A strong parametric h-principle for complete minimal surfaces

Abstract

We prove a parametric h-principle for complete nonflat conformal minimal immersions of an open Riemann surface M into R n , n ≥ 3. It follows that the inclusion of the space of such immersions into the space of all nonflat conformal minimal immersions is a weak homotopy equivalence. When M is of finite topological type, the inclusion is a genuine homotopy equivalence. By a parametric h-principle due to Forstnerič and Lárusson, the space of complete nonflat conformal minimal immersions therefore has the same homotopy type as the space of continuous maps from M to the punctured null quadric. Analogous results hold for holomorphic null curves M → C n and for full immersions in place of nonflat ones.