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 → R^n (n ≥ 3) is homotopic through nonflat conformalminimal immersions M → R^n to a proper one. If 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 → R^n is homotopic to the real part of a proper holomorphic null embedding M → C^n. We also obtain a result for a more general family of holomorphic immersions from an open Riemann surface into C^n directed by Oka cones in C^n.