Eigenvalues of the weighted Laplacian under the extended Ricci flow

Let Delta(phi), = Delta-Delta phi del be a symmetric diffusion operator with an invariant weighted volume measure d mu = e(-phi) dv on an n-dimensional compact Riemannian manifold (M, g), where g = g(t) solves the extended Ricci flow. We study the evolution and monotonicity of the first nonzero eigenvalue of Delta(phi) and we obtain several monotone quantities along the extended Ricci flow and its volume preserving version under some technical assumption. We also show that the eigenvalues diverge in a finite time for n >= 3. Our results are natural extensions of some known results for Laplace-Beltrami operators under various geometric flows.