Monotonicity of eigenvalues of geometric operators along the Ricci–Bourguignon flow

Bin Chen, Qun He and Fanqi Zeng

Vol. 296 (2018), No. 1, 1–20

DOI: 10.2140/pjm.2018.296.1

Abstract

We study monotonicity of eigenvalues of the Schrödinger-type operator $-\Delta +cR$, where $c$ is a constant, along the Ricci–Bourguignon flow. For $c\ne 0$, we derive monotonicity of the lowest eigenvalue of the Schrödinger-type operator $-\Delta +cR$, which generalizes some results of Cao (2008). As an application, we rule out nontrivial compact steady breathers in the Ricci–Bourguignon flow. For $c=0$, we derive monotonicity of the first eigenvalue of the Laplacian, which generalizes some results of Ma (2006).

Keywords

eigenvalue, Laplacian, monotonicity, Ricci–Bourguignon flow, breathers

Mathematical Subject Classification 2010

Primary: 53C21

Secondary: 53C44

Milestones

Received: 23 January 2016

Revised: 7 March 2017

Accepted: 2 March 2018

Published: 1 May 2018