The Gehring-Hayman type theorems on complex domains

2020/05/06 by Liu, Jinsong, Wang, Hongyu, Zhou, Qingshan
#Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2005.02594

Abstract

In this paper we establish Gehring-Hayman type theorems for some complex domains. Suppose that Ω⊂ ℂⁿ is a bounded m-convex domain with Dini-smooth boundary, or a bounded strongly pseudoconvex domain with C²-smooth boundary. Then we prove that the Euclidean length of Kobayashi geodesic [x,y] in Ω is less than c₁|x-y|^c₂. 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.

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