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    The d-bar-Neumann operator and the Kobayashi metric

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    etd-tamu-2003B-2003063012-Kim-1.pdf (258.9Kb)
    Date
    2004-09-30
    Author
    Kim, Mijoung
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    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.
    URI
    http://hdl.handle.net/1969.1/94
    Subject
    d-bar problem
    Kobayashi Metric
    compact Neumann operator
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    • Electronic Theses, Dissertations, and Records of Study (2002– )
    Citation
    Kim, Mijoung (2003). The d-bar-Neumann operator and the Kobayashi metric. Doctoral dissertation, Texas A&M University. Texas A&M University. Available electronically from http : / /hdl .handle .net /1969 .1 /94.

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