Monotonicity formulas of eigenvalues and energy functionals along the rescaled List's extended Ricci flow

TL;DR

This paper establishes new monotonicity formulas for eigenvalues and energy functionals along the rescaled List's extended Ricci flow, extending previous results to more generalized functionals and eigenvalues.

Contribution

It introduces generalized monotonicity formulas for eigenvalues and energy functionals in the context of the rescaled List's extended Ricci flow, expanding on prior work.

Findings

Monotonicity formulas for Laplacian eigenvalues generalizing Li and Cao-Hou-Ling.

Monotonicity of $_k$-functional related to steady Ricci breathers.

Monotonicity of $W_k$-functional related to expanding Ricci breathers.

Abstract

In this paper, we study monotonicity formulas of eigenvalues and entropies along the rescaled List's extended Ricci flow. We derive some monotonicity formulas of eigenvalues of Laplacian which generalize those of Li in [8] and Cao-Hou-Ling in [3]. Moreover, we also consider monotonicity formulas of $\mathcal{F}_k$-functional which can be seen as a generalized $\mathcal{F}$-functional corresponding with steady Ricci breathers, and $\mathcal{W}_k$-functional which generalizes $\mathcal{W}$-functional corresponding with expanding Ricci breathers.