Concerning the $L^4$ norms of typical eigenfunctions on compact surfaces

Let $(M,g)$ be a two-dimensional compact boundaryless Riemannian manifold with Laplacian, $Δ_g$. If $e_λ$ are the associated eigenfunctions of $\sqrt{-Δ_g}$ so that $-Δ_g e_λ= λ^2 e_λ$, then it has been known for some time \cite{soggeest} that $\|e_λ\|_{L^4(M)}\lesssim λ^{1/8}$, assuming that $e_λ$ 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 $\sum_{j = 1}^{2k + 1} \|e_λ\|_{L^4}^4$ 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 $\|e_{λ_{j_k}}\|_{L^4(M)}=o(λ_{j_k}^{1/8})$ for a density one subsequence of eigenvalues $λ_{j_k}$. 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 $\|e_λ\|_{L^4(M)}=o(λ^{1/8})$.

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