THE CLASS O(AD) OF RIEMANNIAN 2-SPACES.
Abstract
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 AD(M) 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 AD(M) 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)
Document Details
- Document Type
- Technical Report
- Publication Date
- Jun 01, 1966
- Accession Number
- AD0633691
Entities
People
- Norman Lee Breazeal
Organizations
- University of California, Los Angeles