Electron. J. Diff. Eqns., Vol. 2000(2000), No. 46, pp. 1-30.

Semilinear parabolic problems on manifolds and applications to the non-compact Yamabe problem

Qi S. Zhang

Abstract:
We show that the well-known non-compact Yamabe equation (of prescribing constant positive scalar curvature) on a manifold with non-negative Ricci curvature and positive scalar curvature behaving like $c/d(x)^2$ near infinity can not be solved if the volume of geodesic balls do not increase "fast enough". Even though both existence and nonexistence results have appeared in the case when the scalar curvature is negative somewhere([J], [AM]), or when the scalar curvature is positive ([Ki], [Zhan5]), the current paper seems to give the first nonexistence result in the case that the scalar curvature is positive and $Ricci \ge 0$, which seems to be the fundamental part of the noncompact Yamabe problem. We also find some complete non-compact manifolds with positive scalar curvature which are conformal to complete manifolds with constant and with zero scalar curvature. This is a new phenomenon which does not happen in the compact case.

Submitted March 8, 1999. Published June 15, 2000.
Math Subject Classifications: 35K55, 58J35.
Key Words: semilinear parabolic equations, critical exponents, noncompact Yamabe problem.

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Qi S. Zhang
Department of Mathematics, University of Memphis
Memphis, TN 38152, USA
e-mail: qizhang@memphis.edu
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