Lie group structures on automorphism groups of real-analytic CR manifolds. (English) Zbl 1165.32017

Lamel, Bernhard; Mir, Nordine; Zaitsev, Dmitri

Am. J. Math. 130, No. 6, 1709-1726 (2008).

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)].

Reviewer: Mauro Nacinovich (Roma)

MSC:

32V40

Real submanifolds in complex manifolds

Keywords:

CR manifold; CR automorphism