Eigen-value Monotonicity for the Ricci-hamilton Flow

∂tgij = −2Rij , on MT := M × [0, T ) where Rij is the Ricci tensor of the metric g := g(t) and T is the maximal existing time for the flow. In [2], R.Hamilton proved the local existence of the flow for the compact manifold case. His argument is much simplified by De Turck [1]. When (M,g0) is a a complete non-compact Riemannian manifold with bounded geometry, W.X.Shi [5] obtained the local existence result for the flow. We study the monotonicity property of the first eigenvalue of the Laplacian operator ∆ := ∆g(t) of the evolving metric (g(t)) along the RicciHamilton flow. More precisely, let D be a compact domain with smooth boundary in the manifold M . Let f := f(x, t) be the first eigen-function of ∆. Then we have −∆f = μf, in D, (1) with Dirichlet boundary condition