Mathematical Sciences: Applications of the d-bar Neumann    Problem to the Study of Weakly Pseudoconvex Domains
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{'SignBlockName': 'John V. Ryff', 'PO_EMAI': None, 'PO_PHON': None}
The central theme of this work is geometric problems in                    several complex variables.  Three questions will be addressed in                the current project, the first being the Kuranishi problem.  Here               one is interested in real manifolds which have structure similar                to boundaries of domains in complex space.  At issue is whether                 or not smooth pseudoconvex C-R manifolds admit embeddings.  The                 analytic side of this question is one of solving a complex                      partial differential equation with mixed boundary conditions.  If               the research proceeds as predicted, the embedding result should                 be expressible in terms of the first two eigenvectors of the Levi               form for the manifold (they must be positive).                                       Another line of work of a similar nature concerns the                      fundamental first order d-bar-Neumann problem on pseudoconvex                   domains.  One is interested in finding solutions of the                         inhomogeneous d-bar equation on a domain which are orthogonal to                the space of holomorphic functions on that domain.  The question                of global regularity of the solutions depends not only on the                   geometry of the domain but also on the metric (which determines                 the volume element.).  Work will be done in establishing                        conditions for global regularity on smoothly bounded pseudoconvex               domains in terms of curvature inequalities which must be                        satisfied by the metric.                                                             The third element of this research concerns automorphisms or               self-maps of bounded domains.  Recent work of Bell and Pinchuk                  has essentially classified all pseudoconvex domains in two-                     dimensional complex space with noncompact automorphism group (the               domains must also be of finite type).  A number of related                      questions remain open which will be analyzed during this                        undertaking.
{'FirstName': 'David', 'LastName': 'Catlin', 'PI_MID_INIT': None, 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'David Catlin', 'EmailAddress': 'catlin@math.purdue.edu', 'NSF_ID': '000207152', 'StartDate': '06/01/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
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{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}
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