Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 40, 2015, 485-501

GUIDING ISOTOPIES AND HOLOMORPHIC MOTIONS

Frederick P. Gardiner, Yunping Jiang and Zhe Wang

CUNY Graduate Center, Department of Mathematics
365 Fifth Avenue, New York, NY 10016, U.S.A.; frederick.gardiner 'at' gmail.com

Queens College of CUNY, Department of Mathematics, Flushing, NY 11367
and CUNY Graduate Center, Department of Mathematics
365 Fifth Avenue, New York, NY 10016, U.S.A.; yunping.jiang 'at' qc.cuny.edu

Bronx Community College of CUNY, Department of Mathematics and Computer Science
2155 University Avenue, Bronx, NY 10453, U.S.A.; wangzhecuny 'at' gmail.com

Abstract. We develop an isotopy principle for holomorphic motions. Our main result concerns the extendability of a holomorphic motion h(t,z) of a finite subset E of the Riemann sphere \overline{C} parameterized by a pointed hyperbolic Riemann surface (X,t₀). We prove that if this holomorphic motion has a guiding quasiconformal isotopy, then it can be extended to a new holomorphic motion of E ∪ {p} for any point p in \overline{C} \ E that follows the guiding isotopy. The proof gives a canonical way to replace a continuous motion of the (n + 1)-st point by a holomorphic motion while leaving unchanged the given holomorphic motion of the first n points.

2010 Mathematics Subject Classification: Primary 30F60; Secondary 32G15, 30C70, 30C75.

Key words: Guiding quasiconformal isotopy, holomorphic motion, continuous motion, holomorphic quadratic differential, heights mapping.

Reference to this article: F. P. Gardiner, Y. Jiang and Z. Wang: Guiding isotopies and holomorphic motions. Ann. Acad. Sci. Fenn. Math. 40 (2015), 485-501.

Full document as PDF file

doi:10.5186/aasfm.2015.4031