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The boundary of a domain with noncompact automorphism group

Title
The boundary of a domain with noncompact automorphism group
Authors
송민주
Date Issued
2011
Publisher
포항공과대학교
Abstract
We study domains with noncompact automorphism group and their boundary. In the line of research, especially related to the theorem of Bedford-Pinchuk, it is meaningful to try weakening the global real analyticity assumption. For this purpose, we consider the Condition (BR) which means that the Bergman representative coordinate system is well-defined near the boundary. We prove that a domain $\Omega$ with smooth boundary in $\mathbb{C}^2$ which satisfies Condition (BR) is biholomorphic to the Thullen domain $\{ (z,w) \in \mathbb{C}^2 \mid
z
^{2m} +
w
^2 < 1 \}$ if it admits an automorphism orbit accumulating at a boundary point of finite D'Angelo type $2m$. An important step of the proof is to show the smooth extension of a certain totally geodesic disc into the given domain. In equi-dimension case, the smooth extension of a biholomorphic mapping has been obtained by C. Fefferman, S. Webster, S. Bell- E. Ligocka, etc. On the other hand, we consider a holomorphic tangent vector field vanishing at a boundary point to establish that the automorphism orbit accumulating point is of finite D'Angelo type. We investigate the relation between the D'Angelo type of the boundary and the holomorphic tangent vector field. As an answer to this question, we classify the holomorphic tangent vector fields vanishing at the boundary point of infinite D'Angelo type.
URI
http://postech.dcollection.net/jsp/common/DcLoOrgPer.jsp?sItemId=000000900566
http://oasis.postech.ac.kr/handle/2014.oak/1094
Article Type
Thesis
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