Holomorphic Legendrian curves in ${\mathbb{CP}}^3$ and superminimal surfaces in ${\mathbb{S}}^4$

Abstract

We obtain a Runge approximation theorem for holomorphic Legendrian curves and immersions in the complex projective $3$-space ${\mathbb{CP}}^3$, both from open and compact Riemann surfaces, and we prove that the space of Legendrian immersions from an open Riemann surface into ${\mathbb{CP}}^3$ 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 ${\mathbb{CP}}^3$ as a complete holomorphic Legendrian curve. Under the twistor projection $\pi :{\mathbb{CP}}^3\to {\mathbb{S}}^4$ onto the $4$-sphere, immersed holomorphic Legendrian curves $M\to {\mathbb{CP}}^3$ 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.

References in corpus (17)

• Null curves and directed immersions of open Riemann surfaces
• Every bordered Riemann surface is a complete proper curve in a ball
• Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves
• Holomorphic Legendrian curves
• New complex analytic methods in the theory of minimal surfaces: a survey
• Interpolation by conformal minimal surfaces and directed holomorphic curves
• Embedded minimal surfaces in ${\mathbb{R}}^n$
• Complete bounded embedded complex curves in C^2
• Darboux charts around holomorphic Legendrian curves and applications
• Complete embedded complex curves in the ball of ${\mathbb{C}}^2$ can have any topology
• The Calabi-Yau property of superminimal surfaces in self-dual Einstein four-manifolds
• Complete minimal surfaces densely lying in arbitrary domains of ${\mathbb{R}}^n$
• The Calabi-Yau problem for Riemann surfaces with finite genus and countably many ends
• A foliation of the ball by complete holomorphic discs
• New complex analytic methods in the study of non-orientable minimal surfaces in ${\mathbb{R}}^n$
• The Oka principle for holomorphic Legendrian curves in ${\mathbb{C}}^{2n+1}$
• Immersions of open Riemann surfaces into the Riemann sphere

Cited by in corpus (4)

• Recent developments on Oka manifolds
• The Calabi-Yau problem for minimal surfaces with Cantor ends
• Proper superminimal surfaces of given conformal types in the hyperbolic four-space
• Minimal surfaces with symmetries