Journal of the
Korean Mathematical Society JKMS

In this paper we consider the monotonicity of the lowest constant $\lambda^{b}_{a}(g)$ under the Ricci-Bourguignon flow and the normalized Ricci-Bourguignon flow such that the equation \begin{equation*} -{\rm \Delta} u + au\log u + bRu= \lambda^{b}_{a}(g) u \end{equation*} with $\int_M u^2\, {\rm dV}=1,$ has positive solutions, where $a$ and $b$ are two real constants. We also construct various monotonic quantities under the Ricci-Bourguignon flow and the normalized Ricci-Bourguignon flow. Moreover, we prove that a compact steady breather which evolves under the Ricci-Bourguignon flow should be Ricci-flat.

Keywords: Ricci-Bourguignon flow, eigenvalue, homogeneous manifold, locally symmetric manifold, breather

MSC numbers: Primary 53C21, 53C44