Holomorphic motions and relationship of quasiconformal mappings with structural stability and bifurcation in rational functions dynamics, holomorphic motions of Julia sets of quadratic polynomials. Lambda lemma of extension of holomorphic motion to the closure of a domain, initiated by Mañé, Sullivan and Sad in 1983; Slodkowski theorem of extension of holomorphic motion of a subset of the Riemann sphere to the whole sphere, of 1991 and 1995; Slodkowski equivariant theorem, conjectured in 1990 by McMullen, proved in 1994 by Earle, Kra and Krushkal and extended in 1995 by Slodkowski; harmonic lambda lemma of extension with uniqueness, of Bers and Royden in 1986, using Teichmüller spaces of hyperbolic regions in the Riemann sphere, Schwarzian derivative with Nehari norm, quadratic differentials, the Ahlfors-Weill section theorem of 1962 and the Bers embedding of a Teichmüller space of a subset of the Riemann sphere in the space of quadratic differentials with the Nehari norm of 1960.