Isotopies of complete minimal surfaces of finite total curvature

TL;DR

This paper proves that the space of complete minimal surfaces of finite total curvature in Euclidean space is topologically equivalent to the space of their algebraic null immersions, revealing deep homotopy properties of these geometric objects.

Contribution

It establishes a weak homotopy equivalence between spaces of minimal surfaces and algebraic null immersions, extending to proper algebraic immersions directed by algebraic cones.

Findings

Weak homotopy equivalence between minimal surfaces and null immersions.

The differential map $ ext{partial}$ is a weak homotopy equivalence.

Analogous results for proper algebraic immersions directed by algebraic cones.

Abstract

Let $M$ be a Riemann surface biholomorphic to an affine algebraic curve. We show that the inclusion of the space $\Re \mathrm{NC}_*(M,\mathbb{C}^n)$ of real parts of nonflat proper algebraic null immersions $M\to\mathbb{C}^n$, $n\ge 3$, into the space $\mathrm{CMI}_*(M,\mathbb{R}^n)$ of complete nonflat conformal minimal immersions $M\to\mathbb{R}^n$ of finite total curvature is a weak homotopy equivalence. We also show that the $(1,0)$-differential $\partial$, mapping $\mathrm{CMI}_*(M,\mathbb{R}^n)$ or $\Re \mathrm{NC}_*(M,\mathbb{C}^n)$ to the space $\mathscr{A}^1(M,\mathbf{A})$ of algebraic $1$-forms on $M$ with values in the punctured null quadric $\mathbf{A} \subset \mathbb{C}^n\setminus\{0\}$, is a weak homotopy equivalence. Analogous results are obtained for proper algebraic immersions $M\to\mathbb{C}^n$, $n\ge 2$, directed by a flexible or algebraically elliptic punctured cone in $\mathbb{C}^n\setminus\{0\}$.