Conformal bounds for the first eigenvalue of the pp-Laplacian

Let MM be a compact, connected, mm-dimensional manifold without boundary and p>1p>1. For 1mp>m, we show that any conformal class of Riemannian metrics on MM contains metrics of volume one with λ1,pλ1,p arbitrarily large. As a consequence, we obtain that in two dimensions λ1,pλ1,p is uniformly bounded on the space of Riemannian metrics of volume one if 12p>2.