Math 2.0

Several complex variables and analytic spaces CR manifolds Theorem

Boutet de Monvel's embedding theorem

Builds on CR manifoldLevi formCR function

Statement

Let MM be a compact smooth CR manifold of hypersurface type, of real dimension 2n+12n + 1 with n≥2n \ge 2. Assume that MM is strongly pseudoconvex, that is, its Levi form is definite at every point. Then there is a smooth CR embedding of MM into ℂN\C^N for some NN.

Remarks

The theorem is due to L. Boutet de Monvel. The proof uses Kohn's analysis of the tangential Cauchy–Riemann operator ∂ˉb\bar\partial_b and of the Kohn Laplacian. On a compact strongly pseudoconvex manifold of real dimension at least 55, this analysis gives good solvability of ∂ˉb\bar\partial_b. This produces many global CR functions, and finitely many of them give an embedding. The boundary of a bounded strongly pseudoconvex domain in ℂn+1\C^{n+1} with smooth boundary is an example where the embedding is given from the start. The bound on the dimension is necessary: in real dimension 33 there are compact strongly pseudoconvex CR manifolds with no CR embedding into any ℂN\C^N, such as Rossi's example.

Going deeper

For a student

The result is a global embedding theorem, in contrast with local realisation questions, where Kuranishi and others worked on embedding small pieces. Combined with results on filling, it shows that such MM bound complex spaces with isolated singularities. The three-dimensional case behaves very differently and is an active subject: there embeddability is a delicate property that is not preserved under small deformations.

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  1. CR function concept
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Further reading

  1. Web https://en.wikipedia.org/wiki/CR_manifold
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Kindstatement, theorem
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AboutCR manifold
UsesCR manifold, Levi form, CR function

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