AI 中文总结
该研究开发了适用于斯坦流形并集的全纯龙格嵌入逼近延拓原理，将其应用于多类斯坦流形相关问题，推广了现有嵌入结果并构造了洛普纳偏微分方程的解。
AI 中文摘要
我们针对递增斯坦流形并集开发了一种全纯龙格嵌入的逼近延拓原理，目标是将其嵌入到具有密度性质的斯坦流形中。基本假设是，在每个穷竭阶段都存在一个压缩该阶段的龙格同痕，且其终端映射可全纯延拓至下一阶段。由此得到的并集全局嵌入可选取为具有龙格像，且固定阶段的每个龙格嵌入都可被该并集的龙格嵌入在紧子集上一致逼近。我们将该原理应用于正时间部分全纯(R,+)作用下不变的区域、带有全局吸引不动点的半完全全纯向量场的斯坦流形。它还给出了(ℂⁿ\{z∈ℂⁿ:f(z)=0})×ℂ到ℂⁿ⁺¹的龙格嵌入，推广了此前(ℂ*)ⁿ×ℂ到ℂⁿ⁺¹的龙格嵌入结果。我们还构造了单射全纯半群作用到全纯(R,+)作用的斯坦全局化。最后，当初始区域允许嵌入到具有密度性质的斯坦流形时，赫尔格洛茨向量场的抽象洛普纳范围存在同维龙格嵌入，这给出了取值于ℂⁿ的洛普纳偏微分方程的对应解。我们还给出了一个非龙格完备双曲区域的例子，其允许取值于ℂⁿ的洛普纳偏微分方程解。
英文摘要
We develop an extension-by-approximation principle for holomorphic Runge embeddings of increasing union of Stein manifolds into Stein manifolds with density property. The basic hypothesis is the existence, on each stage of the exhaustion, of a Runge isotopy which compresses the stage and whose terminal map extends holomorphically to the next stage. The resulting global embedding of the union may be chosen with Runge image, and every Runge embedding of a fixed stage can be approximated uniformly on compact subsets by the Runge embeddings of the union. We apply this principle to domains that are invariant under positive time part of holomorphic $(R,+)$-actions, to Stein manifolds carrying a semicomplete holomorphic vector field with globally attracting fixed point. It also gives a Runge embedding of $(\mathbb{C}^n\setminus \{z\in\mathbb{C}^n: f(z)=0\})\times \mathbb{C}$ in $\mathbb{C}^{n+1}$, which generalizes previous result of Runge embedding of $(\mathbb{C}^*)^n\times\mathbb{C}$ into $\mathbb{C}^{n+1}$. We also construct Stein globalization of an injective holomorphic semigroup action to holomorphic $(R,+)$-action. Finally, the abstract Loewner range of a Herglotz vector field is shown to admit a same-dimensional Runge embedding whenever the initial domain admits a Runge embedding into a Stein domain with density property; this yields a corresponding solution of the Loewner PDE with values in $\mathbb{C}^n$. We also give an example of non-Runge complete hyperbolic domain which admits $\mathbb{C}^n$-valued solution of the Loewner PDE.
Comments31 pages; comments are welcome
