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A relative index on the space of embeddable CR-structures. I. (English) Zbl 0942.32025

Summary: “We study the problem of embeddability of three-dimensional CR-manifolds. Let \((M,{}^0\overline{\partial}_b)\) denote a compact, embeddable, strictly pseudoconvex CR-manifold and \({}^0{\mathcal S}\) the orthogonal projection onto \(\text{ker} {}^0\overline{\partial}_b\). If \({}^1\overline{\partial}_b\) denotes a deformation of this CR-structure, then \({}^1\overline{\partial}_b\) is embeddable if and only if \({}^0{\mathcal S}: \text{ker} {}^1\overline{\partial}_b \to\text{ker} {}^0\overline{\partial}_b\) is a Fredholm operator. We define the relative index \(\text{Ind}({}^0\overline{\partial}_b,{}^1\overline{\partial}_b)\) to be the Fredholm index of this operator. This integer is shown to be independent of the volume form used to define \({}^0{\mathcal S}\) and to be constant along orbits of the group of contact transformations. The relative index therefore defines a stratification of the moduli space of embeddable CR-structures. For small perturbations its value is related to small eigenvalues of the associated \(\square_b\)-operator.”
The proof of the constancy of the index along orbits requires a delicate analysis of how the Szegő projector depends on the perturbation as well as on eigenvalue estimates for the Laplacian \(\square_b\). The stratification reflects the “instability” of the algebra of CR-functions in a functional-analytic way. For example, the author proves the following result: Let \(X\) be a Stein space and suppose that \(M_1\) and \(M_2\) are smooth strictly pseudoconvex hypersurfaces in \(X\) which bound compact domains and are isotopic through a smooth family of strictly pseudoconvex smooth hypersurfaces in \(X\). Then \(\text{Ind}({}^1\overline{\partial}_b,{}^2\overline{\partial}_b)=0\).
The strata are defined as \[ S_n=\{\Phi\in C^\infty(M;\text{Hom}(T^{0,1}M,T^{1,0}M)): \|\Phi\|_{L^\infty}<1\text{ and }\text{Ind}({}^0\overline{\partial}_b,{}^\Phi\overline{\partial}_b)\geq -n\}. \] The spectral analysis of the associated \(\square_b\)-operators and perturbation analysis enable the author to study the topology of the stratification. He proves; For each natural number \(n\) and \(\varepsilon>0\) the intersection of the stratum \(S_n\) with the \(L^\infty\)-ball of radius \(\sqrt{1/2-\varepsilon}\) is closed in the \(C^\infty\) topology.
For Part II, see ibid. 147, No. 1, 61–91 (1998; Zbl 0942.32026). Errata are given in ibid. 154, No. 1, 223–226 (2001; Zbl 0983.32036).
Reviewer: J.S.Joel (Kelly)

MSC:

32V30 Embeddings of CR manifolds
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J20 Index theory and related fixed-point theorems on manifolds
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