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

Let (M , g (t) , e - ϕ d ν) be a measure space and (g (t) , ϕ (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 - Δ ϕ + c S under the rescaled List's extended Ricci flow, where Δ ϕ is the Witten Laplacian, ϕ ∈ C ∞ (M) , S = R - α | ∇ ϕ | 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. [ABSTRACT FROM AUTHOR]