Evolution of Eigenvalues of Geometric Operator Under the Rescaled List’s Extended Ricci Flow

Let $(M,g(t), e^{-\phi}d\nu)$ be a measure space and $(g(t),\phi(t))$ evolve by the rescaled List s extended Ricci flow. In this paper, we derive the evolution equations for first eigenvalue of the geometric operators $-\Delta_{\phi}+cS$ under the rescaled List s extended Ricci flow, where $\Delta_{\phi}$ is the Witten Laplacian, $\phi\in C^{\infty}(M)$, $S=R-\alpha|\nabla \phi|^{2}$ and $R$ is the scalar curvature with respect to the metric $g(t)$. As an application, we obtain several monotonic quantities along the rescaled List s extended Ricci flow. Our results are natural extensions of some known results for Witten Laplace operator under various geometric flows.