C^n上K-双曲区域与开单位球双全纯等价的充要条件

The Necessary and Sufficient Condition for a K-hyperbolic Domain being Biholomorphic to the Unit Ball in C^n

在Krantz证明的定理[1]的基础上，讨论了Cn上K－双曲区域与开单位球双全纯等价的充要条件，并证明了此条件．给出了C－体度量与K－体度量的定义；K－双曲区域的定义，将Krantz的结论推广到K－双曲区域所需的引理即引理1及引理2．

This paper gives the definitions of Caratheodory volume, Kabayashi volume and K-hyperbolic domain. It also gives the lemmas needed in the proof of the main theorem and discusses the problem when Kabayashi metrics equal Caratheodory metrics. By the Schwarz lemma on K-hyperbolic domain in Cn, it proved that a K-hyperbolic domain is biholomorphic to the unit ball Cn if its Caratheodory volume is only equal to its Kabayashi volume.