Conformal bounds for the first eigenvalue of the [formula omitted]-Laplacian

Abstract

Let M be a compact, connected, m-dimensional manifold without boundary and p>1. For 1<p≤m, we prove that the first eigenvalue λ1,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 λ1,p arbitrarily large. As a consequence, we obtain that in two dimensions λ1,p is uniformly bounded on the space of Riemannian metrics of volume one if 1<p≤2, respectively unbounded if p>2.