A microlocal approach to eigenfunction concentration

TL;DR

This paper introduces a microlocal method to analyze high energy eigenfunction concentration on submanifolds, providing new bounds and insights without global manifold assumptions.

Contribution

The authors develop a microlocal approach to eigenfunction averages over submanifolds, improving bounds under non-recurrence conditions and avoiding global assumptions.

Findings

Quantitative bounds on eigenfunction averages over submanifolds.

Method applies uniformly to submanifolds of various codimensions.

Improved understanding of eigenfunction concentration without global manifold restrictions.

Abstract

We describe a new approach to understanding averages of high energy Laplace eigenfunctions, $u_h$, over submanifolds, $$ \Big|\int _H u_hd\sigma_H\Big| $$ 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 $1\leq k\leq n$ and in particular, gives a new approach to understanding $\|u_h\|_{L^\infty(M)}$. 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 $(M,g)$.