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 ${\mathbb{R}}^n$, $n\ge 3$. 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 Forstneric and Larusson, 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 $M\to {\mathbb{C}}^n$ and for full immersions in place of nonflat ones.

To appear in The Journal of Geometric Analysis