Holomorphic Legendrian curves in CP3 and superminimal surfaces in S4 Alarcón López, Antonio Alarcon is supported by the State Research Agency (SRA) and European Regional Development Fund (ERDF) via the grant no. MTM2017-89677-P, MICINN, Proyecto PID2020-117868GB-I00 financiado por MCIN/AEI/10.13039/501100011033/the Junta de Andalucia grant no. P18-FR-4049, and the Junta de Andalucia -FEDER grant no. AFQM-139-UGR18, Spain. Forstneri.c is supported by the research program P1-0291 and the research grant J1-9104 from ARRS, Republic of Slovenia. Larusson is supported by Australian Research Council grant DP150103442. A part of the work on this paper was done while Forstneri.c and Larusson were visiting the University of Granada in September 2019. They wish to thank the university for the invitation and partial support. 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. 2022-02-23T11:27:47Z 2022-02-23T11:27:47Z 2022-01-25 journal article Alarcón, A., Forstnerič, F., & Lárusson, F. (2022). Holomorphic Legendrian curves in CP3 and superminimal surfaces in S4. Geometry & Topology, 25(7), 3507-3553. DOI: [10.2140/gt.2021.25.3507] http://hdl.handle.net/10481/72962 10.2140/gt.2021.25.3507 eng http://creativecommons.org/licenses/by-nc-nd/3.0/es/ open access Atribución-NoComercial-SinDerivadas 3.0 España Mathematical Sciences Publishers