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- W4214844464 abstract "We obtain a Runge approximation theorem for holomorphic Legendrian curves and immersions in the complex projective $3$-space $mathbb{CP}^3$, both from open and compact Riemann surfaces, and we prove that the space of Legendrian immersions from an open Riemann surface into $mathbb{CP}^3$ 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 $mathbb{CP}^3$ as a complete holomorphic Legendrian curve. Under the twistor projection $pi:mathbb{CP}^3to mathbb S^4$ onto the $4$-sphere, immersed holomorphic Legendrian curves $Mto mathbb{CP}^3$ are in bijective correspondence with superminimal immersions $Mtomathbb S^4$ of positive spin according to a result of Bryant. This gives as corollaries the corresponding results on superminimal surfaces in $mathbb S^4$. In particular, superminimal immersions into $mathbb S^4$ satisfy the Runge approximation theorem and the Calabi-Yau property." @default.
- W4214844464 created "2022-03-05" @default.
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- W4214844464 date "2021-12-31" @default.
- W4214844464 modified "2023-10-14" @default.
- W4214844464 title "Holomorphic Legendrian curves in ℂℙ3 and superminimal surfaces in 𝕊4" @default.
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- W4214844464 doi "https://doi.org/10.2140/gt.2021.25.3507" @default.
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