Deformation of big pseudoholomorphic disks and application to the Hanh pseudonorm

We simplify 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. We aim here to prove the following statement, which was proved by another (analogous to the approach of [NW]) and more complicated method in [K1]. Theorem 1. Let (M, J) be an almost complex manifold and f0 : (DR, i) → (M,J), (f0)∗(0)e = v0 6= 0, be a pseudoholomorphic disk. Here e = 1 is the unit vector at 0 ∈ C. For every ε > 0 there exists a neighborhood Vε(v0) of the vector v0 ∈ TM such that for each v ∈ Vε there is a bit smaller pseudoholomorphic disk f : (DR−ε, i) → (M,J), f∗(0)e = v. The approximating curve f can be embedded/immersed if such is the curve f0. This theorem was used in [K1] for the proof of equivalence of two definitions of Kobayashi pseudodistance dM in almost complex category. In the second dM is associated via path integration to the Kobayashi-Royden pseudonorm: FM (v) = inf{1/r | f : (Dr, i) → (M,J), f∗(0)e = v}, v ∈ TM. The above theorem assures FM to be upper semicontinous, implying that