Every conformal minimal surface in $\mathbb{R}^3$ is isotopic to the real part of a holomorphic null curve

Abstract

In this paper, 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). Furthermore, if $u$ is nonflat then $u_1$ can be chosen to have any prescribed flux and to be complete.