Evolution of the first eigenvalue of weighted p-Laplacian along the Ricci-Bourguignon flow.

Let M be an n-dimensional closed Riemannian manifold with metric g, dμ = e-ø(x)dv be the weighted measure and Δp, ø be the weighted p-Laplacian. In this article we will investigate monotonicity for the first eigenvalue problem of the weighted p-Laplace operator act- ing on the space of functions along the Ricci-Bourguignon ow on closed Riemannian manifolds. We find the first variation formula for the eigen-values of the weighted p-Laplacian on a closed Riemannian manifold evolving by the Ricci-Bourguignon ow and we obtain various monotonic quantities. At the end we find some applications in 2-dimensional and 3-dimensional manifolds and give an example. [ABSTRACT FROM AUTHOR]