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].