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

Antonio Alarcón, Franc Forstnerič and Finnur Lárusson

Geometry & Topology 25 (2021) 3507–3553

We obtain a Runge approximation theorem for holomorphic Legendrian curves and immersions in the complex projective 3–space 3, both from open and compact Riemann surfaces, and we prove that the space of Legendrian immersions from an open Riemann surface into 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 3 as a complete holomorphic Legendrian curve. Under the twistor projection π: 3 𝕊4 onto the 4–sphere, immersed holomorphic Legendrian curves M 3 are in bijective correspondence with superminimal immersions M 𝕊4 of positive spin, according to a result of Bryant. This gives as corollaries the corresponding results on superminimal surfaces in 𝕊4. In particular, superminimal immersions into 𝕊4 satisfy the Runge approximation theorem and the Calabi–Yau property.

Riemann surface, Legendrian curve, Runge approximation, superminimal surface
Mathematical Subject Classification 2010
Primary: 53D10
Secondary: 32E30, 32H02, 53A10
Received: 29 October 2019
Revised: 7 September 2020
Accepted: 7 October 2020
Published: 25 January 2022
Proposed: Tobias H Colding
Seconded: Yasha Eliashberg, Paul Seidel
Antonio Alarcón
Departamento de Geometría y Topología e Instituto de Matemáticas (IEMath-GR)
Universidad de Granada
Franc Forstnerič
Faculty of Mathematics and Physics
University of Ljubljana
Institute of Mathematics, Physics and Mechanics
Finnur Lárusson
School of Mathematical Sciences
University of Adelaide
Adelaide SA