Isotopies of complete minimal surfaces of finite total curvature

Abstract

Let $M$ be a Riemann surface biholomorphic to an affine algebraic curve. We show that the inclusion of the space $\Re {\mathrm{NC}}_{\ast }(M,{\mathbb{C}}^n)$ of real parts of nonflat proper algebraic null immersions $M\to {\mathbb{C}}^n$, $n\ge 3$, into the space ${\mathrm{CMI}}_{\ast }(M,{\mathbb{R}}^n)$ 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 ${\mathrm{CMI}}_{\ast }(M,{\mathbb{R}}^n)$ or $\Re {\mathrm{NC}}_{\ast }(M,{\mathbb{C}}^n)$ to the space ${\mathcal{A}}^1(M,\mathbf{A})$ 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 ${\mathbb{C}}^n\setminus \{0\}$.