Evolution of first eigenvalues of some geometric operators under the rescaled List's extended Ricci flow (Q6667510)

Let \((M^n,g_0)\) be a smooth closed Riemannian manifold and \(\phi_0:M\to\mathbb R\) be a smooth function. The authors call \((g(t),\phi(t))\) for \(t\in[0,T)\) a solution of the rescaled List's extended Ricci flow if \N\[\N\begin{cases} \frac{\partial}{\partial t}g(t)=-2\mathcal S(t)+\frac{2r(t)}{n}g(t),\quad&g(0)=g_0, \\\N\frac{\partial}{\partial t}\phi(t)=\Delta_{g(t)}\phi(t),\quad&\phi(0)=\phi_0, \end{cases}\N\]\Nwhere \(r\) a smooth function depending on the time variable \(t\), \(\mathcal S\) denotes the generalized Ricci curvature of the weighted Riemannian manifold \((M,g(t),e^{-\phi(t)}d\nu)\) given by \(\mathcal S=\operatorname{Ric}-\alpha\nabla\phi\otimes\nabla\phi\), \(\alpha>0\) is a coupling constant depending on \(n\), and \(\operatorname{Ric}\) and \(\Delta_{g(t)}\) stand for the Ricci tensor and the Laplace-Beltrami operator of \((M,g(t))\).\N\NThe main goal of the article under review is to provide for a given \((M^n,g(t),\phi(t),d\mu=e^{-\phi(t)}d\nu)\) for \(t\in[0,T)\) solution to the rescaled List's extended Ricci flow, monotonicity results of the first eigenvalue \(\lambda(t)\) of the geometric operator \(-\Delta_\phi +cS^a\), where \(\Delta_\phi\) is the Witten Laplacian, \(a\) and \(c\) are constant satisfying \(0<a\leq 1\) and \(c>\frac14\), and \(S\) is the generalized scalar curvature given by \(S=R-\alpha|\nabla\phi|^2\) (\(R\) is the scalar curvature of \((M,g(t))\)).\N\NTheorem 1.1 ensures under some technical condition on \(\mathcal S\) and \(\nabla\phi\otimes\nabla\phi\), that \(\lambda(t)e^{\frac{2}{n} \int_0^tr(\tau)d\tau}\) is non-decreasing along the flow if \(0<S<a^{\frac{1}{1-a}}\). Moreover, if in addition \(r(t)\leq 0\) for all \(t\), then \(\lambda(t)\) is non-decreasing.\N\NThis article is a continuation of [\textit{S. Azami}; Bull. Iran. Math. Soc. 48 (5), 1265--1279 (2022; Zbl 1500.53095)], where the first author studied the monotonicity of the first eigenvalue of the operator \(-\Delta_\phi+cS\) under the rescaled List's extended Ricci flow.