First eigenvalues evolution for some geometric operators along the Yamabe flow

Let (M, g) be a compact Riemannian manifold without boundary or with smooth boundary having vanishing mean curvature, and \(g=g(t)\) $g=g(t)$ are one-parameter family of Riemannian metrics evolving by the Yamabe flow. In this paper, we derive some evolution equations for the first eigenvalue of the operators \(-\Delta +aR^\alpha \) $-\mathrm{\Delta }+a{R}^{\alpha }$ , \(0<\alpha \le 1\) $0<\alpha \le 1$ for some positive constant a under the (unnormalized and normalized) Yamabe flow. Some monotonic quantities depending on the first eigenvalue are obtained under certain restrictions on the constant a as an application of the evolution equations.