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 ${ℂ\mathbb{P}}^3$, both from open and compact Riemann surfaces, and we prove that the space of Legendrian immersions from an open Riemann surface into ${ℂ\mathbb{P}}^3$ 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 ${ℂ\mathbb{P}}^3$ as a complete holomorphic Legendrian curve. Under the twistor projection $\pi :{ℂ\mathbb{P}}^3\to {\mathbb{𝕊}}^4$ onto the $4$–sphere, immersed holomorphic Legendrian curves $M\to {ℂ\mathbb{P}}^3$ are in bijective correspondence with superminimal immersions $M\to {\mathbb{𝕊}}^4$ of positive spin, according to a result of Bryant. This gives as corollaries the corresponding results on superminimal surfaces in ${\mathbb{𝕊}}^4$. In particular, superminimal immersions into ${\mathbb{𝕊}}^4$ satisfy the Runge approximation theorem and the Calabi–Yau property.