Conformal upper bounds for the first eigenvalue of the p-Laplacian

TL;DR

This paper establishes bounds on the first eigenvalue of the p-Laplacian within conformal classes of Riemannian metrics, showing boundedness for certain p ranges and unboundedness for others, with implications for two-dimensional cases.

Contribution

It provides the first conformal upper bounds for the first eigenvalue of the p-Laplacian on compact manifolds, highlighting differences based on the value of p relative to the dimension.

Findings

p is bounded on each conformal class for 1<p

p can be arbitrarily large for p>m within conformal classes

In 2D, p is bounded if 1<p, unbounded if p>2

Abstract

Let M be a compact, connected, m-dimensional manifold without boundary and p>1. For 1<p\leq m, we prove that the first eigenvalue \lambda_{1,p} of the p-Laplacian is bounded on each conformal class of Riemannian metrics of volume one on M. For p>m, we show that any conformal class of Riemannian metrics on M contains metrics of volume one with \lambda_{1,p} arbitrarily large. As a consequence, we obtain that in two dimensions \lambda_{1,p} is uniformly bounded on the space of Riemannian metrics of volume one if 1<p\leq 2, respectively unbounded if p>2.