Abstract: We first study the Poisson equation $\Delta u =f$ on complete
noncompact
Riemannian manifolds by constructing solutions with precise growth
estimates. In particular we
established the necessary and sufficient conditions on when the equation
admits a solution of
logarithmic growth when the base manifold $M$ has nonnegative Ricci
curvature. Then we applied
the result to the study of plurisubharmonic functions on complete Kahler
manifolds. Combining
with $L^2$ estimate on the power of the canonical (anti-canonical)
holomorphic line bundle over
$M$ we can obtain some optimal curvature decay results on complete Kahler
manifolds with
nonnegative Ricci curvature. In the last part of the paper we also study
the complete Kahler
manifold with pinched curvature using the solution of Poincare-Lelong
equation we constructed.
This is a joint work with Y Shi (Peking) and L-F Tam (Hong Kong).