Gromov hyperbolicity, the Kobayashi metric, and $\mathbb{C}$-convex sets
Gromov hyperbolicity, the Kobayashi metric, and $\mathbb{C}$-convex sets
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格罗莫夫双曲性、小林度量和 $mathbb{C}$-凸集
DOI：
10.1090/tran/6909
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
Andrew M. Zimmer
中科院分区：
文献类型：
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作者：
Andrew M. Zimmer
本文研究复欧氏空间中整环上小林度量的整体几何。我们特别感兴趣的是发展的必要和充分条件的小林度量格罗莫夫双曲。对于一般域，有人认为边界上的非平凡复仿射圆盘是Gromov双曲性的障碍。这是已知的情况下，该集的问题是凸的。本文首先将这一结果推广到边界为C^1 $-光滑的$\mathbb{C}$-凸集.然后，我们将表明，一些边界的正则性是必要的，通过生产在任何维的例子开界$\mathbb{C}$-凸集的小林度量是Gromov双曲，但其边界包含一个复杂的仿射球的复余维1。
In this paper we study the global geometry of the Kobayashi metric on domains in complex Euclidean space. We are particularly interested in developing necessary and sufficient conditions for the Kobayashi metric to be Gromov hyperbolic. For general domains, it has been suggested that a non-trivial complex affine disk in the boundary is an obstruction to Gromov hyperbolicity. This is known to be the case when the set in question is convex. In this paper we first extend this result to $\mathbb{C}$-convex sets with $C^1$-smooth boundary. We will then show that some boundary regularity is necessary by producing in any dimension examples of open bounded $\mathbb{C}$-convex sets where the Kobayashi metric is Gromov hyperbolic but whose boundary contains a complex affine ball of complex codimension one.