THE CLASS O(AD) OF RIEMANNIAN 2-SPACES.
CALIFORNIA UNIV LOS ANGELES DEPT OF MATHEMATICS
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This dissertation is primarily concerned with the development of tests to determine whether a given 2-dimensional Riemannian space M belongs to the class O sub AD to be specified, and the construction of an important class of covering 2-spaces with this property. An investigation is first made of the modulus function h and of a related multiple-valued function h, the conjugate of h. It is shown in particular that every branch h of h is harmonic and that the level lines of h and h form a system of coordinates which are orthogonal except on a set of measure zero. Let ADM be the class of harmonic functions on M with a finite Dirichlet integral on M and with vanishing flux across every 1-cycle. If the class ADM consists of constants then M belongs to the class O sub AD. A modular criterion is established to determine whether a given M belongs to O sub AD. This modular test is used to develop a second test in terms of a conformal metric. Moreover, a relation between the modulus and extremal length is established and used to determine a third test which uses regular chains. Finally Abelian type covering spaces of a compact 2-dimensional Riemannian space are constructed and, using the regular chain test, shown to belong to the degenerate class O sub AD. Author
- Theoretical Mathematics