A relative index on the space of embeddable CR-structures, I


We study the problem of embeddability for three dimensional CR-manifolds. Let $(M,^0\overline\partial_b)$ denote a compact, embeddable, strictly pseudoconvex CR-manifold and ${}^0S$ the orthogonal projection on $\ker {}^0\overline\partial_b$. If ${}^1\overline\partial_b$ denotes a deformation of this CR-structure then $^1\overline\partial_b$ is embeddable if and only if
{}^0S\colon\ker {}^1\overline\partial_b \to \ker {}^0\overline\partial_b
is a Fredholm operator. We define the relative index, $\mathrm{Ind}(^0\overline\partial_b,{}^1\overline\partial_b)$, to be the Fredholm index of this operator. This integer is shown to be independent of the volume form used to define $^0S$ 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.


Charles L. Epstein