Every Nonflat Conformal Minimal Surface is Homotopic to a Proper One

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 .