  In this paper we will prove a maximum principle for the solutions of linear
parabolic equation on complete non-compact manifolds with a time varying
metric. We will prove the convergence of the Neumann Green function of the
conjugate heat equation for the Ricci flow in $B_k\times (0,T)$ to the minimal
fundamental solution of the conjugate heat equation as $k\to\infty$. We will
prove the uniqueness of the fundamental solution under some exponential decay
assumption on the fundamental solution. We will also give a detail proof of the
convergence of the fundamental solutions of the conjugate heat equation for a
sequence of pointed Ricci flow $(M_k\times (-\alpha,0],x_k,g_k)$ to the
fundamental solution of the limit manifold as $k\to\infty$ which was used
without proof by Perelman in his proof of the pseudolocality theorem for Ricci
flow.
