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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 |
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Geometry & Topology 25 (2021) 3507–3553
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Abstract |
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We obtain a Runge approximation theorem for holomorphic Legendrian curves and immersions in the complex projective –space , both from open and compact Riemann surfaces, and we prove that the space of Legendrian immersions from an open Riemann surface into 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 as a complete holomorphic Legendrian curve. Under the twistor projection onto the –sphere, immersed holomorphic Legendrian curves are in bijective correspondence with superminimal immersions of positive spin, according to a result of Bryant. This gives as corollaries the corresponding results on superminimal surfaces in . In particular, superminimal immersions into satisfy the Runge approximation theorem and the Calabi–Yau property. |
Keywords
Riemann surface, Legendrian curve, Runge approximation,
superminimal surface
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Mathematical Subject Classification 2010
Primary: 53D10
Secondary: 32E30, 32H02, 53A10
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References |
Publication
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
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Authors |
| Antonio Alarcón | |
| Departamento de Geometría y
Topología e Instituto de Matemáticas (IEMath-GR) Universidad de Granada Granada Spain |
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| Franc Forstnerič | |
| Faculty of Mathematics and
Physics University of Ljubljana Ljubljana Slovenia |
Institute of Mathematics, Physics
and Mechanics Ljubljana Slovenia |
| Finnur Lárusson | |
| School of Mathematical
Sciences University of Adelaide Adelaide SA Australia |
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