×

Deformation of big pseudoholomorphic disks and application to the Hanh pseudonorm. (English. Abridged French version) Zbl 1041.32015

In the paper [Funct. Anal. Appl. 33, 38–48 (1999; Zbl 0967.32024)], the author proved the following Theorem: Let \((M,J)\) be a quasi-complex manifold and let \(f_0:(D_R,i)\to (M,J)\) be a pseudo-holomorphic disk satisfying \((f_0)_*(0)e=v_0\not=0\), where \(e=1\) is the unit vector based at \(0\) in the complex plane. Then, for every \(\epsilon>0\), there exists a neighborhood \({\mathcal V}_{\varepsilon}(v_0)\) of the vector \(v_0\in TM\) such that for each \(v\in {\mathcal V}_{\varepsilon}(v_0)\) one can find a pseudo-holomorphic disk \(f:(D_{R-\varepsilon},i)\to (M,J)\) satisfying \((f_0)_*(0)e=v_0\). If \(f_0\) is an embedding (respectively an immersion), then \(f\) can also be chosen to be an embedding (respectively an immersion).
In the paper under review, the author gives a simpler proof of that theorem, and he applies this result to show that the Kobayashi and Hahn pseudostances coincide in dimensions greater than four. The result is new even in the case where \(M\) is a complex manifold.

MSC:

32Q65 Pseudoholomorphic curves
32F45 Invariant metrics and pseudodistances in several complex variables
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
32Q60 Almost complex manifolds

Citations:

Zbl 0967.32024
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Jarnicki, W., Kobayashi – royden vs. hanh pseudometric in \(C\^{}\{2\}\), Ann. polon. math., 75, 3, 289-294, (2000) · Zbl 0982.32012
[2] Kruglikov, B.S., Existence of close pseudoholomorphic disks for almost complex manifolds and application to the kobayashi – royden pseudonorm, Funct. anal. appl., 33, 1, 46-58, (1999) · Zbl 0967.32024
[3] B.S. Kruglikov, Tangent and normal bundles in almost complex geometry, Preprint Univ. Tromsø, 2002-46
[4] Kruglikov, B.S.; Overholt, M., The Kobayashi pseudodistance on almost complex manifolds, Differential geom. appl., 11, 265-277, (1999) · Zbl 0954.32019
[5] McDuff, D., Singularities of J-holomorphic curves in almost complex 4-manifolds, J. geom. anal., 2, 3, 249-266, (1992) · Zbl 0758.53019
[6] Moser, J., Pseudo-holomorphic curves on a torus, Proc. roy. irish acad. sect. A, suppl., 95, 13-21, (1995) · Zbl 1280.32015
[7] Nijenhuis, A.; Woolf, W., Some integration problems in almost-complex and complex manifolds, Ann. math., 77, 424-489, (1963) · Zbl 0115.16103
[8] Overholt, M., Injective hyperbolicity of domains, Ann. polon. math., 62, 1, 79-82, (1995) · Zbl 0847.32027
[9] Royden, H.L., The extension of regular holomorphic maps, Proc. amer. math. soc., 43, 2, 306-310, (1974) · Zbl 0292.32019
[10] Sikorav, J.-C., Some properties of holomorphic curves in almost complex manifolds, (), 165-189
[11] Vekua, I.N., Generalized analytic functions, (1962), Pergamon London, (in Russian); Engl. translation: · Zbl 0127.03505
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.