Eigenvalues of Weighted-Laplacian under the extended Ricci flow

Let $Δ_φ= Δ-\nabla φ\nabla$ be a symmetric diffusion operator with an invariant weighted volume measure $dμ= e^{-φ} dv$ 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 $Δ_φ$ 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 $n\geq 3$. Our results are natural extension of some known results for Laplace-Beltrami operator under various geometric flows.

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