Evolution and monotonicity of the first eigenvalue of p-Laplacian under the Ricci-harmonic flow

We study the evolution and monotonicity of the eigenvalues of p-Laplace operator on an m-dimensional compact Riemannian manifold M whose metric g(t) evolves by the Ricci-harmonic flow. The first nonzero eigenvalue is proved to be monotonically nondecreasing along the flow and differentiable almost everywhere. As a corollary, we recover the corresponding results for the usual Laplace-Beltrami operator when p = 2. We also examine the evolution and monotonicity under volume preserving flow and it turns out that the first eigenvalue is not monotone in general.