Let \((M^n, g(t))\) be a compact Riemannian manifold. In this paper, we derive the evolution formula for the geometric constant \(\lambda _{a}^{b} (g)\) as an infimum of a certain energy function when the following partial differential equation:

$$\begin{aligned} -\Delta _{\phi } u + a u \log u + b S u = \lambda _{a}^{b}(g) u \end{aligned}$$

with \(\int _M u^2 d\mu = 1\), has positive solutions, where a and b are real constants along the extended Ricci flow and the normalized extended Ricci flow. In addition, we derive some monotonicity formulas by imposing some conditions along both the extended Ricci flow and the normalized extended Ricci flow.

中文翻译：

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沿扩展 Ricci 流的几何常数的演化和单调性

设\((M^n, g(t))\)是一个紧黎曼流形。在本文中，我们推导出几何常数\(\lambda _{a}^{b} (g)\)作为某个能量函数的下界，当以下偏微分方程为：

$$\begin{aligned} -\Delta _{\phi } u + au \log u + b S u = \lambda _{a}^{b}(g) u \end{aligned}$$

与\(\int _M u^2 d\mu = 1\)，具有正解，其中a和b是沿扩展 Ricci 流和归一化扩展 Ricci 流的实常数。此外，我们通过沿扩展 Ricci 流和归一化扩展 Ricci 流施加一些条件来推导出一些单调性公式。