A microlocal approach to eigenfunction concentration

Abstract

We describe a new approach to understanding averages of high energy Laplace eigenfunctions, $u_h$, over submanifolds, $$\left|{\int }_H{u}_{h}d{\sigma }_H\right|$$ where $H\subset M$ is a submanifold and ${\sigma }_H$ the induced by the Riemannian metric on $M$. This approach can be applied uniformly to submanifolds of codimension $1\le k\le n$ and in particular, gives a new approach to understanding $\Vert u_h{\Vert }_{L^{\infty }(M)}$. The method, developed in the author's recent work together with Y. Canzani and J. Toth, relies on estimating averages by the behavior of $u_h$ microlocally near the conormal bundle to $H$. By doing this, we are able to obtain quantitative improvements on eigenfunction averages under certain uniform non-recurrent conditions on the conormal directions to $H$. In particular, we do not require any global assumptions on the manifold $(M,g)$.

arXiv comment16 pages, 7 figures