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

TL;DR

This paper studies how the first eigenvalue of the weighted p-Laplacian changes monotonically along the Ricci-Bourguignon flow on closed manifolds, providing formulas, monotonic quantities, and applications in low dimensions.

Contribution

It derives the first variation formula for eigenvalues of the weighted p-Laplacian under Ricci-Bourguignon flow and identifies monotonic quantities with applications in 2D and 3D.

Findings

Derived the first variation formula for eigenvalues.

Identified monotonic quantities along the flow.

Provided applications in low-dimensional manifolds.

Abstract

Let $M$ be an $n$-dimensional closed Riemannian manifold with metric $g$, $d\mu=e^{-\phi(x)}d\nu$ be the weighted measure and $\Delta_{p,\phi}$ 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.