Evolution of first eigenvalues of some geometric operators under the rescaled List's extended Ricci flow
DOI:
https://doi.org/10.33044/revuma.3413Abstract
Let $(M,g(t), e^{-\phi}d\nu)$ be a compact weighted Riemannian manifold and let $(g(t),\phi(t))$ evolve by the rescaled List's extended Ricci flow. In this paper, we study the evolution equations for first eigenvalues of the geometric operators, $-\Delta_{\phi}+cS^{a}$, along the rescaled List's extended Ricci flow. Here $\Delta_{\phi}=\Delta-\nabla\phi.\nabla$ is a symmetric diffusion operator, $\phi\in C^{\infty}(M)$, $S=R-\alpha|\nabla \phi|^{2}$, $R$ is the scalar curvature with respect to the metric $g(t)$ and $a, c$ are some constants. As an application, we obtain some monotonicity results under the rescaled List's extended Ricci flow.
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