Suresh Eswarathasan (Cardiff)

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

Let $(M,g)$ be a compact Riemannian surface 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, is complementary to the smooth submanifold $L^p$ restriction results of Burq-Gerard-Tzetkov '06 (and later work of other authors). Our method includes concentration inequalities from probability theory in addition to the analysis of singular Fourier integral operators on fractals.