An optimal parabolic estimate and its applications in prescribing scalar curvature on some open manifolds with Ricci
\ge 0 |
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Authors: | Qi S Zhang |
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Affiliation: | (1) Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA (e-mail: qizhang@memphis.edu) , US |
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Abstract: | By establishing an optimal comparison result on the heat kernel of the conformal Laplacian on open manifolds with nonnegative
Ricci curvature, (a) we show that many manifolds with positive scalar curvature do not possess conformal metrics with scalar
curvature bounded below by a positive constant; (b) we identify a class of functions with the following property: If the manifold
has a scalar curvature in this class, then there exists a complete conformal metric whose scalar curvature is any given function
in this class. This class is optimal in some sense; (c) we have identified all manifolds with nonnegative Ricci curvature,
which are “uniformly” conformal to manifolds with zero scalar curvature. Even in the Euclidean case, we obtain a necessary
and sufficient condition under which the main existence results in Ni1] and KN] on prescribing nonnegative scalar curvature
will hold. This condition had been sought in several papers in the last two decades.
Received: 11 November 1998 / Revised: 7 April 1999 |
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