Mergelyan approximation theorem for holomorphic Legendrian curves

Forstnerič, Franc

doi:10.2140/apde.2022.15.983

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Mergelyan approximation theorem for holomorphic Legendrian curves

Franc Forstnerič

Vol. 15 (2022), No. 4, 983–1010

DOI: 10.2140/apde.2022.15.983

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 $f : S \to X$ is a $ξ$ -Legendrian immersion of class $𝒞^{r + 2} (S, X)$ for some $r \geq 2$ which is holomorphic in the interior of $S$ , then $f$ can be approximated in the $𝒞^{r} (S, X)$ 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 ${ℂ ℙ}^{3} \to S^{4}$ we show that a Mergelyan approximation theorem and the Calabi–Yau property hold for conformal superminimal surfaces in the $4$ -sphere $S^{4}$ .

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

Vol. 15, No. 4, 2022