Abstract. We study some relationships between holomorphic motions, continuous motions, and monodromy. We also study extensions of holomorphic motions over Riemann surfaces and characterize the extendability of holomorphic motions over some planar regions in terms of monodromy.
We review several applications of Douady-Earle section to holomorphic 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 holomorphic motions in complex analysis.
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