#### 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 $\left(X,\xi \right)$. Explicitly, we show that if $S$ is a compact admissible set in a Riemann surface $M$ and $f:S\to X$ is a $\xi$-Legendrian immersion of class ${\mathsc{𝒞}}^{r+2}\left(S,X\right)$ for some $r\ge 2$ which is holomorphic in the interior of $S$, then $f$ can be approximated in the ${\mathsc{𝒞}}^{r}\left(S,X\right)$ 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 ${ℂℙ}^{3}\to {S}^{4}$ 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