论文标题
全态legendrian曲线的仿真近似定理
Mergelyan approximation theorem for holomorphic Legendrian curves
论文作者
论文摘要
在本文中，我们证明了在任意的复杂触点$（x，ξ）$中，用于沉浸式Holomorphic Legendrian曲线的融合类型近似定理。 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 $\mathscr{C}^{r+2}(S,X)$ for some $r\ge 2$ which is holomorphic in the interior of $S$, then $f$ can be approximated in the $ \ mathscr {c}^r（s，x）$ topology by holomorphic legendrian嵌入了$ s $的$ s $ to $ x $。该结果有许多应用，其中一些应用在论文中指示。特别是，通过使用布莱恩特的通信penrose扭曲图$ \ mathbb {cp}^3 \至s^4 $，我们表明，Mergelyan近似定理和Calabi-yau属性在$ 4 $ -SPHERE $ S^4 $中具有超级级别的属性。
In this paper, 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 $\mathscr{C}^{r+2}(S,X)$ for some $r\ge 2$ which is holomorphic in the interior of $S$, then $f$ can be approximated in the $\mathscr{C}^r(S,X)$ 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 $S^4$.
