  Let $(M,g)$ be a two-dimensional compact boundaryless Riemannian manifold
with Laplacian, $\Delta_g$. If $e_\lambda$ are the associated eigenfunctions of
$\sqrt{-\Delta_g}$ so that $-\Delta_g e_\lambda = \lambda^2 e_\lambda$, then it
has been known for some time \cite{soggeest} that
$\|e_\lambda\|_{L^4(M)}\lesssim \lambda^{1/8}$, assuming that $e_\lambda$ is
normalized to have $L^2$-norm one. This result is sharp in the sense that it
cannot be improved on the standard sphere because of highest weight spherical
harmonics of degree $k$. On the other hand, we shall show that the average
$L^4$ norm of the standard basis for the space ${\mathcal H}_k$ of spherical
harmonics of degree $k$ on $S^2$ merely grows like $(\log k)^{1/4}$. We also
sketch a proof that the average of $\sum_{j = 1}^{2k + 1}
\|e_\lambda\|_{L^4}^4$ for a random orthonormal basis of ${\mathcal H}_k$ is
O(1).
  We are not able to determine the maximum of this quantity over all
orthonormal bases of ${\mathcal H}_k$ or for orthonormal bases of
eigenfunctions on other Riemannian manifolds. However, under the assumption
that the periodic geodesics in $(M,g)$ are of measure zero, we are able to show
that for {\it any} orthonormal basis of eigenfunctions we have that
$\|e_{\lambda_{j_k}}\|_{L^4(M)}=o(\lambda_{j_k}^{1/8})$ for a density one
subsequence of eigenvalues $\lambda_{j_k}$. This assumption is generic and it
is the one in the Duistermaat-Gullemin theorem \cite{dg} which gave related
improvements for the error term in the sharp Weyl theorem. The proof of our
result uses a recent estimate of the first author \cite{Sokakeya} that gives a
necessary and sufficient condition that
$\|e_\lambda\|_{L^4(M)}=o(\lambda^{1/8})$.
