The Poincaré lemma and local embeddability

Judith Brinkschulte; C. Denson Hill; Mauro Nacinovich

Bollettino dell'Unione Matematica Italiana (2003)

  • Volume: 6-B, Issue: 2, page 393-398
  • ISSN: 0392-4041

Abstract

top
For pseudocomplex abstract C ⁢ R manifolds, the validity of the Poincaré Lemma for 0 , 1 forms implies local embeddability in C N . The two properties are equivalent for hypersurfaces of real dimension ≥ 5 . As a corollary we obtain a criterion for the non validity of the Poicaré Lemma for 0 , 1 forms for a large class of abstract C ⁢ R manifolds of C ⁢ R codimension larger than one.

How to cite

top

Brinkschulte, Judith, Hill, C. Denson, and Nacinovich, Mauro. "The Poincaré lemma and local embeddability." Bollettino dell'Unione Matematica Italiana 6-B.2 (2003): 393-398. <http://eudml.org/doc/195395>.

@article{Brinkschulte2003,
abstract = {For pseudocomplex abstract $CR$ manifolds, the validity of the Poincaré Lemma for $(0,1)$ forms implies local embeddability in $\mathbb\{C\}^\{N\}$. The two properties are equivalent for hypersurfaces of real dimension $\geq 5$. As a corollary we obtain a criterion for the non validity of the Poicaré Lemma for $(0,1)$ forms for a large class of abstract $CR$ manifolds of $CR$ codimension larger than one.},
author = {Brinkschulte, Judith, Hill, C. Denson, Nacinovich, Mauro},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {393-398},
publisher = {Unione Matematica Italiana},
title = {The Poincaré lemma and local embeddability},
url = {http://eudml.org/doc/195395},
volume = {6-B},
year = {2003},
}

TY - JOUR
AU - Brinkschulte, Judith
AU - Hill, C. Denson
AU - Nacinovich, Mauro
TI - The Poincaré lemma and local embeddability
JO - Bollettino dell'Unione Matematica Italiana
DA - 2003/6//
PB - Unione Matematica Italiana
VL - 6-B
IS - 2
SP - 393
EP - 398
AB - For pseudocomplex abstract $CR$ manifolds, the validity of the Poincaré Lemma for $(0,1)$ forms implies local embeddability in $\mathbb{C}^{N}$. The two properties are equivalent for hypersurfaces of real dimension $\geq 5$. As a corollary we obtain a criterion for the non validity of the Poicaré Lemma for $(0,1)$ forms for a large class of abstract $CR$ manifolds of $CR$ codimension larger than one.
LA - eng
UR - http://eudml.org/doc/195395
ER -

References

top
  1. AKAHORI, T., A new approach to the local embedding theorem of C ⁢ R -structures for n ≥ 4 , Mem. Amer. Math. Soc., 366 (1987), Amer. Math. Soc. Providence R.I. Zbl0628.32025MR888499
  2. ANDREOTTI, A.- FREDRICKS, G.- NACINOVICH, M., On the absence of Poincaré lemma in tangential Cauchy-Riemann complexes, Ann. Sc. Norm. Sup. Pisa, 8 (1981), 365-404. Zbl0482.35061MR634855
  3. ANDREOTTI, A.- HILL, C. D., E.E. Levi convexity and the Hans Lewy problem I, II, Ann. Sc. norm. sup. Pisa, 26 (1972), 325-363, 747-806. Zbl0283.32013
  4. BOUTET DE MONVEL, L., Intégration des équations de Cauchy-Riemann induites formelles, Sem. Goulaouic-Lions-Schwartz (1974-1975). Zbl0317.58003MR409893
  5. HILL, C. D.- NACINOVICH, M., A weak pseudoconcavity condition for abstract almost C ⁢ R manifolds, Invent. Math., 142 (2000), 251-283. Zbl0973.32018MR1794063
  6. HILL, C. D.- NACINOVICH, M., On the failure of the Poincaré lemma for the ∂ ¯ M - complex, Quaderni sez. Geometria Dip. Matematica Pisa, 1.260.1329 (2001), 1-10. 
  7. HÖRMANDER, L., Linear Partial Differential Operators, Springer, Berlin (1963). MR404822
  8. JACOBOWITZ, H.- TREVES, F., Aberrant C ⁢ R structures, Hokkaido Math. Jour., 12 (1983), 276-292. Zbl0564.32010MR719968
  9. KURANISHI, M., Strongly pseudoconvex CR structures over small balls I-III, Ann. of Math., 115, 116 (1982), 451-500, 1-64, 249-330. Zbl0576.32033MR662117
  10. NACINOVICH, M., On the absence of Poincaré lemma for some systems of partial differential equations, Compos. Math., 44 (1981), 241-303. Zbl0487.58026MR662464
  11. NIRENBERG, L., On a problem of Hans Lewy, Uspeki Math. Naut, 292 (1974), 241-251. Zbl0305.35017MR492752
  12. ROSSI, H., LeBrun's nonrealizability theory in higher dimensions, Duke Math. J., 52 (1985), 457-474. Zbl0573.32018MR792182
  13. WEBSTER, S., On the proof of Kuranishi's embedding theorem, Ann. Inst. H. Poincaré, 9 (1989), 183-207. Zbl0679.32020MR995504

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.