Mergelyan approximation theorem for holomorphic Legendrian curves

Abstract

In this paper, we prove a Mergelyan type approximation theorem for immersed holomorphic Legendrian curves in an arbitrary complex contact manifold $(X,\xi )$. 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 ${\mathcal{C}}^{r+2}(S,X)$ for some $r\ge 2$ which is holomorphic in the interior of $S$, then $f$ can be approximated in the ${\mathcal{C}}^r(S,X)$ topology by holomorphic Legendrian embeddings from open neighbourhoods of $S$ into $X$. This result has numerous applications, some of which are indicated in the paper. In particular, by using Bryant's correspondence for the Penrose twistor map ${\mathbb{CP}}^3\to S^4$ we show that a Mergelyan approximation theorem and the Calabi-Yau property hold for superminimal surfaces in the $4$-sphere $S^4$.

arXiv commentAnal. PDE, to appear

References in corpus (7)

• New complex analytic methods in the theory of minimal surfaces: a survey
• Holomorphic Legendrian curves
• Darboux charts around holomorphic Legendrian curves and applications
• The Calabi-Yau property of superminimal surfaces in self-dual Einstein four-manifolds
• Carleman approximation by conformal minimal immersions and directed holomorphic curves
• Hyperbolic complex contact structures on ${\mathbb{C}}^{2n+1}$
• H-principle for complex contact structures on Stein manifolds

Cited by in corpus (3)

• The Calabi-Yau property of superminimal surfaces in self-dual Einstein four-manifolds
• The Calabi-Yau problem for minimal surfaces with Cantor ends
• Proper superminimal surfaces of given conformal types in the hyperbolic four-space