Evolution of the first eigenvalue of a class of geometric operators under the weighted Yamabe flow

Jingyu Xiao; Jiancheng Liu; Jingyu Xiao; Jiancheng Liu

doi:10.3934/math.20261053

AIMS Mathematics

2026, Volume 11, Issue 8: 26260-26272. doi: 10.3934/math.20261053

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Research article

Evolution of the first eigenvalue of a class of geometric operators under the weighted Yamabe flow

Jingyu Xiao ,
Jiancheng Liu ^,

College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Received: 10 June 2026 Revised: 15 August 2026 Accepted: 18 August 2026 Published: 24 August 2026
MSC : 53C21, 53C25

This paper derived the evolution equation for the first eigenvalue of the operator $ -\Delta_\phi+a(R_\phi^m)^\alpha $ under both the unnormalized and normalized weighted Yamabe flows. For constants $ 0 < \alpha < 1 $, and $ a > $0, we then establish monotone quantities associated with this eigenvalue under suitable assumptions on the weighted scalar curvature.
- weighted Laplacian operator,
- first eigenvalue,
- weighted Yamabe flow,
- monotonicity
Citation: Jingyu Xiao, Jiancheng Liu. Evolution of the first eigenvalue of a class of geometric operators under the weighted Yamabe flow[J]. AIMS Mathematics, 2026, 11(8): 26260-26272. doi: 10.3934/math.20261053

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Abstract

This paper derived the evolution equation for the first eigenvalue of the operator $ -\Delta_\phi+a(R_\phi^m)^\alpha $ under both the unnormalized and normalized weighted Yamabe flows. For constants $ 0 < \alpha < 1 $, and $ a > $0, we then establish monotone quantities associated with this eigenvalue under suitable assumptions on the weighted scalar curvature.

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