Czechoslovak Mathematical Journal

Czechoslovak Mathematical Journal, Vol. 71, No. 2, pp. 387-401, 2021

Monotonicity of first eigenvalues along the Yamabe flow

Liangdi Zhang

Received September 6, 2019. Published online October 29, 2020.

Abstract: We construct some nondecreasing quantities associated to the first eigenvalue of $-\Delta_\phi+cR$ $(c\geq\frac12(n-2)/(n-1))$ 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 $-\Delta_\phi+ \frac14 (n/ (n-1))R$ 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.

Keywords: monotonicity; first eigenvalue; Witten-Laplacian operator; Yamabe flow

Classification MSC: 58C40

DOI: 10.21136/CMJ.2020.0392-19

PDF available at: Springer Institute of Mathematics CAS Digital Mathematics Library

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Affiliations: Liangdi Zhang, Center of Mathematical Sciences, Zhejiang University, 38 Zheda Rd., 310027 Hangzhou, P. R. China, e-mail: zhangliangdi@zju.edu.cn

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