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 ℜNC * (M, C n ) of real parts of nonflat proper algebraic null immersions M → C n , n ≥ 3, into the space CMI * (M, R n ) of complete nonflat conformal minimal immersions M → R n of finite total curvature is a weak homotopy equivalence. We also show that the (1, 0)-differential ∂, mapping CMI * (M, R n ) or ℜNC * (M, C n ) to the space A 1 (M, A) of algebraic 1-forms on M with values in the punctured null quadric A ⊂ C n \ {0}, is a weak homotopy equivalence. Analogous results are obtained for proper algebraic immersions M → C n , n ≥ 2, directed by a flexible or algebraically elliptic punctured cone in C n \ {0}.