论文标题
$ \ mathbb {c}^{n} $的某些域上的近似值
Approximations on certain domains of $\mathbb{C}^{n}$
论文作者
论文摘要
在本文中，我们在$ \ mathbb {c}^n $中研究这些域，这些域是在某些全球定义的，完整的全态矢量字段的正面下不变的 在原点上有一个全球吸引的固定点。我们的第一个结果说，这样的域$ω$始终是runge。接下来，有了关于流量收敛速率的额外假设，我们表明 任何Bihololomormormormormism $φ\ colonω\至φ（ω）$，带有$φ（ω）$是runge，可以通过$ \ mathbb {c}^{n} $的自动形态近似近似。即使向量场是线性的，这也将其概括为在此方向上的所有已知定理。为了应用我们的近似结果，在也完全双曲线的此类域上，我们表明，所有loewner pde pde在一个完整的双曲线域中$ω$中的任何loewner pde都接受了一个本质上独特的单价解决方案，具有$ \ mathbb {c}^n $中的值。我们还为在上述域上保留生物形态的体积提供了近似结果。 我们提供了此类域的几个示例。
In this paper, we study the domains in $\mathbb{C}^n$ that are invariant under the positive flows of some globally defined, complete holomorphic vector field with a globally attracting fixed point at the origin. Our first result says that such a domain $Ω$ is always Runge. Next, with an additional assumption on the rate of convergence of the flow, we show that any biholomorphism $Φ\colon Ω\to Φ(Ω)$, with $Φ(Ω)$ is Runge, can be approximated by automorphisms of $\mathbb{C}^{n}$ uniformly on compacts. This generalizes all earlier known theorems in this direction substantially, even when the vector field is linear. As an application of our approximation results, on such domains that are also complete hyperbolic, we show that any Loewner PDE in a complete hyperbolic domain $Ω$ admits an essentially unique univalent solution with values in $\mathbb{C}^n$. We also provide an approximation result for volume preserving biholomorphisms on above domains. We provide several examples of such domains.
