In this paper we introduce a new class of domains— log-type convex domains, which have no boundary regularity assumptions. Then we will localize the Kobayashi metric in log-type convex subdomains. As an application, we prove a local version of continuous extension of rough isometric maps between two bounded domains with log-type convex Dini-smooth boundary points. Moreover we prove that the Teichmüller space $${\mathcal {T}}_{g,n}$$ T g , n is not biholomorphic to any bounded pseudoconvex domain in $$\mathbb C^{3g-3+n}$$ C 3 g - 3 + n which is locally log-type convex near some boundary point.

中文翻译：

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小林指标的本地化和应用

在本文中，我们介绍了一类新的域——对数型凸域，它没有边界规则假设。然后我们将在对数型凸子域中本地化 Kobayashi 度量。作为一个应用，我们证明了一个局部版本的粗糙等距图在两个有界域之间的连续扩展，具有对数型凸 Dini 平滑边界点。此外，我们证明了 Teichmüller 空间 $${\mathcal {T}}_{g,n}$$ T g , n 对 $$\mathbb C^{3g-3+n} 中的任何有界伪凸域都不是双全纯的$$ C 3 g - 3 + n 在某个边界点附近是局部对数型凸面。