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On the realizability of Lewy structures

Author: Abdelhamid Meziani
Journal: Proc. Amer. Math. Soc. 124 (1996), 2767-2772
MSC (1991): Primary 32F25; Secondary 53C40
MathSciNet review: 1350956
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Abstract: We prove that a nondegenerate CR structure with signature
$(p,n-p)$ at $0\in \mathbb {R}^{2n+1}$ and with $n$ first integrals $z_{1},\cdots ,z_{n}$ satisfying

\begin{equation*}dz_{1}\wedge d\bar {z}_{1}\wedge \cdots \wedge dz_{n}\wedge d\bar {z}_{n} \ne 0\end{equation*}

is realizable if and only if an action of the group $O(2p,2n-2p)$ leaves invariant a one-dimensional subbundle of the structure bundle.

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  • [Ak] T. Akahori, A new approach to the local embedding of CR structures for $n\ge 4$ (the local solvability of the operator $\overline {\partial }_{b}$ in the abstract sense), vol. 366, Memoirs Amer. Math. Soc., 1987. MR 88i:32027
  • [AH] A. Andreotti and C.D. Hill, Complex characteristic coordinates and tangential Cauchy Riemann equations, Ann. Scuola Norm. Pisa 26 (1972), 299--324. MR 57:717
  • [BR] M.S. Baouendi and L.P. Rothschild, Embeddability of Abstract CR structures and integrability of related systems, Ann. Inst. Fourier Grenoble 37 (1987), 131--141. MR 89c:32053
  • [Ha] N. Hanges, The missing first integral, J. Diff. Equations 72 (1988), 178-188. MR 89j:32016
  • [HJ] N. Hanges and H. Jacobowitz, A remark on almost complex structures with boundary, Amer. J. Math 111 (1989), 53--64. MR 90m:32038
  • [Ja1] H. Jacobowitz, The canonical bundle and realizable of CR hypersurfaces, Pacific J. Math. 127 (1987), 91--101. MR 88e:32027
  • [Ja2] ------, An introduction to CR structures, Math Surveys and Monographs, vol. 32, Amer. Math. Soc, 1990. MR 93h:32023
  • [JT1] H. Jacobowitz and F. Treves, Non-realizable CR structures, Invent. Math. 66 (1982), 231--249. MR 83f:53022
  • [JT2] ------, Aberrant CR structures, Hokkaido Math. J. 12 (1983), 267--292. MR 85c:32033
  • [JT3] ------, Nowhere solvable homogeneous partial differential equations, Bull. Amer. Math. Soc. 8 (1983), 467--469. MR 85b:32031
  • [Ku] M. Kuranishi, Strongly pseudoconvex CR structures over small balls, I-II-III, Ann. of Math. 115 - 116 (1982), 451--500; 1--64; 249--330. MR 84h:32023c
  • [M1] A. Meziani, Perturbation of a class of CR structures of codimension larger than one, J. Funct. Analysis 116 (1993), 225--244. MR 94i:32024
  • [M2] ------, Classification of germs of Mizohata structures, Comm. PDE 20 (3-4) (1995), 499--539. CMP 95:08
  • [Ni] L. Nirenberg, Lectures on linear partial differential equations, Reg. Conf. Series in Math., vol. 17, Amer. Math. Soc., 1973. MR 56:9048
  • [Tr] F. Treves, Hypo-analytic structures: local theory, vol. 40, Princeton University Press, 1992. MR 94e:35014

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Additional Information

Abdelhamid Meziani
Affiliation: Department of Mathematics, Florida International University, Miami, Florida 33199

Received by editor(s): December 8, 1994
Received by editor(s) in revised form: March 10, 1995
Communicated by: Eric Bedford
Article copyright: © Copyright 1996 American Mathematical Society

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