zbMATH — the first resource for mathematics

Closed geodesics and non-differentiability of the metric in infinite-dimensional Teichmüller spaces. (English) Zbl 0842.30015
Summary: We construct a closed geodesic in any infinite-dimensional Teichmüller space. The construction also leads to a proof of non-differentiability of the metric in infinite-dimensional Teichmüller spaces, which provides a negative answer to a problem of Goldberg.

MSC:
 30C62 Quasiconformal mappings in the complex plane 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 14H15 Families, moduli of curves (analytic)
Keywords:
Teichmüller spaces
Full Text:
References:
 [1] Clifford J. Earle, The Teichmüller distance is differentiable, Duke Math. J. 44 (1977), no. 2, 389 – 397. · Zbl 0352.32006 [2] Clifford J. Earle and James Eells Jr., On the differential geometry of Teichmüller spaces, J. Analyse Math. 19 (1967), 35 – 52. · Zbl 0156.30604 [3] C. J. Earle, I. Kra, and S. L. Krushkal$$^{\prime}$$, Holomorphic motions and Teichmüller spaces, Trans. Amer. Math. Soc. 343 (1994), no. 2, 927 – 948. · Zbl 0812.30018 [4] Frederick P. Gardiner, Teichmüller theory and quadratic differentials, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1987. A Wiley-Interscience Publication. · Zbl 0629.30002 [5] Lisa R. Goldberg, On the shape of the unit sphere in \?(\Delta ), Proc. Amer. Math. Soc. 118 (1993), no. 4, 1179 – 1185. · Zbl 0787.30030 [6] Saul Kravetz, On the geometry of Teichmüller spaces and the structure of their modular groups, Ann. Acad. Sci. Fenn. Ser. A I No. 278 (1959), 35. · Zbl 0168.04601 [7] Li Zhong, Nonuniqueness of geodesics in infinite-dimensional Teichmüller spaces, Complex Variables Theory Appl. 16 (1991), no. 4, 261 – 272. · Zbl 0737.32010 [8] Li Zhong, Non-uniqueness of geodesics in infinite-dimensional Teichmüller spaces (II), Ann. Acad. Sci. Fenn. Series A. I. Math. 18, 355–367. · Zbl 0801.32006 [9] Li Zhong, Non-convexity of spheres in infinite-dimensional Teichmüller spaces, Science in China 37 (1994), 924–932. · Zbl 0810.30037 [10] H. L. Royden, Report on the Teichmüller metric, Proc. Nat. Acad. Sci. U.S.A. 65 (1970), 497 – 499. · Zbl 0189.36401 [11] 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. · Zbl 0318.30022 [12] Kurt Strebel, On quadratic differentials and extremal quasi-conformal mappings, Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 223 – 227. · Zbl 0334.30012 [13] Harumi Tanigawa, Holomorphic families of geodesic discs in infinite-dimensional Teichmüller spaces, Nagoya Math. J. 127 (1992), 117 – 128. · Zbl 0763.32015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.