Removing singularities of minimal surfaces by isotopies

TL;DR

This paper demonstrates that branch points and ends of finite total curvature in minimal surfaces and null holomorphic curves can be eliminated via isotopies, simplifying their structure while preserving key properties.

Contribution

It introduces a method to remove singularities of minimal surfaces and null curves through isotopies, extending the understanding of their geometric and topological flexibility.

Findings

Singularities can be removed by isotopies in minimal surfaces.

The result applies to surfaces with finite total curvature.

Null holomorphic curves also admit similar isotopy-based regularizations.

Abstract

Given an open Riemann surface $M$, we show that the branch points and the complete ends of finite total curvature of a conformal minimal surface $M\to{\mathbb R}^n$, $n\ge 3$, can be removed by an isotopy through such surfaces. The analogous result holds for null holomorphic curves $M\to{\mathbb C}^n$.