Monotonicity Formulas of Eigenvalues and Energy Functionals Along the Rescaled List’s Extended Ricci Flow

In this paper, we study monotonicity formulas of eigenvalues of Laplacian and entropies along the rescaled List’s extended Ricci flow, which is a weakly coupled system of second order and the motivation to study this flow stems from its connection to general relativity. The rescaled List’s extended Ricci flow, which is a generalized version including the extended Hamilton normalized flow, is equivalent to the List’s extended Ricci flow up to a homothetic rescale of the metric at each time. Moreover, we find natural entropies of Fk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}_k$$\end{document}-functional and Wk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {W}_k$$\end{document}-functional and show that they are monotonic.