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

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 $\mathscr C(M,\mathbb S^{4n-1})$ofcontinuousmapsfrom$M$tothesphere$\mathbb S^{4n-1}$\mathrm{.}If$M$ has finite topological type, then these spaces are homotopy equivalent. We determine the homotopy groups of $\mathscr L(M,\mathbb C^{2n+1})$intermsofthehomotopygroupsof$\mathbb S^{4n-1}$. It follows that $\mathcal{L}(M,{\mathbb{C}}^{2n+1})$ is $(4n-3)$-connected.