Isotopies of complete minimal surfaces of finite total curvature

Published 7 Jun 2024 in math.DG and math.CV | (2406.04767v1)

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 $\mathbb{C}^{n\setminus{0}$.}