Uniqueness in Rough Almost Complex Structures, and Differential Inequalities
Annales de l'Institut Fourier, Volume 60 (2010) no. 6, p. 2261-2273
The study of J-holomorphic maps leads to the consideration of the inequations |u z ¯|C|u|, and |u z ¯|ϵ|u 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 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.
L’étude des applications J-holomorphes conduit à l’étude des inéquations |u z ¯|C|u|, et |u z ¯|ϵ|u 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 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.
DOI : https://doi.org/10.5802/aif.2583
Classification:  32Q65,  35R45,  35A02
Keywords: J-holomorphic curves, differential inequalities, uniqueness
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {60},
     number = {6},
     year = {2010},
     pages = {2261-2273},
     doi = {10.5802/aif.2583},
     zbl = {1211.32017},
     mrnumber = {2791657},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2010__60_6_2261_0}
}
Rosay, Jean-Pierre. Uniqueness in Rough  Almost Complex Structures, and Differential Inequalities. Annales de l'Institut Fourier, Volume 60 (2010) no. 6, pp. 2261-2273. doi : 10.5802/aif.2583. https://aif.centre-mersenne.org/item/AIF_2010__60_6_2261_0/

[1] Alinhac, S. Non-unicité pour des opérateurs différentiels à caractéristiques complexes simples, Ann. Sci. École Norm. Sup. (4), Tome 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, University of Jyväskylä, Jyväskylä, Report. University of Jyväskylä Department of Mathematics and Statistics, Tome 118 (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., Tome 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., Tome 101 (2007) no. 2, pp. 293-319 | MR 2379291 | Zbl 1159.32008

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

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

[7] Hörmander, Lars Notions of convexity, Birkhäuser Boston Inc., Boston, MA, Progress in Mathematics, Tome 127 (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 ¯-inequalities and complete hyperbolicity of almost complex manifolds, Ann. Inst. Fourier (Grenoble), Tome 54 (2004) no. 7, p. 2387-2435 (2005) http://aif.cedram.org/item?id=AIF_2004__54_7_2387_0 | 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, American Mathematical Society, Providence, RI, University Lecture Series, Tome 6 (1994) | MR 1286255 | Zbl 0809.53002

[13] Rosay, Jean-Pierre Notes on the Diederich-Sukhov-Tumanov normalization for almost complex structures, Collect. Math., Tome 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, Birkhäuser, Basel (Progr. Math.) Tome 117 (1994), pp. 165-189 | MR 1274929