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Lie group structures on automorphism groups of real-analytic CR manifolds. (English) Zbl 1165.32017
The authors find very general conditions for the group of CR automorphisms of a real-analytic CR manifold to be a Lie group. Their main result is the following. Let $$M$$ be a real-analytic CR manifold with a finite number of connected components. Assume that $$M$$ is minimal everywhere and finitely nondegenerate in the complement of a compact subset $$K\subset{M}$$, where $$M$$ is essentially finite. Then $$\operatorname{Aut}_{\text{CR}}(M)$$ is a Lie group with the compact-open $$\mathcal{C}^{\omega}$$-topology, and the action is real-analytic.
We recall that $$M$$ is essentially finite at $$p$$ if the $$k$$-th Segre map at $$p$$ is a finite to one map near $$p$$ for some $$k$$. This result applies in particular to compact real-analytic real hypersurfaces of Stein manifolds of dimension at least two, and also to compact real-analytic submanifolds of Stein manifolds that are only assumed to be minimal at all points. The special case where $$K=\emptyset$$ had been previously considered by M. S. Baouendi, L. P. Rothschild, J. Winkelmann and D. Zaitsev [Ann. Inst. Fourier 54, No. 5, 1279–1303 (2004; Zbl 1062.22046)].

##### MSC:
 32V40 Real submanifolds in complex manifolds
##### Keywords:
CR manifold; CR automorphism
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