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Title:听Restriction of eigenfunctions to sparse sets on manifolds

Abstract: Given a compact Riemannian manifold $(M, g)$ without boundary, we consider the restriction of Laplace-Beltrami eigenfunctions to certain subsets $\Gamma$ of the manifold. How do the Lebesgue $L^p$ norms of these restricted eigenfunctions grow? Burq, Gerard, Szvetkov and independently Hu studied this question when $\Gamma$ is a submanifold. In ongoing joint work with Suresh Eswarathasan, we extend earlier results to the setting where $\Gamma$ is an arbitrary Borel subset of $M$. Here differential geometric methods no longer apply. Using methods from geometric measure theory, we obtain sharp growth estimates for the restricted eigenfunctions that rely only on the size of $\Gamma$. Our results are sharp for large $p$, and are realized for large families of sets $\Gamma$ that are random and Cantor-like.

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