Some Results for the Perelman Lyh-type Inequality

Let (M, g(t)), 0 ≤ t ≤ T , ∂M 6= φ, be a compact n-dimensional manifold, n ≥ 2, with metric g(t) evolving by the Ricci flow such that the second fundamental form of ∂M with respect to the unit outward normal of ∂M is uniformly bounded below on ∂M × [0, T ]. We will prove a global Li-Yau gradient estimate for the solution of the generalized conjugate heat equation on M × [0, T ]. We will give another proof of Perelman’s Li-Yau-Hamilton type inequality for the fundamental solution of the conjugate heat equation on closed manifolds without using the properties of the reduced distance. We will also prove various gradient estimates for the Dirichlet fundamental solution of the conjugate heat equation. In [P] Perelman stated a differential Li-Yau-Hamilton type inequality for the fundamental solution of the conjugate heat equation on closed manifolds evolving by the Ricci flow. More precisely let M be a closed manifold with metric g(t), 0 ≤ t ≤ T , evolving by the Ricci flow, ∂ ∂t gij = −2Rij (0.1) in M × (0, T ). Let p ∈M and u = e−f (4πτ) n 2 (0.2) be the fundamental solution of the conjugate heat equation