 53–69. MR 306485, DOI 10.1007/BF02566788
Brian O’Byrne, On Finsler geometry and applications to 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. 317–328. MR 0286141
Edgar Reich and Kurt Strebel, Extremal quasiconformal mappings with given boundary values, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 375–391. MR 0361065

Teichmüller mappings which keep the boundary pointwise fixed

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
Kurt Strebel, On quasiconformal mappings of open Riemann surfaces, Comment. Math. Helv. 53 (1978), no. 3, 301–321. MR 505549, DOI 10.1007/BF02566081