Scalar curvature on noncompact complete Riemannian manifolds
Kim, Seongtag
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https://hdl.handle.net/2142/20163
Description
Title
Scalar curvature on noncompact complete Riemannian manifolds
Author(s)
Kim, Seongtag
Issue Date
1994
Doctoral Committee Chair(s)
Bishop, Richard L.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
Let (M,g) be a noncompact complete Riemannian manifold whose scalar curvature S(x) is positive for all x in M. In this thesis, we study a conformal deformation of the given metric g, which makes the scalar curvature of the deformed metric a positive constant. We also study the existence of a complete conformal metric with positive constant scalar curvature. We obtain a sufficient condition for the existence of a conformal metric with positive constant scalar curvature by studying the conformal structure at infinity. To study the existence of a complete solution, we calculate the Sobolev Quotient on a special admissible set whose elements are candidates for complete solutions. By studying the conformal structure at infinity, the behavior of scalar curvature and the Ricci curvatue in radial direction, we also obtain a sufficient condition for the existence of a complete conformal metric with positive constant scalar curvature. This study finds an interesting obstruction for the existence of an injective conformal immersion from the given m-dimensional complete Riemannian manifold to an m-dimensional compact Riemannian manifold.
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