Communications in Analysis and Geometry

Volume 27 (2019)

Number 1

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

Pages: 1 – 45



Franc Forstnerič (Faculty of Mathematics and Physics, University of Ljubljana, Slovenia; and Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia)

Finnur Lárusson (School of Mathematical Sciences, University of Adelaide, SA, Australia)


Let $M$ be an open Riemann surface. It was proved by Alarcón and Forstnerič 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 $\iota : \mathfrak{RN}_{\ast} (M, \mathbb{C}^n) \hookrightarrow \mathfrak{M}_{\ast} (M, \mathbb{R}^n)$ 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 (see Theorem 4.1). In particular, $\iota$ is a weak homotopy equivalence (see Theorem 1.1). We prove analogous results for several other related maps (see Theorems 1.2 and 5.6 and Corollary 1.3), and we describe the rough shape of the space of all holomorphic immersions $M \to \mathbb{C}^n$ (Theorem 1.4). 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 (see Corollary 6.2 and Remark 6.3).

F. Forstnerič is supported in part by research program P1-0291 and grants J1-5432 and J1-7256 from ARRS, Republic of Slovenia. F. Lárusson is supported in part by Australian Research Council grant DP150103442.

A part of the work on this paper was done while F. Forstnerič was visiting the School of Mathematical Sciences at the University of Adelaide in January and February 2016. He would like to thank the University of Adelaide for hospitality and the Australian Research Council for financial support.

The authors wish to thank Antonio Alarcón for helpful discussions.

Received 21 February 2016

Published 7 May 2019