Restriction of Laplace-Beltrami eigenfunctions on subsets of manifolds


The study of eigenfunctions of Laplacians lies at the interface of several areas of mathematics, including analysis, geometry, mathematical physics and number theory. These special functions arise in physics and in partial differential equations as modes of periodic vibration of drums and membranes.

A fundamental question surrounding Laplace-Beltrami eigenfunctions targets their concentration phenomena, via high-energy asymptotics or high-frequency behaviour. One popular approach to this question involves studying the growth of the $L^p$ norms of these eigenfunctions on the ambient manifold, or on submanifolds thereof. What can one say about the behaviour of eigenfunctions on rougher sets? We will discuss answers to this question, based on joint work with Suresh Eswarathasan.