Guiding Isotopies and Holomorphic Motions

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 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 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.