Every Nonflat Conformal Minimal Surface is Homotopic to a Proper One

Abstract Given an open Riemann surface M, we prove that every nonflat conformal minimal immersion M Rⁿ M → R n (n 3 n ≥ 3) is homotopic through nonflat conformal minimal immersions M Rⁿ M → R n to a proper one. If n 5 n ≥ 5, it may be chosen in addition injective, hence a proper conformal minimal embedding. Prescribing its flux, as a consequence, every nonflat conformal minimal immersion M Rⁿ M → R n is homotopic to the real part of a proper holomorphic null embedding M Cⁿ M → C n. We also obtain a result for a more general family of holomorphic immersions from an open Riemann surface into Cⁿ C n directed by Oka cones in Cⁿ C n.