Suresh Eswarathasan, Cardiff – Analysis & PDE Seminar

Title: $L^p$ restriction of eigenfunctions to random Cantor-type sets

Abstract: Let $(M,g)$ be a compact Riemannian n-manifold without boundary. Consider the corresponding $L^2$-normalized Laplace-Beltrami eigenfunctions. Eigenfunctions of this type arise in physics as modes of periodic vibration of drums and membranes. They also represent stationary states of a free quantum particle on a Riemannian manifold. In the first part of the lecture, I will give a survey of results which demonstrate how the geometry of $M$ affects the behaviour of these special functions, particularly their “size” which can be quantified by estimating $L^p$ norms.

In joint work with Malabika Pramanik (U. British Columbia), I will present in the second part of my lecture a result on the $L^p$ restriction of these eigenfunctions to random Cantor-type subsets of $M$. This, in some sense, interpolates between the standard eigenfunction bounds of Sogge ’88 and the smooth submanifold $L^p$ restriction results of Burq-Gérard-Tzetkov ’06. Our method includes concentration inequalities from probability theory in addition to the analysis of singular Fourier integral operators on fractals.