  We consider the problem of proving $L^p$ bounds for eigenfunctions of the
Laplacian in the high frequency limit in the presence of nonpositive curvature
and more generally, manifolds without conjugate points. In particular, we prove
estimates at the "critical exponent" $p_c = \frac{2(d+1)}{d-1}$, where a
spectrum of scenarios for phase space concentration must be ruled out. Our work
establishes a gain of an inverse power of the logarithm of the frequency in the
bounds relative to the classical $L^p$ bounds of the second author.
