The Oka principle for holomorphic Legendrian curves in C²ⁿ⁺¹

Abstract

Let M be a connected open Riemann surface. We prove that the space L(M,C2n+1) of all holomorphic Legendrian immersions of M to C2n+1, n≥1, endowed with the standard holomorphic contact structure, is weakly homotopy equivalent to the space C(M,S4n−1) of continuous maps from M to the sphere S4n−1. If M has finite topological type, then these spaces are homotopy equivalent. We determine the homotopy groups of L(M,C2n+1) in terms of the homotopy groups of S4n−1. It follows that L(M,C2n+1) is (4n−3)-connected.