Let $(M,g(t))$ be a compact Riemannian manifold and the metric $g(t)$ evolve by the Ricci flow. In the paper we derive the evolution equation for a geometric constant $\lambda$ under the Ricci flow and the normalized Ricci flow, such that there exist positive solutions to the nonlinear equation\[-\Delta_{\phi} f + af \: \ln \, f + bRf = \lambda f \: \textrm{,}\]where $\Delta \phi$ is the Witten–Laplacian operator, $\phi \in C^\infty (M)$, $a$ and $b$ are both real constants, and $R$ is the scalar curvature with respect to the metric $g(t)$. As an application, we obtain the monotonicity of the geometric constant along the Ricci flow coupled to a heat equation for manifold $M$ with some Ricci curvature condition when $b \gt \frac{1}{4}$.

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Ricci流下几何常数的演化和单调性

令$（M，g（t））$为紧凑的黎曼流形，度量$ g（t）$由Ricci流演化。在本文中，我们导出了Ricci流和归一化Ricci流下的几何常数$ \ lambda $的演化方程，因此存在非线性方程\ [-\ Delta _ {\ phi} f + af \的正解： \ ln \，f + bRf = \ lambda f \：\ textrm {，} \]其中$ \ Delta \ phi $是Witten–Laplacian运算符，$ \ phi \ in C ^ \ infty（M）$，$ a $和$ b $都是实常数，$ R $是相对于度量$ g（t）$的标量曲率。作为应用程序，我们获得了当$ b \ gt \ frac {1} {4} $时，在具有某些Ricci曲率条件的流形$ M $的热方程耦合下，沿着Ricci流的几何常数的单调性。