The parametric h-principle for minimal surfaces in Rⁿ and null curves in Cⁿ

Abstract

Let M be an open Riemann surface. It was proved by Alarcón and Forstnerič (arXiv:1408.5315) that every conformal minimal immersion M→R3 is isotopic to the real part of a holomorphic null curve M→C3. In this paper, we prove the following much stronger result in this direction: for any n≥3, the inclusion ι of the space of real parts of nonflat null holomorphic immersions M→Cn into the space of nonflat conformal minimal immersions M→Rn 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→Cn. For an open Riemann surface M of finite topological type, we obtain optimal results by showing that ι and several related maps are inclusions of strong deformation retracts; in particular, they are homotopy equivalences.