Stationary disks and Green functions in almost complex domains
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 12 (2013) no. 4, p. 975-1000

Using generalized Riemann maps, normal forms for almost complex domains (D,J) with singular foliations by stationary disks are defined. Such normal forms are used to construct counterexamples and to determine intrinsic conditions under which the stationary disks are extremal disks for the Kobayashi metric or determine solutions to almost complex Monge-Ampère equation.

Published online : 2019-02-21
Classification:  32Q60,  32Q65,  32U35,  32G05
@article{ASNSP_2013_5_12_4_975_0,
     author = {Patrizio, Giorgio and Spiro, Andrea},
     title = {Stationary disks and Green functions  in almost complex domains},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 12},
     number = {4},
     year = {2013},
     pages = {975-1000},
     zbl = {1295.32040},
     mrnumber = {3184576},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2013_5_12_4_975_0}
}
Patrizio, Giorgio; Spiro, Andrea. Stationary disks and Green functions  in almost complex domains. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 12 (2013) no. 4, pp. 975-1000. http://www.numdam.org/item/ASNSP_2013_5_12_4_975_0/

[1] E. Bedford, Survey of pluri-potential theory, In: “Several Complex Variables (Stockholm, 1987/1988)”, Math. Notes 38, Princeton Univ. Press, Princeton, NJ, 1993. | MR 1207855 | Zbl 0786.31001

[2] A. Besse, “Einstein Manifolds”, Springer, 1987. | MR 867684 | Zbl 1147.53001

[3] F. Bracci and G. Patrizio, Monge-Ampère foliations with singularities at the boundary of strongly convex domains, Math. Ann. 332 (2005), 499–522. | MR 2181760 | Zbl 1086.32028

[4] B. Coupet, H. Gaussier and A. Sukhov, Riemann maps in almost complex manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), 761–785. | Numdam | MR 2040642 | Zbl 1170.32310

[5] B. Coupet, H. Gaussier and A. Sukhov, Fefferman’s mapping theorem on almost complex manifolds in complex dimension two, Math. Z. 250 (2005), 59–90. | MR 2136668 | Zbl 1076.32028

[6] B. Coupet, H. Gaussier and A. Sukhov, Some aspects of analysis on almost complex manifolds with boundary, J. Math. Sci. (N.Y.) 154 (2008), 923–986. | MR 2731964

[7] J.-P.  Demailly,  “Complex  Analytic  and  Differential  Geometry”, 1997, posted on http://www-fourier. ujf-grenoble.fr/demailly/books.html.

[8] K. Diederich and A. Sukhov, Plurisubharmonic exhaustion functions and almost complex Stein Structures, Michigan Math. J. 56 (2008), 331–355. | MR 2492398 | Zbl 1161.32012

[9] H. Gaussier and A.-C. Joo, Extremal discs in almost complex spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 (2010), 759–783. | Numdam | MR 2789474 | Zbl 1209.32003

[10] H. Gaussier and A. Sukhov, On the geometry of model almost complex manifolds with boundary, Math. Z. 254 (2006), 567–589. | MR 2244367 | Zbl 1107.32009

[11] L. Lempert, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France 109 (1981), 427–474. | Numdam | MR 660145 | Zbl 0492.32025

[12] L. Lempert, Intrinsic distances and holomorphic retracts, In: “Complex Analysis and Applications ’81 (Varna, 1981)”, Publ. House Bulgar. Acad. Sci., Sofia, 1984, 341–364. | MR 883254 | Zbl 0583.32060

[13] S. Mikhlin and S. Prosdorf, “Singular Integral Operators”, Springer-Verlag, Berlin, 1986. | MR 881386 | Zbl 0612.47024

[14] M.-Y. Pang, Smoothness of the Kobayashi metric of non-convex domains, Internat. J. Math. 4 (1993), 953–987. | MR 1250257 | Zbl 0795.32008

[15] G. Patrizio, Parabolic exhaustions for strictly convex domains, Manuscripta Math. 47 (1984), 271–309. | MR 744324 | Zbl 0581.32018

[16] G. Patrizio, A characterization of complex manifolds biholomorphic to a circular domain, Math. Z. 189 (1985), 343–363. | MR 783561 | Zbl 0591.32030

[17] G. Patrizio, Disques extrémaux de Kobayashi et équation de Monge-Ampère complexe, C. R. Acad. Sci. Paris, Sér. I Math. 305 (1987), 721–724. | MR 920051 | Zbl 0629.32021

[18] G. Patrizio and A. Spiro, Monge-Ampère equations and moduli spaces of manifolds of circular type, Adv. Math. 223 (2010), 174–197. | MR 2563214 | Zbl 1202.32012

[19] G. Patrizio and A. Spiro, Foliations by stationary disks of almost complex domains, Bull. Sci. Math. 134 (2010), 215–234. | MR 2607931 | Zbl 1204.32017

[20] A. Spiro and A. Sukhov, An existence theorem for stationary disks in almost complex manifolds, J. Math. Anal. Appl. 327 (2007), 269–286. | MR 2277411 | Zbl 1126.32021

[21] A. Spiro, Total reality of conormal bundles of hypersurfaces in almost complex manifolds, Int. J. Geom. Methods Mod. Phys. 3 (2006), 1255–1262. | MR 2264414 | Zbl 1112.53057

[22] A. Tumanov, Extremal disks and the regularity of CR mappings in higher codimension, Amer. J. Math. 123 (2001), 445–473. | MR 1833148 | Zbl 0995.32024

[23] W. Wendland, “Elliptic Systems in the Plane”, Pitman Publ., 1979. | MR 518816 | Zbl 0396.35001

[24] K. Yano and S. Ishihara, “Tangent and Cotangent Bundles: Differential Geometry”, Pure and Applied Mathematics, Vol. 16, Marcel Dekker Inc., New York, 1973. | MR 350650 | Zbl 0262.53024