We study the relationship between growth of eigenfunctions and their 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 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.
"Eigenfunction scarring and improvements in $L^\infty$ bounds." Anal. PDE 11 (3) 801 - 812, 2018. https://doi.org/10.2140/apde.2018.11.801