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

TL;DR

This paper proves that every conformal minimal surface in three-dimensional space can be smoothly deformed into a surface that is the real part of a holomorphic null curve, preserving key properties like flux and completeness.

Contribution

It establishes that all conformal minimal surfaces are isotopic to holomorphic null curves, extending the understanding of their geometric and complex-analytic structure.

Findings

Any conformal minimal immersion can be isotoped to a holomorphic null curve.

The isotopy can preserve flux and completeness if the surface is nonflat.

All conformal minimal surfaces in  are connected through smooth isotopies to holomorphic null curves.

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.