Mathematical Sciences: Differential Geometry and Function   Theory
{'Code': '03040102', 'Directorate': {'Abbreviation': 'MPS', 'LongName': 'Direct For Mathematical & Physical Scien'}, 'Division': {'Abbreviation': 'DMS', 'LongName': 'Division Of Mathematical Sciences'}}
{'SignBlockName': 'John V. Ryff', 'PO_EMAI': None, 'PO_PHON': None}
This project continues mathematical studies on the role of                 hyperbolic geometry in the theyro of complex function theory.                   The goals of the work are to relate hyperbolic geometry to                      Euclidean quantities such as Euclidean estimates of the                         hyperbolic metric and descriptions of Euclidean properties of                   hyperbolic geodesics.  Results of this type have application to                 questions on distortion of univalent functions and estimates for                Bloch and Landau constants.  Geometric notions of symmetry and                  reflection play a central role in this research.                                     Typical of the type of problems to be undertaken is                        establishing the existence and exact values of natural constants                associated with classes of functions.  The Bloch constant, for                  instance, measures the largest schlicht disk in the Riemann image               surface of a function, whereas the Marden constant measures the                 size of the largest disk in the domain where a function must be                 univalent.  These fundamental values have been obtained for some                classes but not for some of the more important univalent                        functions.                                                                           Work also continues in the study of hyperbolic geometry of                 regions where results point to appropriate generalizations to                   multiply connected regions.  Two main goals are pursued here, the               first being one of finding lower bounds for the density of the                  hyperbolic metric in terms of the distance to the boundary.  The                second goal is to obtain estimates for the Euclidean curvature                  and center of curvature for a hyperbolic geodesic.  Much of this                work will concentrate initially on special domains such as those                which are spherically k-convex or hyperbolically k-convex before                moving on to multiply connected ones.
{'FirstName': 'C. David', 'LastName': 'Minda', 'PI_MID_INIT': None, 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'C. David Minda', 'EmailAddress': 'minda@ucbeh.san.uc.edu', 'NSF_ID': '000273633', 'StartDate': '07/01/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
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{'Code': '126100', 'Text': 'CLASSICAL ANALYSIS'}
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