Schwarz Reflection Principle, Boundary Regularity and Compactness for $J$-Complex Curves

Ivashkovich, Sergey; Sukhov, Alexandre

Annales de l'Institut Fourier, Tome 60 (2010) no. 4, pp. 1489-1513.

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 ${𝒞}^{k,\alpha }$.

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 ${𝒞}^{k,\alpha }$-classes.

[1] Alexander, H. Continuing $1$-dimensional Analytic Sets., Math. Ann., Volume 191 (1971), pp. 143-144 | DOI | MR | Zbl

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

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

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

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

[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., Volume 254 (2006), pp. 567-589 | DOI | MR | Zbl

[8] Ivashkovich, S.; Rosay, J.-P. Schwarz-type lemmas for solutions of $\overline{\partial }$-inequalities and complete hyperbolicity of almost complex manifolds, Annales Inst. Fourier, Volume 54 (2004), pp. 2387-2435 | DOI | Numdam | MR | Zbl

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

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

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

[12] McDuff, D.; Salamon, D. $J$-holomorphic curves and symplectic topology, AMS Colloquium Publ., 52, AMS, Providence, RI, 2004 | MR | Zbl

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

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

[15] Schwarz, H. A. Über einige Abbildungsaufgaben, Journal für reine und angewandte Mathematik, Volume 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) | DOI

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

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

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

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