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.

J. reine angew. Math. (Crelles J.), in press. Official version is available at http://dx.doi.org/10.1515/crelle-2015-0069

References in corpus (1)

• Null curves and directed immersions of open Riemann surfaces

Cited by in corpus (2)

• Every meromorphic function is the Gauss map of a conformal minimal surface
• The Oka principle for holomorphic Legendrian curves in ${\mathbb{C}}^{2n+1}$