Endpoint eigenfunction restriction estimates in codimension two

We investigate the optimal endpoint $L^2$ restriction estimates of Laplace eigenfunctions on submanifolds of codimension 2 in a smooth closed Riemannian manifold $M$. For every fixed smooth codimension-two submanifold $\mathrm{\Sigma }$, we prove a little-oh improvement $o({\lambda }^{1\mathrm{/}2}\sqrt{\log \lambda })$ on the classical estimate $O({\lambda }^{1\mathrm{/}2}\sqrt{\log \lambda })$ of Burq–Gérard–Tzvetkov and Hu. Our proof uses the Gaussian transform and Taylor approximation to reduce the problem to polynomial models that can be handled by Stein–Street’s estimate on singular Radon transforms. The three-dimensional case can be handled directly by Ricci–Stein’s estimate. Moreover, we construct explicit examples to show that this improvement is optimal in general. These are closely related to earlier counterexamples for endpoint Strichartz estimates by Montgomery–Smith and Carbery–Hofmann. In particular, we give a negative answer to the log-removal question on the endpoint eigenfunction restriction estimates, as the log-free estimate $O({\lambda }^{1\mathrm{/}2})$ cannot hold in general.