We provide a necessary and sufficient condition that $L^p$-norms, $2<p<6$, of
eigenfunctions of the square root of minus the Laplacian on 2-dimensional
compact boundaryless Riemannian manifolds $M$ are small compared to a natural
power of the eigenvalue $/lambda$. The condition that ensures this is that
their $L^2$ norms over $O(/lambda^{-1/2})$ neighborhoods of arbitrary unit
geodesics are small when $/lambda$ is large (which is not the case for the
highest weight spherical harmonics on $S^2$ for instance). We also discuss
connections between our results and Quantum Ergodicity. Time permitting,