Eigenvalues of Weighted-Laplacian under the extended Ricci flow

Let ${\mathrm{\Delta }}_{\phi }=\mathrm{\Delta }-\mathrm{\nabla }\phi \mathrm{\nabla }$ 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. In this article we study the evolution and monotonicty of the first nonzero eigenvalue of ${\mathrm{\Delta }}_{\phi }$ and we obatin 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 the case $n\ge 3$. Our results are natural extension of some known results for Laplace-Beltrami operator under various geometric flows.