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Scalar curvature on noncompact complete Riemannian manifolds

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Title: Scalar curvature on noncompact complete Riemannian manifolds
Author(s): Kim, Seongtag
Doctoral Committee Chair(s): Bishop, Richard L.
Department / Program: Mathematics
Discipline: Mathematics
Degree Granting Institution: University of Illinois at Urbana-Champaign
Degree: Ph.D.
Genre: Dissertation
Subject(s): Mathematics
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.
Issue Date: 1994
Type: Text
Language: English
URI: http://hdl.handle.net/2142/20163
Rights Information: Copyright 1994 Kim, Seongtag
Date Available in IDEALS: 2011-05-07
Identifier in Online Catalog: AAI9503236
OCLC Identifier: (UMI)AAI9503236
 

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