论文标题
庞加莱定理的公制版本，涉及域的生物形态不等式
A metric version of Poincaré's theorem concerning biholomorphic inequivalence of domains
论文作者
论文摘要
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 $j=1,\ldots,p$,然后，产品域$ \ prod_ {j = 1}^p x_j \ subset \ mathbb {c}^m $不能等距地嵌入$ \ prod_ {j = 1}^q y_j \ subset \ subset \ subset \ mathbb {c}^n $下的kobayashi decorts of kobayashi descorts of $ p> q $ p> q $。该结果概括了Poincaré的定理，该定理说，$ \ Mathbb {c}^n $ in Polydisc没有Biholomormormormormormormormormorthic Map to $ n \ geq 2 $。 证明的方法仅依赖于空间的度量几何形状，并将源自带有SUP-Metric的适当地测量公制空间的产品的结果。实际上，本文的主要目的是就单个度量空间的某些渐近几何特性建立一般标准，这会导致构图在产品度量空间之间存在等轴测嵌入。
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.
