Elsevier

Advances in Mathematics

Volume 308, 21 February 2017, Pages 438-482
Advances in Mathematics

Characterizing domains by the limit set of their automorphism group

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Abstract

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.

Keywords

Convex domains
Kobayashi metric
Holomorphic maps
Wolff–Denjoy theorem
Automorphism group

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