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-△φ+ c R under the Ricci flow and the normalized Ricci flow, where △φis 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 >14.
