Every Nonflat Conformal Minimal Surface is Homotopic to a Proper One Vrhovnik, Tjaša Proper minimal surface Minimal surfaces Riemann surface Null curve Oka manifold Directed immersion This research is partially supported by the State Research Agency (AEI) via the grant no. PID2023-150727NB-I00, funded by MICIU/AEI/10.13039/501100011033 and ERDF/EU, Spain. Funding for open access publishing: Universidad de Granada/CBUA. 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. 2026-03-18T07:24:42Z 2026-03-18T07:24:42Z 2026-03-12 journal article Vrhovnik, T. Every Nonflat Conformal Minimal Surface is Homotopic to a Proper One. J Geom Anal 36, 136 (2026). https://doi.org/10.1007/s12220-026-02389-x https://hdl.handle.net/10481/112223 10.1007/s12220-026-02389-x eng http://creativecommons.org/licenses/by-nc-nd/4.0/ open access Attribution-NonCommercial-NoDerivatives 4.0 Internacional Springer Nature