Schwarz Reflection Principle, Boundary Regularity and Compactness for J-Complex Curves
[Principe de symétrie de Schwarz, régularité au bord et compacité pour les courbes J-complexes]
Ivashkovich, Sergey ; Sukhov, Alexandre
Annales de l'Institut Fourier, Tome 60 (2010), p. 1489-1513 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

Nous établissons le Principe de Symétrie de Schwarz pour les disques complexes attachés à une sous-variété analytique réelle et totalement réelle d’une variété presque complexe munie d’une structure presque complexe analytique réelle. Nous prouvons également la régularité au bord précise de ces disques et nous en déduisons la convergence exacte dans le théorème de compacité de Gromov dans les classes 𝒞 k,α .

We establish the Schwarz Reflection Principle for J-complex discs attached to a real analytic J-totally real submanifold of an almost complex manifold with real analytic J. We also prove the precise boundary regularity and derive the precise convergence in Gromov compactness theorem in 𝒞 k,α -classes.

Publié le : 2010-01-01
DOI : https://doi.org/10.5802/aif.2562
Classification:  32Q65,  32H40
Mots clés: structure presque complexe, variété totalement réelle, disque analytique, principe de symétrie 
@article{AIF_2010__60_4_1489_0,
     author = {Ivashkovich, Sergey and Sukhov, Alexandre},
     title = {Schwarz Reflection Principle, Boundary Regularity and Compactness for $J$-Complex Curves},
     journal = {Annales de l'Institut Fourier},
     volume = {60},
     year = {2010},
     pages = {1489-1513},
     doi = {10.5802/aif.2562},
     zbl = {1208.32026},
     mrnumber = {2722249},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2010__60_4_1489_0}
}

Ivashkovich, Sergey; Sukhov, Alexandre. Schwarz Reflection Principle, Boundary Regularity and Compactness for $J$-Complex Curves. Annales de l'Institut Fourier, Tome 60 (2010) pp. 1489-1513. doi : 10.5802/aif.2562. http://gdmltest.u-ga.fr/item/AIF_2010__60_4_1489_0/

[1] Alexander, H. Continuing 1-dimensional Analytic Sets., Math. Ann., Tome 191 (1971), p. 143-144 | Article | MR 283236 | Zbl 0211.10204

[2] Bers, L.; John, F.; Schechter, M. Partial differential equations, J. Wiley and Sons (1964) | MR 162045 | Zbl 0126.00207

[3] Carathéodory, C. Zum Schwarzschen Spiegelungsprinzip, Comm. Math. Helv., Tome 19 (1946) no. 1, pp. 263-278 | Article | MR 20144

[4] Chirka, E. Regularity of boundaries of analytic sets, Math. USSR Sbornik, Tome 43 (1983), pp. 291-335 | Article | MR 648411 | Zbl 0525.32005

[5] Coupet, B.; Gaussier, H.; Sukhov, A. Fefferman’s mapping theorem on almost complex manifold in complex dimension two, Math. Z., Tome 250 (2005), pp. 59-90 | Article | MR 2136668 | Zbl 1076.32028

[6] Fromm, V. Gromov Compactness in Hölder Spaces and Minimal Connections on Jet Bundles, math. SG/0808.0415

[7] Gaussier, H.; Sukhov, A. On the geometry of model almost complex manifolds with boundary, Math. Z., Tome 254 (2006), pp. 567-589 | Article | MR 2244367 | Zbl 1107.32009

[8] Ivashkovich, S.; Rosay, J.-P. Schwarz-type lemmas for solutions of ∂ ¯-inequalities and complete hyperbolicity of almost complex manifolds, Annales Inst. Fourier, Tome 54 (2004), pp. 2387-2435 | Article | Numdam | MR 2139698 | Zbl 1072.32007

[9] Ivashkovich, S.; Shevchishin, V. Gromov Compactness Theorem for J-Complex Curves with Boundary, Int. Math. Res. Notices, Tome 22 (2000), pp. 1167-1206 | Article | MR 1807156 | Zbl 0994.53010

[10] Ivashkovich, S.; Shevchishin, V. Reflection Principle and J-Complex Curves with Boundary on Totally Real Immersions, Communications in Contemporary Mathematics, Tome 4 (2002), pp. 65-106 | Article | MR 1890078 | Zbl 1025.32024

[11] Lempert, L.; Szöke, R. The tangent bundle of an almost complex manifold, Canad. Math. Bull., Tome 44 (2001), pp. 70-79 | Article | MR 1816050 | Zbl 0984.53029

[12] Mcduff, D.; Salamon, D. J -holomorphic curves and symplectic topology, AMS, Providence, RI, AMS Colloquium Publ., Tome 52 (2004) | MR 2045629 | Zbl 1064.53051

[13] Monakhov, V. Boundary-value problems with free boundary for elliptic systems of equations, AMS, Providence, RI, Translations of Mathematical Monographs, Tome 57 (1983) (522 pp. (Originally published by Nauka, Novosibirsk, 1977)) | MR 717387 | Zbl 0532.35001

[14] Morrey, C. Multiple integrals in the calculus of variations, Springer Verlag (1966) | MR 202511 | Zbl 0142.38701

[15] Schwarz, H. A. Über einige Abbildungsaufgaben, Journal für reine und angewandte Mathematik, Tome 70 (1869), p. 105-120 (see pages 106–107) (See also Gesammelte mathematische Abhandlungen, Springer (1892), 66-67. Or the Second Edition, Bronx, N.Y., Chelsea Pub. Co. (1972)) | Article

[16] Sikorav, J.-C.; M. Audin, J. Lafontaine Some properties of holomorphic curves in almost complex manifolds, Holomorphic curves in symplectic geometry, Birkhäuser (Progress in Mathematics) Tome 117 (1994), pp. 165-189 (Ch. V) | MR 1274929

[17] Triebel, H. Theory of Function Spaces, Birkhäuser (1983) | MR 781540

[18] Vekua, I. N. Generalized analytic functions, Fizmatgiz, Moscow (1959) (English translation - Pergamon Press, London, and Addison-Welsey, Reading, Massachusetts (1962)) | MR 108572 | Zbl 0092.29703

[19] Yano, K.; Ishihara, Sh. Tangent and cotangent bundles, Marcel Dekker, NY (1973) | MR 350650 | Zbl 0262.53024