New examples of non-locally embeddable CR structures (with no non-constant CR distributions)
Annales de l'Institut Fourier, Volume 39 (1989) no. 3, p. 811-823

We construct examples of non-locally embeddable CR structures. These examples may show some improvement on previous examples by Nirenberg, and Jacobowitz and Trèves. They are based on a simple construction which consists in gluing two embedded structures. And (this is our main point) we believe that these examples are very transparent, therefore easy to work with.

De nouveaux exemples de structures CR non réalisables sont donnés. Ils sont basés sur une construction simple qui consiste à recoller deux structures plongées. Ces exemples semblent améliorer en partie des exemples anciens de Nirenberg, et Jacobowitz et Trèves, mais l’avantage principal en est peut-être le caractère transparent, qui en rend l’étude facile.

@article{AIF_1989__39_3_811_0,
     author = {Rosay, Jean-Pierre},
     title = {New examples of non-locally embeddable $CR$ structures (with no non-constant $CR$ distributions)},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {39},
     number = {3},
     year = {1989},
     pages = {811-823},
     doi = {10.5802/aif.1189},
     mrnumber = {1030851},
     zbl = {0674.32008},
     mrnumber = {91f:32020},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1989__39_3_811_0}
}
Rosay, Jean-Pierre. New examples of non-locally embeddable $CR$ structures (with no non-constant $CR$ distributions). Annales de l'Institut Fourier, Volume 39 (1989) no. 3, pp. 811-823. doi : 10.5802/aif.1189. http://www.numdam.org/item/AIF_1989__39_3_811_0/

[1] T. Akahori, A new approach to the local embedding theorem for CR structures for n ≥ 4, Memoirs of the AMS no. 336, Providence, RI 1987. | Zbl 0628.32025

[2] D. Hill, What is the notion of a complex manifold with boundary, Prospect in Algebraic Analysis [M. Saito 60th birthday vol.].

[3] H. Jacobowitz, The canonical bundle and realizable CR hypersurfaces, Pacific J. Math., 127 (1987), 91-101. | MR 88e:32027 | Zbl 0583.32050

[4] H. Jacobowitz, F. Trèves, Nonrealizable CR structures, Inventions Math., 66 (1982), 321-249. | Zbl 0487.32015

[5] M. Kuranishi, Strongly pseudoconvex CR structures over small balls, Ann. of Math., I 115 (1982), 451-500, II 116 (1982), 1-64, III 116 (1982), 249-330. | MR 84h:32023a | Zbl 0576.32033

[6] G. Lupacciolu A theorem on holomorphic extension for CR functions, Pacific J. Math., 124 (1986), 177-191. | MR 87k:32026 | Zbl 0597.32014

[7] L. Nirenberg, Lectures on linear partial differential equations, Conference Board of Math. Sc., Regional Conference Series in mathematics No. 17, AMS, 1973. | MR 56 #9048 | Zbl 0267.35001

[8] L. Nirenberg, On a question of Hans Lewy, Russian Math. Surveys, 29, (1974), 251-262. | MR 58 #11823 | Zbl 0305.35017

[9] J.P. Rosay, E.L. Stout, Rado's theorem for CR functions, to appear in Proc. AMS. | Zbl 0674.32007

[10] M.C. Shaw, Hypoellipticity of a system of complex vector fiels, Duke Math. J., 50 no. 3 (1983), 713-728. | MR 85e:35028 | Zbl 0542.35021

[11] F. Trèves, Approximation and representation of functions and distributions annhilated by a system of complex vector fields, Ecole polytechnique (1981). | Zbl 0515.58030

[12] F. Trèves, Introduction to pseudodifferential and Fourier Integral operators, Plenum (1980). | Zbl 0453.47027

[13] S. Webster, On the proof of Kuranishi's embedding theorem, (preprint). | Numdam | Zbl 0679.32020

[14] D. Catlin, A Newlander Nirenberg theorem for manifolds with boundary, Mich. Math. J., 35 (1988), 233-240. | MR 89j:32026 | Zbl 0679.53029