@misc{10481/72962,
year = {2022},
month = {1},
url = {http://hdl.handle.net/10481/72962},
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.},
organization = {State Research Agency (SRA)},
organization = {European Commission	MTM2017-89677-P},
organization = {Spanish Government},
organization = {European Commission	PID2020-117868GB-I00},
organization = {Junta de Andalucia	P18-FR-4049},
organization = {Junta de Andalucia -FEDER, Spain	AFQM-139-UGR18},
organization = {Slovenian Research Agency - Slovenia	P1-0291	
J1-9104},
organization = {Australian Research Council	DP150103442},
publisher = {Mathematical Sciences Publishers},
title = {Holomorphic Legendrian curves in CP3 and superminimal surfaces in S4},
doi = {10.2140/gt.2021.25.3507},
author = {Alarcón López, Antonio},
}