We simplify the proof of the theorem that close to any pseudoholomorphic disk there passes a pseudoholomorphic disk of arbitrary close size with any pre-described sufficiently close direction. We apply these results to the Kobayashi and Hanh pseudodistances. It is shown they coincide in dimensions higher than four. The result is new even in the complex case.
Nous simplifions la preuve du théorème montrant que près de tout disque pseudo-holomorphe il passe un disque pseudo-holomorphe de taille proche quelconque et avec une direction pré-fixée suffisamment proche. Nous appliquons ces résultats aux pseudodistances de Kobayashi et Hanh. Nous montrons qu'elles coı̈ncident en dimensions supérieures à quatre. Le résultat est nouveau, même dans le cas complexe.
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Boris Kruglikov 1
@article{CRMATH_2004__338_4_295_0, author = {Boris Kruglikov}, title = {Deformation of big pseudoholomorphic disks and application to the {Hanh} pseudonorm}, journal = {Comptes Rendus. Math\'ematique}, pages = {295--299}, publisher = {Elsevier}, volume = {338}, number = {4}, year = {2004}, doi = {10.1016/j.crma.2003.12.010}, language = {en}, }
Boris Kruglikov. Deformation of big pseudoholomorphic disks and application to the Hanh pseudonorm. Comptes Rendus. Mathématique, Volume 338 (2004) no. 4, pp. 295-299. doi : 10.1016/j.crma.2003.12.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.12.010/
[1] Kobayashi–Royden vs. Hanh pseudometric in , Ann. Polon. Math., Volume 75 (2000) no. 3, pp. 289-294
[2] Existence of close pseudoholomorphic disks for almost complex manifolds and application to the Kobayashi–Royden pseudonorm, Funct. Anal. Appl., Volume 33 (1999) no. 1, pp. 46-58
[3] B.S. Kruglikov, Tangent and normal bundles in almost complex geometry, Preprint Univ. Tromsø, 2002-46
[4] The Kobayashi pseudodistance on almost complex manifolds, Differential Geom. Appl., Volume 11 (1999), pp. 265-277
[5] Singularities of J-holomorphic curves in almost complex 4-manifolds, J. Geom. Anal., Volume 2 (1992) no. 3, pp. 249-266
[6] Pseudo-holomorphic curves on a torus, Proc. Roy. Irish Acad. Sect. A, Suppl., Volume 95 (1995), pp. 13-21
[7] Some integration problems in almost-complex and complex manifolds, Ann. Math., Volume 77 (1963), pp. 424-489
[8] Injective hyperbolicity of domains, Ann. Polon. Math., Volume 62 (1995) no. 1, pp. 79-82
[9] The extension of regular holomorphic maps, Proc. Amer. Math. Soc., Volume 43 (1974) no. 2, pp. 306-310
[10] Some properties of holomorphic curves in almost complex manifolds (M. Audin; J. Lafontaine, eds.), Holomorphic Curves in Symplectic Geometry, Progr. Math., vol. 117, Birkhäuser, 1994, pp. 165-189
[11] Generalized Analytic Functions, Fiz.-Mat. Literatury, Moscow, 1959 (in Russian); Engl. translation:, 1962, Pergamon, London
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