The Oka principle for holomorphic Legendrian curves in ${\mathbb{C}}^{2n+1}$

Abstract

Let $M$ be a connected open Riemann surface. We prove that the space $\mathcal{L}(M,{\mathbb{C}}^{2n+1})$ of all holomorphic Legendrian immersions of $M$ into ${\mathbb{C}}^{2n+1}$, $n\ge 1$, endowed with the standard holomorphic contact structure, is weakly homotopy equivalent to the space $\mathcal{C}(M,{\mathbb{S}}^{4n-1})$ of continuous maps from $M$ to the sphere ${\mathbb{S}}^{4n-1}$. If $M$ has finite topological type, then these spaces are homotopy equivalent. We determine the homotopy groups of $\mathcal{L}(M,{\mathbb{C}}^{2n+1})$ in terms of the homotopy groups of ${\mathbb{S}}^{4n-1}$. It follows that $\mathcal{L}(M,{\mathbb{C}}^{2n+1})$ is $(4n-3)$-connected.

Updated references. To appear in Mathematische Zeitschrift

References in corpus (2)

• Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves
• Hyperbolic complex contact structures on ${\mathbb{C}}^{2n+1}$

Cited by in corpus (4)

• Darboux charts around holomorphic Legendrian curves and applications
• Holomorphic Legendrian curves in ${\mathbb{CP}}^3$ and superminimal surfaces in ${\mathbb{S}}^4$
• Existence of Horizontal Immersions in Fat Distributions
• On Horizontal Immersions of Discs in Fat Distributions of Type $(4,6)$