Poincaré关于区域双全纯不等价定理的度量版本
A metric version of Poincaré's theorem concerning biholomorphic inequivalence of domains
AI总结：
本文推广了Poincaré定理，证明在Kobayashi距离下，当因子数更多时，凸有界域乘积无法等距嵌入强凸域乘积，并基于度量几何建立了一般性等距嵌入障碍准则。
摘要：
我们证明，若对于 $j=1,\dots,q$，$Y_j\subset \mathbb{C}^{n_j}$ 是具有 $C^3$-边界的强凸有界域，且对于 $j=1,\ldots,p$，$X_j\subset \mathbb{C}^{m_j}$ 是凸有界域，则当 $p>q$ 时，在 Kobayashi 距离下，乘积域 $\prod_{j=1}^p X_j\subset \mathbb{C}^m$ 不能等距嵌入到 $\prod_{j=1}^q Y_j\subset \mathbb{C}^n$ 中。该结果推广了 Poincaré 定理，即当 $n\geq 2$ 时，不存在从多圆柱到 $\mathbb{C}^n$ 中欧氏球的双全纯映射。证明方法仅依赖于空间的度量几何，并将从具有 sup-度量的真测地度量空间乘积的一个结果中推导得出。事实上，本文的主要目标是建立一个一般性准则，基于各个度量空间的某些渐近几何性质，给出乘积度量空间之间存在等距嵌入的障碍。
英文摘要：
We show that if $Y_j\subset \mathbb{C}^{n_j}$ is a bounded strongly convex domain with $C^3$-boundary for $j=1,\dots,q$, and $X_j\subset \mathbb{C}^{m_j}$ is a bounded convex domain for $j=1,\ldots,p$, then the product domain $\prod_{j=1}^p X_j\subset \mathbb{C}^m$ cannot be isometrically embedded into $\prod_{j=1}^q Y_j\subset \mathbb{C}^n$ under the Kobayashi distance, if $p>q$. This result generalises Poincaré's theorem which says that there is no biholomorphic map from the polydisc onto the Euclidean ball in $\mathbb{C}^n$ for $n\geq 2$. The method of proof only relies on the metric geometry of the spaces and will be derived from a result for products of proper geodesic metric spaces with the sup-metric. In fact, the main goal of the paper is to establish a general criterion, in terms of certain asymptotic geometric properties of the individual metric spaces, that yields an obstruction for the existence of an isometric embedding between product metric spaces.
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