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

TL;DR

This paper proves a parametric h-principle for minimal surfaces and null curves in higher dimensions, showing a deep homotopy equivalence between spaces of minimal immersions and null holomorphic immersions in complex space.

Contribution

It extends previous results by establishing the parametric h-principle for all dimensions n≥3, and describes the homotopy type of the space of holomorphic immersions from Riemann surfaces.

Findings

The inclusion of null holomorphic immersions into minimal immersions satisfies the parametric h-principle.

For finite topological type surfaces, the inclusion is a strong deformation retract.

The homotopy type of the space of all holomorphic immersions is explicitly described.

Abstract

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 $n\geq 3$, 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.