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A relative index on the space of 3‐dimensional embeddable CR‐structures of finite type
Author(s) -
Greiner Peter,
Staubach Wolfgang,
Wang Wei
Publication year - 2005
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200310247
Subject(s) - mathematics , bar (unit) , type (biology) , projection (relational algebra) , space (punctuation) , operator (biology) , eigenvalues and eigenvectors , combinatorics , pure mathematics , deformation (meteorology) , index (typography) , mathematical analysis , physics , chemistry , algorithm , ecology , linguistics , philosophy , biochemistry , repressor , quantum mechanics , meteorology , gene , transcription factor , world wide web , computer science , biology
Abstract For 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)