Monotonicity of eigenvalues of geometric operaters along the Ricci-Bourguignon flow

TL;DR

This paper investigates how the eigenvalues of Laplacian-type operators change monotonically along the Ricci-Bourguignon flow, extending previous results and analyzing their behavior on three-manifolds with positive Ricci curvature.

Contribution

It generalizes existing monotonicity results for eigenvalues of Laplacian-type operators along the Ricci-Bourguignon flow, including the case of zero constant and positive Ricci curvature.

Findings

Monotonicity of the lowest eigenvalue for $- riangle + cR$ when $c eq 0$

Monotonicity of the first eigenvalue of the Laplacian when $c=0$

Eigenvalues diverge on closed three-manifolds with positive Ricci curvature as time approaches maximal interval end

Abstract

In this paper, we study monotonicity of eigenvalues of Laplacian-type operator $-\Delta+cR$, where $c$ is a constant, along the Ricci-Bourguignon flow. For $c\neq0$, We derive monotonicity of the lowest eigenvalue of Laplacian-type operator $-\Delta+cR$ which generalizes some results of Cao \cite{Cao2007}. For $c=0$, We derive monotonicity of the first eigenvalue of Laplacian which generalizes some results of Ma \cite{Ma2006}. Moreover, we prove that when $(M_{3}, g_{0})$ is a closed three manifold with positive Ricci curvature, the eigenvalue of the Laplacian diverges as $t \rightarrow T$ on a limited maximal time in terval $[0, T)$, which generalizes some results of Cerbo and Fabrizio \cite{Fabrizio2007}.