Proper superminimal surfaces of given conformal types in the hyperbolic four-space

Franc Forstnerič

doi:10.5802/afst.1732

Franc Forstnerič ¹

¹ Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI–1000 Ljubljana, Slovenia, and, Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI–1000 Ljubljana, Slovenia

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 32 (2023) no. 1, pp. 145-172.

Abstract

Let $H^{4}$ denote the hyperbolic four-space. Given a bordered Riemann surface, $M$ , we prove that every smooth conformal superminimal immersion $\bar{M} \to H^{4}$ can be approximated uniformly on compacts in $M$ by proper conformal superminimal immersions $M \to H^{4}$ . In particular, $H^{4}$ contains properly immersed conformal superminimal surfaces normalised by any given open Riemann surface of finite topological type without punctures. The proof uses the analysis of holomorphic Legendrian curves in the twistor space of $H^{4}$ .

Received: 2020-05-11
Accepted: 2021-02-22
Published online: 2023-03-27

DOI: 10.5802/afst.1732

Classification: 53A10, 53C28, 32E30, 37J55
Keywords: superminimal surface, hyperbolic space, twistor space, complex contact manifold, holomorphic Legendrian curve

Author's affiliations:

Franc Forstnerič ¹

¹ Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI–1000 Ljubljana, Slovenia, and, Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI–1000 Ljubljana, Slovenia

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Franc Forstnerič. Proper superminimal surfaces of given conformal types in the hyperbolic four-space. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 32 (2023) no. 1, pp. 145-172. doi : 10.5802/afst.1732. https://afst.centre-mersenne.org/articles/10.5802/afst.1732/

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