The parametric h-principle for minimal surfaces in ${\mathbb{R}}^n$ and null curves in ${\mathbb{C}}^n$

Let $M$ be an open Riemann surface. It was proved by Alarc'on and Forstneri\v{c} (arXiv:1408.5315) that every conformal minimal immersion $M\to {\mathbb{R}}^3$ is isotopic to the real part of a holomorphic null curve $M\to {\mathbb{C}}^3$. In this paper, we prove the following much stronger result in this direction: for any $n\ge 3$, the inclusion $\iota$ of the space of real parts of nonflat null holomorphic immersions $M\to {\mathbb{C}}^n$ into the space of nonflat conformal minimal immersions $M\to {\mathbb{R}}^n$ satisfies the parametric h-principle with approximation; in particular, it is a weak homotopy equivalence. We prove analogous results for several other related maps, and we describe the homotopy type of the space of all holomorphic immersions $M\to {\mathbb{C}}^n$. For an open Riemann surface $M$ of finite topological type, we obtain optimal results by showing that $\iota$ and several related maps are inclusions of strong deformation retracts; in particular, they are homotopy equivalences.