论文标题
复杂域上的gehring-hayman类型定理
The Gehring-Hayman type theorems on complex domains
论文作者
论文摘要
在本文中，我们为某些复杂域建立了gehring-hayman型定理。假设$ω\ subset \ mathbb {c}^n $是带有dini-smooth边界的界限$ m $ -convex域，或一个有限制的具有$ c^2 $ -smooth边界的界面pseudoconvex域。然后，我们证明了$ω$中的Kobayashi Geodesic $ [x，y] $的欧几里得长度小于$ C_1 | x-y |^{C_2} $。此外，如果符合Kobayashi度量的$ω$是Gromov双曲线，那么对于Bergman Metric，CarathéodoryMetric或Kähler-Einstein Metric，我们可以将此结果推广到准地理。 作为应用，我们证明了欧几里得边界与格罗莫夫边界之间的比尔德尔等效性。此外，通过使用此边界对应关系，我们可以显示一些针对生物形态的扩展结果，以及相对于域之间的Kobayashi指标，更通用的粗糙准静态。
In this paper we establish Gehring-Hayman type theorems for some complex domains. Suppose that $Ω\subset \mathbb{C}^n$ is a bounded $m$-convex domain with Dini-smooth boundary, or a bounded strongly pseudoconvex domain with $C^2$-smooth boundary. Then we prove that the Euclidean length of Kobayashi geodesic $[x,y]$ in $Ω$ is less than $c_1|x-y|^{c_2}$. Furthermore, if $Ω$ endowed with the Kobayashi metric is Gromov hyperbolic, then we can generalize this result to quasi-geodesics with respect to Bergman metric, Carathéodory metric or Kähler-Einstein metric. As applications, we prove the bi-Hölder equivalence between the Euclidean boundary and the Gromov boundary. Moreover, by using this boundary correspondence, we can show some extension results for biholomorphisms, and more general rough quasi-isometries with respect to the Kobayashi metrics between the domains.
