A class of monotonic quantities along the Yamabe flow

Abstract

Let $(M,g(t))$ be a closed Riemannian manifold of dimension $n\geq 2$. In this paper we obtain the evolution formula for the lowest constant $\lambda^{b}_{a}(g)$ under the normalized and unnormalized Yamabe flow such that the equation \begin{equation*} -{\rm \Delta} f + af\log f + bRf= \lambda^{b}_{a}(g) f, \end{equation*} with $\int_M f^2\, {\rm dV}=1,$ has positive solutions, where $a$ and $b$ are two real constants. Then we construct various monotonic quantities under the normalized and unnormalized Yamabe flow. We also show that the scalar curvature of a steady Yamabe breather with nonnegative scalar curvature is identically zero.