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Keywords:
monotonicity; first eigenvalue; Witten-Laplacian operator; Yamabe flow
monotonicity; first eigenvalue; Witten-Laplacian operator; Yamabe flow
Summary:
We construct some nondecreasing quantities associated to the first eigenvalue of $-\Delta _\phi +cR$ $(c\geq \frac 12(n-2)/(n-1))$ along the Yamabe flow, where $\Delta _\phi $ is the Witten-Laplacian operator with a $C^2$ function $\phi $. We also prove a monotonic result on the first eigenvalue of $-\Delta _\phi + \frac 14 (n/ (n-1))R$ along the Yamabe flow. Moreover, we establish some nondecreasing quantities for the first eigenvalue of $-\Delta _\phi +cR^a$ with $a\in (0,1)$ along the Yamabe flow.
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