Holomorphic Legendrian curves in convex domains

Andrej Svetina
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Abstract

We prove several results on approximation and interpolation of holomorphic Legendrian curves in convex domains in $\mathbb{C}^{2n+1}$, $n \geq 2$, with the standard contact structure. Namely, we show that such a curve, defined on a compact bordered Riemann surface $M$, whose image lies in the interior of a convex domain $\mathscr{D} \subset \mathbb{C}^{2n+1}$, may be approximated uniformly on compacts in the interior $\mathrm{Int} \, M$ by holomorphic Legendrian curves $\mathrm{Int} \, M \to \mathscr{D}$ such that the approximants are proper, complete, agree with the starting curve on a given finite set in $\mathrm{Int} \, M$ to a given finite order, and hit a specified diverging discrete set in the convex domain. We first show approximation of this kind on bounded strongly convex domains and then generalise it to arbitrary convex domains. As a consequence we show that any bordered Riemann surface properly embeds into a convex domain as a complete holomorphic Legendrian curve under a suitable geometric condition on the boundary of the codomain.
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凸域中的全态 Legendrian 曲线
我们证明了在 $\mathbb{C}^{2n+1}$, $n \geq 2$ 的凸域中具有标准接触结构的全形黎曼曲线的逼近和插值的几个结果。也就是说,我们证明了这样一条曲线,它定义在一个紧凑的有边黎曼曲面 $M$上,其图像位于一个凸域 $\mathscr{D} 的内部。\子集$mathbb{C}^{2n+1}$上的曲线可以在内部$mathrm{Int}, M$的紧凑体上通过全角近似得到\M$ 可以在内部的紧凑的 $\mathrm{Int}\M 到 Mathscr{D}$ 这样的近似值是合适的 完整的 与$mathrm{Int}中给定无限集上的起始曲线一致的\M$ 中给定有限阶的起始曲线一致,并击中凸域中指定的发散离散集。我们首先展示了在有界强凸域上的这种近似,然后将其推广到任意凸域。因此,我们证明了在凸域边界上的适当几何条件下,任何有界黎曼曲面都可以作为一条完整的全形黎曼曲线嵌入凸域。
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