On the Spectrum of the p-Biharmonic Operator Under the Ricci Flow

Abstract

This paper is devoted to the study of the behaviour of the spectrum of the p-biharmonic operator on a complete closed Riemannian manifold evolving by the Ricci flow. In particular, evolution formulas, monotonicity properties and differentiability for the least nonzero eigenvalue are derived along the flow. As a by-product, several monotone quantities involving the first eigenvalue are obtained under the flow. These monotone quantities depend on the Euler characteristics of compact surfaces in the case $$n=2$$. Furthermore, the spectrum diverges in a finite time under some geometric condition on the curvature.