Monotonicity of functionals along conformal ricci flow

The main purpose of this note is to construct two functionals of positive solutions to the conjugate heat equation associated to the metrics evolving by the conformal Ricci flow on closed manifolds. We show that they are nondecreasing by calculating the explicit evolution formulas of these functionals. For the entropy functional we give another proof of the monotonicity by establishing a pointwise formula. Moreover, we show that the increase is strict unless the metrics are Einstein.