A strong parametric h-principle for complete minimal surfaces

TL;DR

This paper establishes a parametric h-principle for complete nonflat conformal minimal immersions of open Riemann surfaces into Euclidean spaces, revealing their homotopy equivalence with certain continuous map spaces.

Contribution

It proves a strong parametric h-principle for these minimal immersions, showing their space's homotopy type aligns with continuous maps into the punctured null quadric, extending to holomorphic null curves.

Findings

The inclusion of minimal immersions into all conformal minimal immersions is a weak homotopy equivalence.

For finite topological type surfaces, this inclusion is a homotopy equivalence.

The space of such immersions shares the homotopy type with continuous maps into the punctured null quadric.

Abstract

We prove a parametric h-principle for complete nonflat conformal minimal immersions of an open Riemann surface $M$ into $\mathbb R^n$, $n\geq 3$. 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.