Every conformal minimal surface in R-3 is isotopic to the real part of a holomorphic null curve

We show that for every conformal minimal immersion u : M -> R-3 from an open Riemann surface M to R-3 there exists a smooth isotopy u(t) : M -> R-3 (t is an element of[0,1]) of conformal minimal immersions, with u(0) = u, such that u(1) is the real part of a holomorphic null curve M -> C-3 (i.e. u(1) has vanishing flux). If furthermore u is nonflat, then u(1) can be chosen to have any prescribed flux and to be complete.