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BIRS Workshop Lecture Videos

Characterizing domains by their automorphism group Zimmer, Andrew


It is generally believed that (up to biholomorphism) very few domains have a large automorphism group and a nice boundary. For instance the Wong-Rosay Ball theorem says that a strongly pseudoconvex domain with non-compact automorphism group must be bi-holomorphic to the ball. Later, Bedford and Pinchuk proved that a convex domain of finite type and non-compact automorphism group must be bi-holomorphic to a domain defined by a polynomial. I will discuss a recent result which removes the finite type condition from the Bedford-Pinchuk result but at the cost of assuming that the automorphism group is slightly larger than non-compact. In particular, a smoothly bounded convex domain is biholomorphic to a domain defined by a polynomial if and only if an orbit of the automorphism group accumulates on at least two different complex faces of the set. The proof of this result combines rescaling arguments and ideas from the theory of metric spaces of non-positively curvature.

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