Isotopies of complete minimal surfaces of finite total curvature

Let M be a Riemann surface biholomorphic to an affine algebraic curve. We show that the inclusion of the space NC_*(M,Cⁿ) of real parts of nonflat proper algebraic null immersions Mⁿ, n≥ 3, into the space CMI_*(M,Rⁿ) of complete nonflat conformal minimal immersions Mⁿ of finite total curvature is a weak homotopy equivalence. We also show that the $(1,0)$-differential ∂, mapping CMI_*(M,Rⁿ) or NC_*(M,Cⁿ) to the space A¹(M,A) of algebraic $1$-forms on M with values in the punctured null quadric A ⊂ Cⁿ\0\, is a weak homotopy equivalence. Analogous results are obtained for proper algebraic immersions Mⁿ, n≥ 2, directed by a flexible or algebraically elliptic punctured cone in Cⁿ\0\.