11. Holomorphically invariant functions in special classes of domains
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Polish supervisor:
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Włodzimierz Zwonek
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Cooperating partners:
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Nikolai Nikolov (Bulgarian Academy of Sciences, Sophia)
Peter Pflug (Carl von Ossietzky Universitat, Oldenburg)
Pascal J. Thomas (Universite Paul Sabatier, Toulouse)
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It is often desirable to concentrate on problems appearing in the
general case on a smaller class of domains. Often such an attitude
may lead to the discovery of phenomena that could help to give an
answer in a more general setting or it would deliver appropriate
(counter)examples. This attitude can be applied to the study of
(bi)holomorphically invariant functions such as Caratheodory,
Kobayashi and Bergman distances which play an important role in
Several Complex Variables (SCV).
The suggested classes of domains that could be considered in the
project are the following: smooth pseudoconvex, balanced and
C-convex domains and more special ones (symmetrized polydisc,
spectral ball). There are many important problems of
holomorphically invariant functions in the above mentioned classes
on which the PhD student may work. We list some of them below.
The Lempert Theorem states that on all bounded convex domains the
holomorphically invariant functions coincide. It is open whether
the result may be extended to the class of bounded C-convex
domains.
It is still not known whether all smooth bounded pseudoconvex
domains are Kobayashi (and even Caratheodory) complete. Recent
results have shown that a more subtle attitude to this problem is
necessary than the one used up to now. Namely we cannot hope to
prove the Kobayashi completeness of smooth pseudoconvex domains
proving at first that the Kobayashi-Royden metric explodes in the
normal direction as one over the distance to the boundary. A more
subtle attitude seems to be necessary. The problem is old and
important because the knowledge of the boundary behaviour of
invariant functions finds applications in other problems of SCV -
for instance, in the study of the regular extension of holomorphic
mappings onto the boundary.
Although it is known that there is a bounded balanced domain of
holomorphy with continuous Minkowski functional which is not
Kobayashi complete it is still an open problem to find a good
sufficient condition for such a domain to be Kobayashi complete.
Moreover, it is not known whether such an example exists in
dimension two (an example is known in the three-dimensional case).
The Bergman distance has been very extensively studied in recent
years. Its behaviour near the boundary is closely related to the
behaviour of the pluricomplex Green function (and to the
pluripotential theory in general). This relation allowed to prove
in nineties one of the most important results in this direction.
Namely, all bounded hyperconvex domains are Bergman complete. But
many problems remain open. For example it would be desirable to
get results on the quantative behaviour of the Bergman distance
(and metric) near the boundary of a domain admitting good defining
functions (or domains with sufficiently smooth boundary).
The problems outlined above are difficult and probably even some
partial results in these directions or solutions of some other
related problems which definitely appear while studying them would
be sufficient for the content of a good PhD Thesis.
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