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Holomorphic motions and related topics. (English) Zbl 1198.30019
Gardiner, Frederick P. (ed.) et al., Geometry of Riemann surfaces. Proceedings of the Anogia conference to celebrate the 65th birthday of William J. Harvey, Anogia, Crete, Greece, June–July 2007. Cambridge: Cambridge University Press (ISBN 978-0-521-73307-6/pbk). London Mathematical Society Lecture Note Series 368, 156-193 (2010).
The \(\lambda\)-lemma for holomorphic motions was proved by Mane, Sad and Sullivan in the paper [R. Mane, P. Sad, and D. P. Sullivan, Ann. Sci. Éc. Norm. Supér. (4) 16, 193–217 (1983; Zbl 0524.58025)]. The lemma says that any holomorphic motion of a subset \(E\) of the extended complex plane \(\overline{\mathbb{C}}\) parametrized by the unit disk \(\Delta\) and with basepoint \(0\) can be extended uniquely to a holomorphic motion of the closure \(\overline{E}\) of \(E\), again parametrized by \(\Delta\), and with the same basepoint. In the paper [Acta Math. 157, 243–286 (1986; Zbl 0619.30026)], D. P. Sullivan and W. P. Thurston proved an important extension of the \(\lambda\)-lemma, namely that any holomorphic motion of \(E\) parametrized by \(\Delta\) and with basepoint \(0\) can be extended to a holomorphic motion of \(\overline{\mathbb{C}}\), but parametrized by a smaller disk \(\Delta_r\) centered at the origin and of radius \(r\), for some universal number \(0<r<1\). The question was then raised whether one has \(r=1\). In the paper [Proc. Am. Math. Soc. 111, No. 2, 347–355 (1991; Zbl 0741.32009)], Z. Slodkowski gave a positive answer to that question.
In the paper under review, the authors give an expository account of a proof of Slodkowski’s theorem given by E. M. Chirka [Dokl. Akad. Nauk. 397, 37–40 (2004; Zbl 1198.37071)].
It is well known that the \(\lambda\)-lemma has applications to Teichmüller theory. The authors give a proof of the fact (due to Royden for surfaces of finite type, with imporvements by various people) that the Kobayashi and the Teichmüller metric on the Teichmüller space of a Riemann surface coincide.
For the entire collection see [Zbl 1182.30003].

MSC:
30C62 Quasiconformal mappings in the complex plane
30F60 Teichmüller theory for Riemann surfaces
32F45 Invariant metrics and pseudodistances in several complex variables
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