Restriction bounds for Laplace eigenfunctions

McGill University

Let $(M,g)$ be a compact Riemannian manifold, $\phi_{\lambda}$ an $L^{2}$-normalized eigenfunction with eigenvalue $\lambda^{2}$ and let $H \subset M$ be a hypersurface. I will review recent results on the asymptotics of the $L^{2}$-restrictions, $ \int_{H} |\phi_{\lambda}|^{2} d\vol_H$ both in the completely integrable and ergodic settings. This question is closely related to the issue of "eigenfunction scarring" and is heavily influenced by the geometry of $M$ and also the relative geometry of the hypersurface, $H.$

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