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

Zbl 0967.32024
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References:

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