Characterizing domains by the limit set of their automorphism group

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• 作者：Andrew M. Zimmer ^{aazimmer@uchicago.edu}
• 关键词：Convex domains ; Kobayashi metric ; Holomorphic maps ; Wolff&ndash ; Denjoy theorem ; Automorphism group
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：21 February 2017
• 年：2017
• 卷：308
• 期：Complete
• 页码：438-482
• 全文大小：592 K
• 卷排序：308

文摘

In this paper we study the automorphism group of smoothly bounded convex domains. We show that such a domain is biholomorphic to a “polynomial ellipsoid” (that is, a domain defined by a weighted homogeneous balanced polynomial) if and only if the limit set of the automorphism group intersects at least two closed complex faces of the set. The proof relies on a detailed study of the geometry of the Kobayashi metric and ideas from the theory of non-positively curved metric spaces. We also obtain a number of other results including the Greene–Krantz conjecture in the case of uniform non-tangential convergence, new results about continuous extensions (of biholomorphisms and complex geodesics), and a new Wolff–Denjoy theorem.