Uniqueness in Rough Almost Complex Structures, and Differential Inequalities
[Unicité dans les structures presque complexes non-lisses et inégalités différentielles]
Annales de l'Institut Fourier, Tome 60 (2010) no. 6, pp. 2261-2273.

L’étude des applications $J$-holomorphes conduit à l’étude des inéquations $|\frac{\partial u}{\partial \overline{z}}|\le C|u|$, et $|\frac{\partial u}{\partial \overline{z}}|\le ϵ|\frac{\partial u}{\partial z}|$. La première inéquation est facile à utiliser. La seconde, qui intervient naturellement dans les structures non lisses, est plus difficile. De façon intéressante, le cas d’applications vectorielles $u$ est différent du cas scalaire. Les questions étudiées ont trait à l’unicité de prolongement et aux zéros isolés. Parmi les résultats, il est démontré que, pour les structures presque complexes de classe Hölderienne $\frac{1}{2}$, toute courbe $J$-holomorphe constante sur un ouvert non vide, est constante. Ceci est en contraste avec des exemples immédiats de non-unicité.

The study of $J$-holomorphic maps leads to the consideration of the inequations $|\frac{\partial u}{\partial \overline{z}}|\le C|u|$, and $|\frac{\partial u}{\partial \overline{z}}|\le ϵ|\frac{\partial u}{\partial z}|$. The first inequation is fairly easy to use. The second one, that is relevant to the case of rough structures, is more delicate. The case of $u$ vector valued is strikingly different from the scalar valued case. Unique continuation and isolated zeroes are the main topics under study. One of the results is that, in almost complex structures of Hölder class $\frac{1}{2}$, any $J$-holomorphic curve that is constant on a non-empty open set, is constant. This is in contrast with immediate examples of non-uniqueness.

DOI : https://doi.org/10.5802/aif.2583
Classification : 32Q65,  35R45,  35A02
Mots clés : courbes $J$-holomophes, inégalités différentielles, unicité
@article{AIF_2010__60_6_2261_0,
author = {Rosay, Jean-Pierre},
title = {Uniqueness in {Rough}  {Almost} {Complex} {Structures,} and {Differential} {Inequalities}},
journal = {Annales de l'Institut Fourier},
pages = {2261--2273},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {60},
number = {6},
year = {2010},
doi = {10.5802/aif.2583},
mrnumber = {2791657},
zbl = {1211.32017},
language = {en},
url = {http://www.numdam.org/articles/10.5802/aif.2583/}
}
TY  - JOUR
AU  - Rosay, Jean-Pierre
TI  - Uniqueness in Rough  Almost Complex Structures, and Differential Inequalities
JO  - Annales de l'Institut Fourier
PY  - 2010
DA  - 2010///
SP  - 2261
EP  - 2273
VL  - 60
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2583/
UR  - https://www.ams.org/mathscinet-getitem?mr=2791657
UR  - https://zbmath.org/?q=an%3A1211.32017
UR  - https://doi.org/10.5802/aif.2583
DO  - 10.5802/aif.2583
LA  - en
ID  - AIF_2010__60_6_2261_0
ER  - 
Rosay, Jean-Pierre. Uniqueness in Rough  Almost Complex Structures, and Differential Inequalities. Annales de l'Institut Fourier, Tome 60 (2010) no. 6, pp. 2261-2273. doi : 10.5802/aif.2583. http://www.numdam.org/articles/10.5802/aif.2583/

[1] Alinhac, S. Non-unicité pour des opérateurs différentiels à caractéristiques complexes simples, Ann. Sci. École Norm. Sup. (4), Volume 13 (1980) no. 3, pp. 385-393 | Numdam | MR 597745 | Zbl 0456.35002

[2] Bojarski, B. V. Generalized solutions of a system of differential equations of the first order and elliptic type with discontinuous coefficients, Report. University of Jyväskylä Department of Mathematics and Statistics, 118, University of Jyväskylä, Jyväskylä, 2009 (Translated from the 1957 Russian original, With a foreword by Eero Saksman) | MR 2488720 | Zbl 1173.35403

[3] Floer, Andreas; Hofer, Helmut; Salamon, Dietmar Transversality in elliptic Morse theory for the symplectic action, Duke Math. J., Volume 80 (1995) no. 1, pp. 251-292 | Article | MR 1360618 | Zbl 0846.58025

[4] Gong, Xianghong; Rosay, Jean-Pierre Differential inequalities of continuous functions and removing singularities of Rado type for $J$-holomorphic maps, Math. Scand., Volume 101 (2007) no. 2, pp. 293-319 | MR 2379291 | Zbl 1159.32008

[5] Hörmander, Lars The analysis of linear partial differential operators. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 256, Springer-Verlag, Berlin, 1990 (Distribution theory and Fourier analysis) | MR 1065993

[6] Hörmander, Lars The analysis of linear partial differential operators. III, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 274, Springer-Verlag, Berlin, 1994 (Pseudo-differential operators, Corrected reprint of the 1985 original) | MR 1313500 | Zbl 0601.35001

[7] Hörmander, Lars Notions of convexity, Progress in Mathematics, 127, Birkhäuser Boston Inc., Boston, MA, 1994 | MR 1301332 | Zbl 0835.32001

[8] Ivashkovich, Sergey; Rosay, Jean-Pierre Boundary values and boundary uniqueness of $J$-holomorphic mappings (arXiv:0902.4800)

[9] Ivashkovich, Sergey; Rosay, Jean-Pierre Schwarz-type lemmas for solutions of $\overline{\partial }$-inequalities and complete hyperbolicity of almost complex manifolds, Ann. Inst. Fourier (Grenoble), Volume 54 (2004) no. 7, p. 2387-2435 (2005) | Article | Numdam | MR 2139698 | Zbl 1072.32007

[10] Ivashkovich, Sergey; Shevchishin, V. Local properties of $J$-complex curves in Lipschitz-continuous structures (arXiv:0707.0771)

[11] Malgrange, Bernard Lectures on the theory of several complex variables. Notes by Raghavan Narasimhan, Tata Institute of Fundamental Research, Bombay, 1958

[12] McDuff, Dusa; Salamon, Dietmar $J$-holomorphic curves and quantum cohomology, University Lecture Series, 6, American Mathematical Society, Providence, RI, 1994 | MR 1286255 | Zbl 0809.53002

[13] Rosay, Jean-Pierre Notes on the Diederich-Sukhov-Tumanov normalization for almost complex structures, Collect. Math., Volume 60 (2009) no. 1, pp. 43-62 | Article | MR 2490749 | Zbl 1175.32016

[14] Sikorav, Jean-Claude Some properties of holomorphic curves in almost complex manifolds, Holomorphic curves in symplectic geometry (Progr. Math.), Volume 117, Birkhäuser, Basel, 1994, pp. 165-189 | MR 1274929

Cité par Sources :