First eigenvalues of geometric operator under the Ricci-Bourguignon flow

Shahroud Azami


Let $(M,g(t))$ be a compact Riemannian manifold  and  the metric $g(t)$ evolve by the Ricci-Bourguignon flow. We find the formula variation of the eigenvalues of  geometric operator $-\Delta_{\phi}+cR$ under  the Ricci-Bourguignon flow, where  $\Delta_{\phi}$  is the Witten-Laplacian operator and $R$ is the scalar curvature. In the final  we show that some quantities dependent to the eigenvalues of  the geometric operator are  nondecreasing along the Ricci-Bourguignon flow on  closed manifolds  with nonnegative curvature.


Laplace, Ricci-Bourguignon flow


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