On the realizability of Lewy structures

Meziani, Abdelhamid

doi:10.1090/S0002-9939-96-03592-7

On the realizability of Lewy structures
HTML articles powered by AMS MathViewer

by Abdelhamid Meziani

Proc. Amer. Math. Soc. 124 (1996), 2767-2772

DOI: https://doi.org/10.1090/S0002-9939-96-03592-7

PDF | Request permission

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.

References

Takao Akahori, A new approach to the local embedding theorem of CR-structures for $n\geq 4$ (the local solvability for the operator $\overline \partial _b$ in the abstract sense), Mem. Amer. Math. Soc. 67 (1987), no. 366, xvi+257. MR 888499, DOI 10.1090/memo/0366
Aldo Andreotti and C. Denson Hill, Complex characteristic coordinates and tangential Cauchy-Riemann equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 26 (1972), 299–324. MR 460724
M. S. Baouendi and L. P. Rothschild, Embeddability of abstract CR structures and integrability of related systems, Ann. Inst. Fourier (Grenoble) 37 (1987), no. 3, 131–141 (English, with French summary). MR 916277, DOI 10.5802/aif.1101
Nicholas Hanges, The missing first integral, J. Differential Equations 72 (1988), no. 1, 178–188. MR 929203, DOI 10.1016/0022-0396(88)90154-4
Nicholas Hanges and Howard Jacobowitz, A remark on almost complex structures with boundary, Amer. J. Math. 111 (1989), no. 1, 53–64. MR 980299, DOI 10.2307/2374479
Howard Jacobowitz, The canonical bundle and realizable CR hypersurfaces, Pacific J. Math. 127 (1987), no. 1, 91–101. MR 876018, DOI 10.2140/pjm.1987.127.91
Howard Jacobowitz, An introduction to CR structures, Mathematical Surveys and Monographs, vol. 32, American Mathematical Society, Providence, RI, 1990. MR 1067341, DOI 10.1090/surv/032
Howard Jacobowitz and François Trèves, Nonrealizable CR structures, Invent. Math. 66 (1982), no. 2, 231–249. MR 656622, DOI 10.1007/BF01389393
Howard Jacobowitz and François Trèves, Aberrant CR structures, Hokkaido Math. J. 12 (1983), no. 3, 276–292. MR 719968, DOI 10.14492/hokmj/1470081006
Howard Jacobowitz and François Trèves, Nowhere solvable homogeneous partial differential equations, Bull. Amer. Math. Soc. (N.S.) 8 (1983), no. 3, 467–469. MR 693964, DOI 10.1090/S0273-0979-1983-15123-6
Masatake Kuranishi, Strongly pseudoconvex CR structures over small balls. I. An a priori estimate, Ann. of Math. (2) 115 (1982), no. 3, 451–500. MR 657236, DOI 10.2307/2007010
Abdelhamid Meziani, Perturbation of a class of CR structures of codimension larger than one, J. Funct. Anal. 116 (1993), no. 1, 225–244. MR 1237994, DOI 10.1006/jfan.1993.1112
—, Classification of germs of Mizohata structures, Comm. PDE 20 (3-4) (1995), 499–539.
Louis Nirenberg, Lectures on linear partial differential equations, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 17, American Mathematical Society, Providence, RI, 1973. Expository Lectures from the CBMS Regional Conference held at the Texas Technological University, Lubbock, Tex., May 22–26, 1972. MR 450755, DOI 10.1090/cbms/017
François Trèves, Hypo-analytic structures, Princeton Mathematical Series, vol. 40, Princeton University Press, Princeton, NJ, 1992. Local theory. MR 1200459