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Abstract
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The classification of bounded domains in
, with
,
is related to the geometric properties of the boundary. A conjecture of
Greene and Krantz relates the geometry of the boundary with the group of
biholomorphic self mappings of the domain. The Greene–Krantz conjecture, if true,
can tell us much about the classification of smoothly bounded domains in
.
Much work has been done to attempt to solve this conjecture, though it has
yet to be proved or disproved. However, there are numerous partial results
which support the conjecture. In this paper, we prove a special case of the
conjecture:
Theorem: Suppose
is a
bounded convex domain with
boundary. Suppose there exists
and
such that for the
sequence of iterates
we have
nontangentially. Then
is of finite type.
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Keywords
several complex variables, Greene Krantz conjecture, finite
type, convex domains, automorphism orbit accumulation
points, iterated automorphisms
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Mathematical Subject Classification 2010
Primary: 32-02
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Milestones
Received: 5 July 2017
Revised: 16 July 2018
Accepted: 10 August 2018
Published: 8 March 2019
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