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

TL;DR

This paper establishes a homotopy equivalence between the space of holomorphic Legendrian immersions of a Riemann surface into complex Euclidean space and the space of continuous sphere maps, revealing topological properties of these spaces.

Contribution

It proves the Oka principle for holomorphic Legendrian curves in complex Euclidean spaces, connecting complex geometry with topological homotopy theory.

Findings

The space of Legendrian immersions is weakly homotopy equivalent to continuous sphere maps.

For finite topological type surfaces, the spaces are homotopy equivalent.

The homotopy groups of Legendrian immersions are determined by those of spheres, showing $(4n-3)$-connectivity.

Abstract

Let $M$ be a connected open Riemann surface. We prove that the space $\mathscr L(M,\mathbb C^{2n+1})$ of all holomorphic Legendrian immersions of $M$ into $\mathbb C^{2n+1}$, $n\geq 1$, endowed with the standard holomorphic contact structure, is weakly homotopy equivalent to the space $\mathscr C(M,\mathbb S^{4n-1})$ 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 $\mathscr L(M,\mathbb C^{2n+1})$ in terms of the homotopy groups of $\mathbb S^{4n-1}$. It follows that $\mathscr L(M,\mathbb C^{2n+1})$ is $(4n-3)$-connected.