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The boundary of a domain with noncompact automorphism group

The boundary of a domain with noncompact automorphism group
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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 $\{ (z,w) \in \mathbb{C}^2 \mid
^{2m} +
^2 < 1 \}$ 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.
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