  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). The proof exploits
Gauss' lemma and the fact that the bilinear oscillatory integrals in
H\"ormander's proof of the Carleson-Sj\"olin theorem become better and better
behaved away from the diagonal. Our results are related to a recent work of
Bourgain who showed that $L^2$ averages over geodesics of eigenfunctions are
small compared to a natural power of the eigenvalue $\lambda$ provided that the
$L^4(M)$ norms are similarly small. Our results imply that QUE cannot hold on a
compact boundaryless Riemannian manifold $(M,g)$ of dimension two if
$L^p$-norms are saturated for a given $2<p<6$. We also show that eigenfunctions
cannot have a maximal rate of $L^2$-mass concentrating along unit portions of
geodesics that are not smoothly closed.
