A relative index on the space of 3‐dimensional embeddable CR‐structures of finite type

Abstract

AbstractFor a small deformation Φ of a 3‐dimensional pseudoconvex embeddable CR‐structure of finite type, we prove that Φ defines an embeddable CR‐structure if and only if the Szegö projection from ker $ ^{\rm \Phi} {\bar \partial} _{b} $ to ker $ {\bar \partial} _{b} $ is a Fredholm operator. This defines a relative index which is in this case equal to the negative of number of small eigenvalues of the operator $ ^{\rm \Phi} \bar \partial _{b}^{*} \, ^{\rm \Phi} \bar \partial _{b}^{} $. We also prove a cocycle formula for the relative index. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)