Monotonicity of Eigenvalues and Functionals Along the Ricci–Bourguignon Flow

Abstract

In this paper we first prove the monotonicity of the lowest eigenvalue of the Schrödinger operator

$$\begin{aligned} \frac{(1-(n-1)\rho )^2}{1-2(n-1)\rho }R-4\varDelta \end{aligned}$$

along the Ricci–Bourguignon flow

$$\begin{aligned} \frac{\partial \mathrm {g}}{\partial t}=-2(\mathrm {Ric}-\rho R\mathrm {g}) \end{aligned}$$

based on an evolving formula of the $\mathcal {F}_{\rho }$-functional, and rule out nontrivial steady breathers. Then we prove the monotonicity of the lowest eigenvalue of the Schrödinger operator $BR-4\varDelta$ along the Ricci–Bourguignon flow for any constant B satisfying

$$\begin{aligned} B\ge \frac{4(1-(n-1)\rho )^2-n\rho }{4(1-2(n-1)\rho )}>0 \end{aligned}$$

for the case that $\rho \le 0.$ We also study the evolving formula of the $\mathcal {W}_{\rho }$-functional and get the monotonicity of the infimum of the $\mathcal {W}_{\rho }$-functional, based on which we can prove that a shrinking breather should be an Einstein metric for the case that $\rho <0.$