Extremal discs of domains in almost complex spaces.

The construction of holomorphic discs attached to the boundary
of a domain is an important problem in both complex and symplectic
geometry. In this talk, we will consider the Kobayashi extremal discs in
the almost complex case.
L.Lempert introduced the notion of stationary maps to describe extremal
discs for strongly convex domains in the complex Euclidean space.
A.Tumanov succeeded to convert the definition of stationary maps into an
intrinsic one, which is still valid for almost complex spaces. These can
be used to prove the existence of foliation by stationay maps on almost
complex domains in case the almost complex structure is sufficiently close
to the standard
one.
I will present a result on a variational formula of the Kobayashi
extremal problem which implies that the stationary condition is a natural
necessary condition for extremal discs even in almost complex cases.