Removing singularities of minimal surfaces by isotopies

TL;DR

The paper addresses removing singularities of conformal minimal surfaces in $\mathbb{R}^n$ for $n\ge3$ by isotopies on open Riemann surfaces $M$. It develops three technical tools—Weierstrass interpolation with parameters, abelian differentials with values in complex cones, and a parametric period-control interpolation—to construct isotopies that eliminate branch points and complete ends of finite total curvature while controlling the Gauss map and periods. The main contributions are (i) showing branch points are removable by isotopy, (ii) showing complete ends of finite total curvature are removable by isotopy (under fullness), and (iii) combining these to deduce surjectivity on path components of the natural inclusions between spaces of conformal minimal immersions, with extensions to generalized null curves. These results imply path-connectedness and richness of the moduli of minimal surfaces, and they enable isotopic realization of immersed, proper, or nonflat minimal surfaces starting from singular ones, via Runge and Oka theory-based deformations of the Weierstrass data $f\theta$ and related abelian differentials.

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 $M\to{\mathbb C}^n$.

Theorems & Definitions (17)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Corollary 1.4
• Lemma 2.1
• proof
• Proposition 2.2
• Lemma 2.3
• proof
• proof : Proof of Proposition \ref{['pr:multipliers']}