# Eigenfunction scarring and improvements in $L^{\infty}$ bounds

## TL;DR

This paper investigates how the concentration and diffusion of eigenfunction defect measures influence their maximum amplitude growth, revealing that both highly concentrated and overly diffuse measures limit eigenfunction growth.

## Contribution

It establishes a link between defect measure concentration and $L^
Infty$ eigenfunction bounds, showing that certain types of measure concentration or diffusion prevent maximal growth.

## Key findings

- Scarring (measure concentration) is incompatible with maximal $L^
Infty$ growth.
- Diffuse measures like Liouville measure also prevent maximal eigenfunction growth.
- Eigenfunction growth is constrained by the nature of their defect measures.

## Abstract

We study the relationship between $L^\infty$ growth of eigenfunctions and their $L^2$ concentration as measured by defect measures. In particular, we show that scarring in the sense of concentration of defect measure on certain submanifolds is incompatible with maximal $L^\infty$ growth. In addition, we show that a defect measure which is too diffuse, such as the Liouville measure, is also incompatible with maximal eigenfunction growth.