Conformal bounds for the first eigenvalue of the $(p,q)$-Laplacian system

TL;DR

This paper studies bounds on the first eigenvalue of the $(p,q)$-Laplacian system on compact manifolds and shows conformal metrics on spheres can achieve arbitrarily large eigenvalues.

Contribution

It provides conformal bounds for the first eigenvalue of the $(p,q)$-Laplacian and demonstrates the existence of metrics with large eigenvalues on spheres.

Findings

Conformal bounds established for the first eigenvalue.

Existence of metrics with arbitrarily large eigenvalues on spheres.

Results apply to $p,q > n$ cases.

Abstract

Consider $\left(M,g\right)$ as an $m$-dimensional compact connected Riemannian manifold without boundary. In this paper, we investigate the first eigenvalue $\lambda_{1,p,q}$ of the $\left(p,q\right)$-Laplacian system on $M$. Also, in the case of $p,q >n$ we will show that for arbitrary large $\lambda_{1,p,q}$ there exists a Riemannian metric of volume one conformal to the standard metric of $\mathbb{S}^{m}$.