Arnold's seminar, December 1, 1998

B.Kruglikov
"Entire pseudoholomorphic curves in the tamed manifolds and Kobayashi
pseudodistance".

Gromov's theory of pseudoholomorphic curves resulted in many other attempts
to describe moduli spaces of prescribed curves in almost complex manifolds.
Usually the almost complex structure is assumed to be tamed by some
symplectic form. A particular case of this approach is the investigation of
pseudoholomorphic 2-tori in a tamed $2n$-dimensional torus. Such torus
$T^2$ can be thinked as $Z^2$-covered entire line $C$ in the torus T^{2n}$.
$Sometimes we can talk just about unfolded tori, i.e. entire
pseudoholomorphic lines. The existence of such lines in $T^{2n}$ was proved
recently by V.Bangert. We present a sketch of another proof connected with
the notion of Kobayashi pseudodistance. Such an object (called the Kobayashi
distance in the case of hyperbolic domains) was generalized to almost
complex case by M.Overholt and the author (B.K.). It turns that the
Bangert's result follows from the Brody-type result for the pseudodistance.
We discuss the behaviour of pseudoholomorphic curves and explain the theory
of pseudoholomorphic jets.