\(G\)-equivariant embedding theorems for CR manifolds of high codimension. (English) Zbl 1510.32082
A theorem of L. Boutet de Monvel [Sémin. Goulaouic-Lions-Schwartz 1974-1975, Équat. dériv. part. lin. non- lin., Exposé IX, 13 p. (1975; Zbl 0317.58003)] asserts that compact strongly pseudoconvex CR manfifolds of codimension one and dimension greater than or equal to five can be embedded by CR maps into the complex Euclidean space.
Now, let \((X,T^{1,0}X)\) be a \((d+1)\)-codimensional CR manifolds of dimension \(2n+1+d\), \(n\geq 2\), and \(G\) a \(d\)-dimensional Lie groups acting locally free on \(X\) such that \[ TX \otimes \mathbb{C}=T^{(1,0)}X\oplus T^{(0,1)}X \oplus \mathbb{C}\,T \oplus \mathbb{C}\,\mathfrak{g}_X \] where \(T\) be a globally defined vector field on \(X\), \(\mathfrak{g}\) is the Lie algebra of \(G\) and \(\mathfrak{g}_X\) is the corresponding algebra of fundamental vector fields. In this paper, the authors prove that, if \(X\) is strongly pseudoconvex in the direction of \(T\), then there exists a \(G\)-equivariant CR embedding of \(X\) into \(\mathbb{C}^N\) for some \(N\in \mathbb{N}\). The embedding map is defined by smooth CR functions which are \(G\)-finite, which means they are finite sums of functions lying in the isotypes induced by the group representation. The proof makes use of the microlocal properties of the Szegő kernel.
Now, let \((X,T^{1,0}X)\) be a \((d+1)\)-codimensional CR manifolds of dimension \(2n+1+d\), \(n\geq 2\), and \(G\) a \(d\)-dimensional Lie groups acting locally free on \(X\) such that \[ TX \otimes \mathbb{C}=T^{(1,0)}X\oplus T^{(0,1)}X \oplus \mathbb{C}\,T \oplus \mathbb{C}\,\mathfrak{g}_X \] where \(T\) be a globally defined vector field on \(X\), \(\mathfrak{g}\) is the Lie algebra of \(G\) and \(\mathfrak{g}_X\) is the corresponding algebra of fundamental vector fields. In this paper, the authors prove that, if \(X\) is strongly pseudoconvex in the direction of \(T\), then there exists a \(G\)-equivariant CR embedding of \(X\) into \(\mathbb{C}^N\) for some \(N\in \mathbb{N}\). The embedding map is defined by smooth CR functions which are \(G\)-finite, which means they are finite sums of functions lying in the isotypes induced by the group representation. The proof makes use of the microlocal properties of the Szegő kernel.
Reviewer: Andrea Galasso (Taipei)
MSC:
| 32V05 | CR structures, CR operators, and generalizations |
| 32V20 | Analysis on CR manifolds |
| 32V30 | Embeddings of CR manifolds |
Citations:
Zbl 0317.58003References:
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