Vol. 15, No. 4, 2022

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Mergelyan approximation theorem for holomorphic Legendrian curves

Franc Forstnerič

Vol. 15 (2022), No. 4, 983–1010
Abstract

We prove a Mergelyan-type approximation theorem for immersed holomorphic Legendrian curves in an arbitrary complex contact manifold (X,ξ). Explicitly, we show that if S is a compact admissible set in a Riemann surface M and f : S X is a ξ-Legendrian immersion of class 𝒞r+2(S,X) for some r 2 which is holomorphic in the interior of S, then f can be approximated in the 𝒞r(S,X) topology by holomorphic Legendrian embeddings from open neighbourhoods of S into X. This has numerous applications, some of which are indicated in the paper. In particular, by using Bryant’s correspondence for the Penrose twistor map 3 S4 we show that a Mergelyan approximation theorem and the Calabi–Yau property hold for conformal superminimal surfaces in the 4-sphere S4.

Keywords
complex contact manifold, Legendrian curve, Mergelyan theorem, superminimal surface
Mathematical Subject Classification 2010
Primary: 53D35, 32E30, 34D10
Secondary: 37J55, 53A10
Milestones
Received: 14 January 2020
Revised: 9 July 2020
Accepted: 11 December 2020
Published: 3 September 2022
Authors
Franc Forstnerič
Faculty of Mathematics and Physics
University of Ljubljana
Ljubljana
Slovenia
Institute of Mathematics, Physics and Mechanics
Ljubljana
Slovenia