Evolution of the First Eigenvalue of a (<i>p</i>,<i>q</i>)-Laplacian Under a Harmonic Ricci Flow

Abstract

The properties of the first eigenvalue of a class of ( p , q ) Laplacian are investigated. A variational formulation for the first eigenvalue of the Laplacian on a closed Riemannian manifold is obtained. This eigenvalue corresponds to a nonlinear, coupled system of p -Laplacian partial differential equations. The main idea is to investigate the evolution of the first eigenvalue of the system under the Ricci harmonic flow. It is also possible to construct monotonic quantities based on them and study their evolution which is done.