Abstract |
The Slodkowski theorem shows that a holomorphic motion of a subset in
the Riemann sphere over the unit disk is extendable. Several proofs of
the Slodkowski theorem are given by applying several complex
variables. The idea from Chirka uses generalized Beltrami equations,
which returns the study to the original idea of Bers and Royden. We
have combined both ideas to give a complete theory for the extension
problem of holomorphic motions of subsets in the Riemann sphere over
hyperbolic Riemann surfaces. In this talk, I will provide an overview
of this work. I will discuss the extension problem's zero-winding
number and trivial monodromy conditions. They are both necessary. I
will give a counterexample to show that the zero-winding number
condition is not sufficient and prove that the trivial monodromy
condition is sufficient. I will explain how the study of the lifting
problem, which used to be called one of the most important problems in
Teichmueller theory by Bers and Royden, plays an important role in
studying the extension problem.
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