A relative index on the space of 3‐dimensional embeddable CR‐structures of finite type
A relative index on the space of 3‐dimensional embeddable CR‐structures of finite type
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DOI：
10.1002/mana.200310247
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发表时间：
2005-03
影响因子：
1
通讯作者：
P. Greiner;W. Staubach;Wei Wang
中科院分区：
文献类型：
--
作者：
P. Greiner;W. Staubach;Wei Wang
对于有限类型的三维伪凸可嵌入CR-结构的小变形Φ，我们证明了Φ定义可嵌入CR-结构的充要条件是从Ker${\rm\Phi}{\BAR\PARTIAL}{\BAR\PARTIAL}_{b}$到KER${\BAR\PARTIAL}_{b}$的Szegö投影是Fredholm算子.这定义了一个相对指数，在这种情况下，它等于算子$^{\rm\phi}\bar\Partial_{b}^{*}\bar\Partial_{b}^{}$的小特征值个数的负数。我们还证明了相对指数的余循环公式。(2015Wiley-VCH Verlag GmbH&Co.KGaA，Weinheim)
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)