Monotonicity of functionals along conformal Ricci flow

TL;DR

This paper constructs and analyzes two functionals related to the conjugate heat equation under conformal Ricci flow, proving their monotonicity and characterizing Einstein metrics through strict increase conditions.

Contribution

It introduces new functionals for solutions to the conjugate heat equation under conformal Ricci flow and proves their monotonicity, providing insights into the flow's geometric properties.

Findings

The two functionals are nondecreasing along the flow.

Monotonicity is strict unless the metrics are Einstein.

An alternative proof of entropy functional monotonicity is provided.

Abstract

The main purpose of this note is to construct two functionals of the 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 are strict unless the metrics are Einstein.