#### Vol. 296, No. 1, 2018

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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
##### 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
Primary: 53C21
Secondary: 53C44