Stably-interior points and the Semicontinuity of the Automorphism group |
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Authors: | Robert E Greene Kang-Tae Kim |
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Institution: | 1. Department of Mathematics, University of California, Los Angeles, CA, 90095, USA 2. Center for Geometry and its Applications and Department of Mathematics, Pohang University of Science and Technology, 790-784, Pohang, Republic of Korea
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Abstract: | We present the new semicontinuity theorem for automorphism groups: If a sequence \(\{\Omega _j\}\) of bounded pseudoconvex domains in \(\mathbb C^2\) converges to \(\Omega _0\) in \({\mathcal C}^\infty \) -topology, where \(\Omega _0\) is a bounded pseudoconvex domain in \(\mathbb C^2\) with its boundary \({\mathcal C}^\infty \) and of the D’Angelo finite type and with \(\text {Aut}\,(\Omega _0)\) compact, then there is an integer \(N>0\) such that, for every \(j > N\) , there exists an injective Lie group homomorphism \(\psi _j:\text {Aut}\,(\Omega _j) \rightarrow \text {Aut}\,(\Omega _0)\) . The method of our proof of this theorem is new that it simplifies the proof of the earlier semicontinuity theorems for bounded strongly pseudoconvex domains. |
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