Holomorphic Legendrian curves in CP3 and superminimal surfaces in S4

Abstract

We obtain a Runge approximation theorem for holomorphic Legendrian curves and immersions in the complex projective 3-space CP3, both from open and compact Riemann surfaces, and we prove that the space of Legendrian immersions from an open Riemann surface into CP3 is path-connected. We also show that holomorphic Legendrian immersions from Riemann surfaces of finite genus and at most countably many ends, none of which are point ends, satisfy the Calabi-Yau property. Coupled with the Runge approximation theorem, we infer that every open Riemann surface embeds into CP3 as a complete holomorphic Legendrian curve. Under the twistor projection pi : CP3 -> S-4 onto the 4-sphere, immersed holomorphic Legendrian curves M -> CP3 are in bijective correspondence with superminimal immersions M -> S-4 of positive spin, according to a result of Bryant. This gives as corollaries the corresponding results on superminimal surfaces in S-4. In particular, superminimal immersions into S-4 satisfy the Runge approximation theorem and the Calabi-Yau property.