Maximum Principle and Convergence of Fundamental Solutions for the Ricci Flow

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 Bk × (0, T ) to the minimal fundamental solution of the conjugate heat equation as k → ∞. 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 (Mk × (−α, 0], xk, gk) to the fundamental solution of the limit manifold as k → ∞ which was used without proof by Perelman in his proof of the pseudolocality theorem for Ricci flow [P]. Maximum principle for the heat equation on complete non-compact manifold with a fixed metric was proved by P. Li, L. Karp [LK] and J. Wang [W] (cf. [CLN]). Maximum principle for parabolic equations on complete non-compact manifold with a metric with uniformly bounded Riemannian curvature and evolving by the Ricci flow, ∂ ∂t gij = −2Rij (0.1) was proved byW.X. Shi [S1], [S2], [S3] under either a uniform boundedness condition on the solution or some structural conditions on the parabolic equation or positivity assumption on the Riemannian curvature operator. 1991 Mathematics Subject Classification. Primary 58J35, 53C43.