Evolution and monotonicity of eigenvalues under the Ricci flow

Abstract

Let (M, g(t)) be a compact Riemannian manifold and the metric g(t) evolve by the Ricci flow. We derive the evolution equation for the eigenvalues of geometric operator −Δϕ + cR under the Ricci flow and the normalized Ricci flow, where Δϕ is the Witten-Laplacian operator, ϕ ∈ C∞(M), and R is the scalar curvature with respect to the metric g(t). As an application, we prove that the eigenvalues of the geometric operator are nondecreasing along the Ricci flow coupled to a heat equation for manifold M with some Ricci curvature condition when \(c > \tfrac{1} {4}\).