J -holomorphic discs and real analytic hypersurfaces

William Alexandre[1]; Emmanuel Mazzilli[1]

  • [1] Laboratoire Paul Painlevé U.M.R. CNRS 8524 U.F.R. de Mathématiques cité scientifique Université Lille 1 F59 655 Villeneuve d’Ascq Cedex, France.

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 5, page 2223-2250
  • ISSN: 0373-0956

Abstract

top
We give in ℝ 6 a real analytic almost complex structure J , a real analytic hypersurface M and a vector v in the Levi null set at 0 of M , such that there is no germ of J -holomorphic disc γ included in M with γ ( 0 ) = 0 and ∂ γ ∂ x ( 0 ) = v , although the Levi form of M has constant rank. Then for any hypersurface M and any complex structure J , we give sufficient conditions under which there exists such a germ of disc.

How to cite

top

Alexandre, William, and Mazzilli, Emmanuel. "$J$-holomorphic discs and real analytic hypersurfaces." Annales de l’institut Fourier 64.5 (2014): 2223-2250. <http://eudml.org/doc/275494>.

@article{Alexandre2014,
abstract = {We give in $\mathbb\{R\}^6$ a real analytic almost complex structure $J$, a real analytic hypersurface $M$ and a vector $v$ in the Levi null set at $0$ of $M$, such that there is no germ of $J$-holomorphic disc $\gamma $ included in $M$ with $\gamma (0)=0$ and $\frac\{\partial \gamma \}\{\partial x\}(0)=v$, although the Levi form of $M$ has constant rank. Then for any hypersurface $M$ and any complex structure $J$, we give sufficient conditions under which there exists such a germ of disc.},
affiliation = {Laboratoire Paul Painlevé U.M.R. CNRS 8524 U.F.R. de Mathématiques cité scientifique Université Lille 1 F59 655 Villeneuve d’Ascq Cedex, France.; Laboratoire Paul Painlevé U.M.R. CNRS 8524 U.F.R. de Mathématiques cité scientifique Université Lille 1 F59 655 Villeneuve d’Ascq Cedex, France.},
author = {Alexandre, William, Mazzilli, Emmanuel},
journal = {Annales de l’institut Fourier},
keywords = {almost complex structure; $J$-holomorphic disc; hypersurface; -holomorphic disc; Levi form},
language = {eng},
number = {5},
pages = {2223-2250},
publisher = {Association des Annales de l’institut Fourier},
title = {$J$-holomorphic discs and real analytic hypersurfaces},
url = {http://eudml.org/doc/275494},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Alexandre, William
AU - Mazzilli, Emmanuel
TI - $J$-holomorphic discs and real analytic hypersurfaces
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 5
SP - 2223
EP - 2250
AB - We give in $\mathbb{R}^6$ a real analytic almost complex structure $J$, a real analytic hypersurface $M$ and a vector $v$ in the Levi null set at $0$ of $M$, such that there is no germ of $J$-holomorphic disc $\gamma $ included in $M$ with $\gamma (0)=0$ and $\frac{\partial \gamma }{\partial x}(0)=v$, although the Levi form of $M$ has constant rank. Then for any hypersurface $M$ and any complex structure $J$, we give sufficient conditions under which there exists such a germ of disc.
LA - eng
KW - almost complex structure; $J$-holomorphic disc; hypersurface; -holomorphic disc; Levi form
UR - http://eudml.org/doc/275494
ER -

References

top
  1. J.-F. Barraud, E. Mazzilli, Regular type of real hyper-surface in (almost) complex manifolds, Math. Z. 248 (2004), 757-772 Zbl1082.32017MR2103540
  2. M. Freeman, Local Complex foliation of Real Submanifolds, Math. Ann. 209 (1974), 1-30 Zbl0267.32006MR346185
  3. S. Ivashkovich, J.-P. Rosay, Schwarz-type lemmas for solutions of ∂ ¯ -inequalities and complete hyperbolicity of almost complex manifolds, Ann. Inst. Fourier (Grenoble) 54 (2004), 2387-2435 Zbl1072.32007MR2139698
  4. S. Ivashkovich, A. Sukhov, Schwarz reflection principle, boundary regularity and compactness for J -complex curves, Ann. Inst. Fourier (Grenoble) 60 (2010), 1489-1513 Zbl1208.32026MR2722249
  5. N. Kruzhilin, A. Sukhov, Pseudoholomorphic discs attached to CR-submanifolds of almost complex spaces, Bull. Sci. Math. 129 (2005), 398-414 Zbl1073.32021MR2146158
  6. L. H. Loomis, An introduction to abstract harmonic analysis, (1953), D. Van Nostrand Company, Inc., Toronto-New York-London Zbl0052.11701MR54173
  7. T. Nagano, Linear differential systems with singularities and an application to transitive Lie algebras, J. Math. Soc. Japan 18 (1966), 398-404 Zbl0147.23502MR199865
  8. J.-C. Sikorav, Some properties of holomorphic curves in almost complex manifolds, Holomorphic curves in symplectic geometry 117 (1994), 165-189, Birkhauser, Basel MR1274929

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.