Every conformal minimal surface in ℝ3 is isotopic to the real part of a holomorphic\break null curve

Abstract

Abstract We show that for every conformal minimal immersion {u:M\to\mathbb{R}^{3}} from an open Riemann surface M to {\mathbb{R}^{3}} there exists a smooth isotopy {u_{t}:M\to\mathbb{R}^{3}} ( {t\in[0,1]} ) of conformal minimal immersions, with {u_{0}=u} , such that {u_{1}} is the real part of a holomorphic null curve {M\to\mathbb{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.