Holomorphic motions, soon after they were introduced, became an important subject in complex analysis. It is now an important tool in the study of complex dynamical systems and in the study of Teichmuller theory. This thesis serves on two purposes: an expository of the past developments and a discovery of new theories.
First, I give an expository account of Slodkowski's theorem based on the proof given by Chirka. Then I present a result about infinitesimal holomorphic motions. I prove the |ϵ log ϵ| modulus of continuity for any infinitesimal holomorphic motion. This proof is a very well application of Schwarz's lemma and the estimate of Agard's formula for the hyperbolic metric on the thrice punctured sphere. One application of this result is that, after the integration of an infinitesimal holomorphic motion, it leads to the Holder continuity property of a quasiconformal homeomorphism. This will be presented in Chapter 3.
Second, I compare the proofs given by both Slodkowski and Chirka. Then I construct a different extension of a holomorphic motion in the frame work of Slodkowsk's proof by using the method in Chirka's proof. This gives some opportunity for me to discuss the uniqueness in the extension problem for a holomorphic motion. This will be presented in Chapter 4.
Third, I discuss the universal holomorphic motion for a closed subset of the Riemann sphere and the lifting property in the Teichmuller theory. One application of this discussion is the proof of the coincidence of Teichmuller's metric and Kobayashi's metric, a result due to Royden and Gardiner, given by Earle, Kra, and Krushkal by using Slodkowski's theorem. This will be presented in Chapters 5 and 6.
Fourth, I study the complex structure of the universal asymptotically conformal Teichmuller space. I give a direct and new proof of the coincidence of Teichmuller's metric and Kobayashi's metric on the universal asymptotically conformal Teichmuller space, a result previously proved by Earle, Gardiner, and Lakic. The main technique that I have used in this proof is Strebel's frame mapping theorem. This will be presented in Chapter 7.
Finally, in Chapter 8, I study extremal annuli on a Riemann sphere with four points removed. By using the measurable foliation theory, the Weierstrass P-function, and the variation formula for the modulus of an annulus, I prove that the Mori annulus maximize the modulus for the two army problem in the chordal distance on the Riemann sphere. Gardiner and Masur's minimum axis is also discussed in this chapter.
Most of the results in this thesis have been published in several research papers jointly with Fred Gardiner, Jun Hu, Yunping Jiang, and Sudeb Mitra.