The d-bar-Neumann operator and the Kobayashi metric
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Date
2004-09-30
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Texas A&M University
Abstract
We study the ∂-Neumann
operator and the Kobayashi metric. We observe that under certain
conditions, a higher-dimensional domain fibered over Ω can
inherit noncompactness of the d-bar-Neumann
operator from the base domain Ω. Thus we have a domain
which has noncompact d-bar-Neumann operator but
does not necessarily have the standard conditions which usually
are satisfied with noncompact d-bar-Neumann operator.
We define the property K which is related to the Kobayashi metric and gives
information about holomorphic structure of fat subdomains. We
find an equivalence between compactness of the d-bar-Neumann operator and the property K in any convex domain.
We also find a local property of the Kobayashi metric [Theorem IV.1], in
which the domain is not necessary pseudoconvex.
We find a more
general condition than finite type for the local regularity of the
d-bar-Neumann operator with the vector-field
method. By this generalization, it is possible for an analytic
disk to be on the part of boundary where we have local
regularity of the d-bar-Neumann operator. By Theorem V.2, we show that an isolated infinite-type point in the
boundary of the domain is not an obstruction for the local
regularity of the d-bar-Neumann operator.
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Keywords
d-bar problem, Kobayashi Metric, compact Neumann operator