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

Abstract

Let $M$ be an $n$-dimensional closed Riemannian manifold with metric $g$, $d\mu =e^{-\varphi (x)}d\nu$ be the weighted measure and ${\Delta }_{p,\varphi }$ be the weighted $p$-Laplacian. In this article we will investigate monotonicity for the first eigenvalue problem of the weighted $p$-Laplace operator acting on the space of functions along the Ricci-Bourguignon flow on closed Riemannian manifolds. We find the first variation formula for the eigenvalues of the weighted $p$-Laplacian on a closed Riemannian manifold evolving by the Ricci-Bourguignon flow and we obtain various monotonic quantities. At the end we find some applications in $2$-dimensional and $3$-dimensional manifolds and give an example.