Scalar curvature on noncompact complete Riemannian manifolds

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2011

Author dc:creator

• Kim, Seongtag

Contributors dc:contributor

• Bishop, Richard L.

• Mathematics

Statement dc:rights

• Copyright 1994 Kim, Seongtag

Language dc:language

eng

Identifier

AAI9503236
(UMI)AAI9503236

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/20163