The d-bar-Neumann operator and the Kobayashi metric [electronic resource] /
We study the d-bar-Neumann operator and the Kobayashi metric. We observe that under certain conditions, a higher-dimensional domain fibered over [omega] can inherit noncompactness of the d-bar-Neumann operator from the base domain [omega]. 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.
School:Texas A&M International University
School Location:USA - Texas
Source Type:Master's Thesis
Keywords:major mathematics d bar problem compact neumann operator kobayashi metric
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