The Journal of Geometric Analysis

, Volume 29, Issue 2, pp 1116–1135

# Monotonicity of Eigenvalues and Functionals Along the Ricci–Bourguignon Flow

• Lin Feng Wang
Article

## 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 $$\rho <0.$$

## Keywords

Ricci–Bourguignon flow Eigenvalue $$\mathcal {F}_{\rho }$$-functional $$\mathcal {W}_{\rho }$$-functional Evolving formula Breather

53C21

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