This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. Contemporary Mathematics Volume 575, 2012 http://dx.doi.org/10.1090/conm/575/11400 Douady-Earle section, holomorphic motions, and some applications Yunping Jiang and Sudeb Mitra For Professor Clifford Earle on his 75th birthday Abstract. We review several applications of Douady-Earle section to holo- morphic motions over infinite dimensional parameter spaces. Using Douady- Earle section we study group-equivariant extensions of holomorphic motions. We also discuss the relationship between extending holomorphic motions and lifting holomorphic maps. Finally, we discuss several applications of holomor- phic motions in complex analysis. Introduction This is a survey article on holomorphic motions and Teichm¨ uller spaces, and some applications of holomorphic motions in complex analysis. Our paper is divided into two parts. In Part 1, we study the applications of Douady-Earle section to holomorphic motions over infinite dimensional parameter spaces. It is well-known that holomorphic motions were first introduced in the study of the dynamics of rational maps in the paper [30]. Since its inception, a fundamental topic in this subject has been about extending holomorphic motions. In their famous paper [39], Sullivan and Thurston asked two important questions on extending holomorphic motions over the open unit disk. We use Douady-Earle section to study these two questions over infinite dimensional parameter spaces. There is an intimate relationship between extending holomorphic motions and lifting holomorphic maps into appropriate Teichm¨ uller spaces, first observed by Bers and Royden in [5]. We study that in the fullest generality, which is another application of Douady-Earle section. In particular, we discuss some new results on group-equivariant extensions of holomorphic motions. In Part 2, we focus on holomorphic motions over the open unit disk to study some problems in complex analysis. We first review a proof of a theorem on gluing holomorphic germs on the Riemann sphere. Using the same idea, we give outlines of new proofs of K¨ onig’s theorem, B¨ottcher’s theorem, and 2010 Mathematics Subject Classification. Primary 32G15; Secondary 37F30, 37F45. Key words and phrases. Teichm¨ uller spaces, holomorphic motions, quasiconformal motions, continuous motions. The authors want to thank PSC-CUNY grants for supporting this research. The first author also wants to thank a Simons Collaboration grant and CUNY collaborative grant for supporting this research. He also wants to thank the Academy of Mathematics and Systems Science and the Morningside Center of Mathematics at the Chinese Academy of Sciences for their hospitality. c 2012 American Mathematical Society 219