Involutive structures on compact manifolds. (English) Zbl 0826.32014
A vector bundle \(V\) is called involutive if \([V,V] \subset V\). This paper discusses questions about involutive structure \(V\) on compact manifolds \(M\), in particular, questions related to the realizability or embeddability. An important result of this sort is due to L. Boutet de Monvel [‘Intégration des équations de Cauchy-Riemann induites formelles’, Semin. Goulaouic-Lion-Schwartz 1974-1975 (1975; Zbl 0317.58003)], which states that a compact strictly pseudoconvex CR manifold (of hypersurface type) of dimension at least five can be embedded in \(\mathbb{C}^n\) for \(N\) sufficiently large. The principal result of this paper can be regarded as a generalization of the above result.
From the assumption that \(V\) is involutive, the authors construct local basis for \(V\), which enables them to introduce the Levi form \(\mathcal L\), and then the notion of elliptic structures. One main object of study is the generic nonelliptic case, or structures of hypersurface type. So let \(\Sigma\) be the set of points where the structure is not elliptic. If \(\mathcal L\) is nondegenerate, \(\Sigma\) is a smooth submanifold. In the special case where \(V\) is a CR structure, \(\Sigma\) is the whole space. A corollary from their main theorem states the following. Suppose that \(V\), of rank at least two, is of hypersurface type on \(M\), and is strictly pseudoconvex at each point of \(\Sigma\). Then at each \(p \in \Sigma\), there exists a smooth function \(z\), defined on all of \(M\), with \(dz(p) \neq 0\) such that \(Lz = 0\) for any local section \(L\) of \(V\). This result turns out to take care of the embedding of \(M\) at points of \(\Sigma\), but not at those points away from \(\Sigma\), i.e. points that \(V\) is elliptic. Some further discussions along this line, including global embedding results, are also done in some detail. For the proof the main idea follows that of Boutet de Monvel, via a differential complex on \(V\) and the associated Laplacian \(L_V\). The proof is finally accomplished by the study of “Hodge theory” for \(L_V\) on compact manifold \(M\).
From the assumption that \(V\) is involutive, the authors construct local basis for \(V\), which enables them to introduce the Levi form \(\mathcal L\), and then the notion of elliptic structures. One main object of study is the generic nonelliptic case, or structures of hypersurface type. So let \(\Sigma\) be the set of points where the structure is not elliptic. If \(\mathcal L\) is nondegenerate, \(\Sigma\) is a smooth submanifold. In the special case where \(V\) is a CR structure, \(\Sigma\) is the whole space. A corollary from their main theorem states the following. Suppose that \(V\), of rank at least two, is of hypersurface type on \(M\), and is strictly pseudoconvex at each point of \(\Sigma\). Then at each \(p \in \Sigma\), there exists a smooth function \(z\), defined on all of \(M\), with \(dz(p) \neq 0\) such that \(Lz = 0\) for any local section \(L\) of \(V\). This result turns out to take care of the embedding of \(M\) at points of \(\Sigma\), but not at those points away from \(\Sigma\), i.e. points that \(V\) is elliptic. Some further discussions along this line, including global embedding results, are also done in some detail. For the proof the main idea follows that of Boutet de Monvel, via a differential complex on \(V\) and the associated Laplacian \(L_V\). The proof is finally accomplished by the study of “Hodge theory” for \(L_V\) on compact manifold \(M\).
Reviewer: I-Hsun Tsai (Taipei)
MSC:
| 32V05 | CR structures, CR operators, and generalizations |
