Holomorphic Legendrian curves in CP3 and superminimal surfaces in S4
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Alarcón López, AntonioEditorial
Mathematical Sciences Publishers
Date
2022-01-25Referencia bibliográfica
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]
Sponsorship
State Research Agency (SRA); European Commission MTM2017-89677-P; Spanish Government; European Commission PID2020-117868GB-I00; Junta de Andalucia P18-FR-4049; Junta de Andalucia -FEDER, Spain AFQM-139-UGR18; Slovenian Research Agency - Slovenia P1-0291 J1-9104; Australian Research Council DP150103442Abstract
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.