  In \cite{P1}, Perelman established a differential Li-Yau-Hamilton (LYH) type
inequality for fundamental solutions of the conjugate heat equation
corresponding to the Ricci flow on compact manifolds (also see \cite{N2}). As
an application of the LYH inequality, Perelman proved a pseudolocality result
for the Ricci flow on compact manifolds. In this article we provide the details
for the proofs of these results in the case of a complete non-compact
Riemannian manifold. Using these results we prove that under certain
conditions, a finite time singularity of the Ricci flow must form within a
compact set. We also prove a long time existence result for the \KRF flow on
complete non-negatively curved \K manifolds.
