Estimates of the Kobayashi-Royden metric in almost complex manifolds
Bulletin de la Société Mathématique de France, Volume 133 (2005) no. 2, p. 259-273

We establish a lower estimate for the Kobayashi-Royden infinitesimal pseudometric on an almost complex manifold (M,J) admitting a bounded strictly plurisubharmonic function. We apply this result to study the boundary behaviour of the metric on a strictly pseudoconvex domain in M and to give a sufficient condition for the complete hyperbolicity of a domain in (M,J).

Nous établissons une estimée inférieure pour la métrique de Kobayashi-Royden sur une variété presque complexe (M,J) admettant une fonction bornée strictement pluri-sous-harmonique. Nous appliquons ce résultat à l’étude du comportement de la métrique au bord d’un domaine strictement pseudoconvexe dans M et donnons une condition suffisante d’hyperbolicité complète d’un domaine dans (M,J).

DOI : https://doi.org/10.24033/bsmf.2486
Classification:  32V40,  32V15,  32H40,  32T15,  53C15
Keywords: almost complex manifolds, Kobayashi-Royden metric, J-holomorphic discs
@article{BSMF_2005__133_2_259_0,
     author = {Gaussier, Herv\'e and Sukhov, Alexandre},
     title = {Estimates of the Kobayashi-Royden metric in almost complex manifolds},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {133},
     number = {2},
     year = {2005},
     pages = {259-273},
     doi = {10.24033/bsmf.2486},
     zbl = {1083.32011},
     mrnumber = {2172267},
     language = {en},
     url = {http://www.numdam.org/item/BSMF_2005__133_2_259_0}
}
Gaussier, Hervé; Sukhov, Alexandre. Estimates of the Kobayashi-Royden metric in almost complex manifolds. Bulletin de la Société Mathématique de France, Volume 133 (2005) no. 2, pp. 259-273. doi : 10.24033/bsmf.2486. http://www.numdam.org/item/BSMF_2005__133_2_259_0/

[1] F. Berteloot - « Attraction des disques analytiques et continuité höldérienne d'applications holomorphes propres », Topics in complex analysis (Warsaw, 1992), Banach Center Publ., vol. 31, Polish Acad. Sci., Warsaw, 1995, p. 91-98. | MR 1341379 | Zbl 0831.32012

[2] -, « Principe de Bloch et estimations de la métrique de Kobayashi des domaines de 2 », J. Geom. Anal. 13 (2003), p. 29-37. | MR 1967034 | Zbl 1040.32011

[3] E. Chirka, B. Coupet & A. Sukhov - « On boundary regularity of analytic discs », Michigan Math. J. 46 (1999), p. 271-279. | MR 1704142 | Zbl 0985.32009

[4] R. Debalme - « Kobayashi hyperbolicity of almost complex manifolds », Preprint IRMA, arXiv: math.CV/9805130, 1999.

[5] R. Debalme & S. Ivashkovich - « Complete hyperbolic neighborhoods in almost-complex surfaces », Internat. J. Math. 12 (2001), p. 211-221. | MR 1823575 | Zbl 1110.32306

[6] K. Diederich & J. Fornæss - « Proper holomorphic maps onto pseudoconvex domains with real-analytic boundary », Ann. of Math. (2) 110 (1979), p. 575-592. | MR 554386 | Zbl 0394.32012

[7] I. Graham - « Boundary behavior of the Carathéodory and Kobayashi metrics on strongly pseudoconvex domains in n with smooth boundary », Trans. Amer. Math. Soc. 207 (1975), p. 219-240. | MR 372252 | Zbl 0305.32011

[8] F. Haggui - « Fonctions FSH sur une variété presque complexe », C.R. Math. Acad. Sci. Paris 335 (2002), p. 509-514. | MR 1936821 | Zbl 1013.32019

[9] S. Ivashkovich & J.-P. Rosay - « Schwarz-type lemmas for solutions of ¯-inequalities and complete hyperbolicity of almost complex manifolds », arXiv: math.CV/0310474, 2003. | Numdam | MR 2139698 | Zbl 1072.32007

[10] N. Kerzman & J.-P. Rosay - « Fonctions plurisousharmoniques d'exhaustion bornées et domaines taut », Math. Ann. 257 (1981), p. 171-184. | MR 634460 | Zbl 0451.32012

[11] S. Kobayashi - « Almost complex manifolds and hyperbolicity », Results Math. 40 (2001), p. 246-256, Dedicated to Shiing-Shen Chern on his 90th birthday. | MR 1860372 | Zbl 1004.53052

[12] B. Kruglikov - « Existence of close pseudoholomorphic disks for almost complex manifolds and their application to the Kobayashi-Royden pseudonorm », Funktsional. Anal. i Prilozhen. 33 (1999), p. 46-58, 96. | MR 1711878 | Zbl 0967.32024

[13] A. Nijenhuis & W. Woolf - « Some integration problems in almost-complex and complex manifolds », Ann. of Math. (2) 77 (1963), p. 424-489. | MR 149505 | Zbl 0115.16103

[14] S. Pinchuk - « The scaling method and holomorphic mappings », Several complex variables and complex geometry, Part 1 (Santa Cruz, CA, 1989), Proc. Sympos. Pure Math., vol. 52, Amer. Math. Soc., Providence, RI, 1991, p. 151-161. | MR 1128522 | Zbl 0744.32013

[15] N. Sibony - « A class of hyperbolic manifolds », Recent developments in several complex variables (Proc. Conf., Princeton Univ., Princeton, N.J., 1979), Ann. of Math. Stud., vol. 100, Princeton Univ. Press, Princeton, N.J., 1981, p. 357-372. | MR 627768 | Zbl 0476.32033

[16] J.-C. Sikorav - « Some properties of holomorphic curves in almost complex manifolds », Holomorphic curves in symplectic geometry, Progr. Math., vol. 117, Birkhäuser, Basel, 1994, p. 165-189. | MR 1274929