在複n維歐氏空間中有關凸域之不變度量與測度

Abstract

本文中我們將證明Kobayashi擬度量在凸域中的三角不等式成立，任一Cⁿ中不包含複仿射線之凸域皆可解析嵌入n維單位多重圓板，在凸域中的Carathéodory距離函數產生原來的拓樸以及在凸域中的hyperbolicity和measure hyperbolicity是等價的概念，進而推論到任一體積有限的凸域必須是hyperbolic，因此，當然是measure hyperbolic。

In this thesis , we prove that the triangle inequality of the Kobayashi pseudometric holds in any convex domain. Also , for a convex domain Q containing no complex affine line , we prove that Ω is biholomorphic to a subdomain of the unit polydisc Dⁿ and the topology induced by the Carathéodory distance function coincides with the Euclidean topology of Ω. Finally , we prove that hyperbolicity and measure hyperbolicity in a convex domain are equivalent. Moreover, any convex domain with finite Euclidean volume must be hyperbolic, therefore , it is measure hyperbolic.

摘要\r\nAbstract\r\nContent\r\n§0 Introduction-----1\r\n§1 The Poincáre-Bergman Metric in the Unit Disc-----3\r\n§2 The Kobayashi Pseudodistance and Pseudometric-----6\r\n§3 The Carathéodory Pseudodistance and Pseudometric-----11\r\n§4 An Imbedding of Convex Domain into Unit Polydisc-----14\r\n§5 The Topology Induced by the Carathéodory Distance Function-----20\r\n§6 Measure Hyperbolicity and Convexity-----23