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

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 $c\ne 0$, 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 $t\to T$ on a limited maximal time in terval $[0,T)$, which generalizes some results of Cerbo and Fabrizio \cite{Fabrizio2007}.

17 pages. arXiv admin note: text overlap with arXiv:1507.00324, arXiv:0912.4775 by other authors