Darboux charts around holomorphic Legendrian curves and applications

Abstract

In this paper, we find a holomorphic Darboux chart around any immersed noncompact holomorphic Legendrian curve in a complex contact manifold $(X,\xi )$. By using such a chart, we show that every holomorphic Legendrian immersion $R\to X$ from an open Riemann surface can be approximated on relatively compact subsets by holomorphic Legendrian embeddings, and every holomorphic Legendrian immersion $M\to X$ from a compact bordered Riemann surface is a uniform limit of topological embeddings $M\hookrightarrow X$ such that $\mathring{M}\hookrightarrow X$ is a complete holomorphic Legendrian embedding. We also establish a contact neighborhood theorem for isotropic Stein submanifolds, and we find a holomorphic Darboux chart around any contractible isotropic Stein submanifolds in an arbitrary complex contact manifold.

Internat. Math. Res. Not. (IMRN), to appear

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• Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves
• The parametric h-principle for minimal surfaces in ${\mathbb{R}}^n$ and null curves in ${\mathbb{C}}^n$
• Hyperbolic complex contact structures on ${\mathbb{C}}^{2n+1}$
• New complex analytic methods in the study of non-orientable minimal surfaces in ${\mathbb{R}}^n$
• The Oka principle for holomorphic Legendrian curves in ${\mathbb{C}}^{2n+1}$

Cited by in corpus (8)

• Recent developments on Oka manifolds
• New complex analytic methods in the theory of minimal surfaces: a survey
• The Calabi-Yau property of superminimal surfaces in self-dual Einstein four-manifolds
• Mergelyan approximation theorem for holomorphic Legendrian curves
• Holomorphic Legendrian curves in ${\mathbb{CP}}^3$ and superminimal surfaces in ${\mathbb{S}}^4$
• H-principle for complex contact structures on Stein manifolds
• Holomorphic Distributions and Connectivity by Integral Curves of Distributions
• Proper superminimal surfaces of given conformal types in the hyperbolic four-space