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

We study the relationship between $L^{\mathrm{\infty }}$ 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 $L^{\mathrm{\infty }}$ 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.