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\.