13. Holomorphic discs in higher dimensional complex
analysis
|
Polish supervisor:
|
Armen Edigarian
|
|
Cooperating partners:
|
Ragnar Sigurdsson (University of Iceland, Reykjavik)
Jan Wiegerinck (University of Amsterdam)
|
In the beginning of 90s of 20th century Evgeny Poletsky in a
series of papers proposed a method of constructing extremal
plurisubharmonic functions via holomorphic discs. His main ideas
are presented in a long paper "Holomorphic currents". In
particular, Poletsky showed how to use holomorphic discs in the
study of the following questions: the solution of the Dirichlet
problem in the given domain, description of the polynomial hull of
a given compact set, properties of the pluricomplex Green function
and so on. These ideas were used later by Levenberg and Poletsky
in the study of small sets (pluripolar, plurithin, etc).
In the middle of 90s of the last century, several people tried to
complete and develop the ideas of Poletsky (Edigarian, Larusson,
Sigurdsson, etc). In particular, Edigarian gave a complete (and
simplified) proof of the description of the pluricomplex Green
function. Larusson and Sigurdsson showed how to extend the results
for a class of complex manifolds. Recently, Rosay gave a proof
(which was later simplified by Edigarian and extended by Larusson
and Sigurdsson) that the theory holds on any complex manifold.
At the same time, we also worked on applications of the theory.
Edigarian and Wiegerinck at the end of 90s gave a counterexample
to a problem posed by Levenberg, Martin and Poletsky regarding an
extension property of holomorphic functions in the disc. This
example turned out to be a starting point for many studies and
results.
The main aim of the project will be to develop analogous theory in
almost complex spaces, to deepen the understanding of the
relations between Poletsky's theory and Gromov's theory on
holomorphic curves. The student should develop new tools to obtain
important results in geometry of almost complex manifolds and
different type of hulls in complex analysis, which are the main
interest of the project. Possible application of the obtained
results to mathematical models of physics are a long term aim of
the project.
|
|
|
