Defect measures of eigenfunctions with maximal $ L^\infty $ growth

TL;DR

This paper investigates the connection between the maximum possible growth of eigenfunctions in the supremum norm and their concentration properties, providing characterizations of defect measures for such eigenfunctions and new proofs of related geometric results.

Contribution

It characterizes defect measures of eigenfunctions with maximal $L^$ growth, revealing they are neither more nor less concentrated than zonal harmonics, and offers new proofs of geometric eigenfunction growth results.

Findings

Defect measures of maximally growing eigenfunctions are characterized.

Eigenfunctions with maximal growth are comparable to zonal harmonics in concentration.

New proofs of geometric eigenfunction growth results are provided.

Abstract

We study the relationship between $L^\infty$ growth of eigenfunctions and their $L^2$ concentration as measured by defect measures. In particular, we characterize the defect measures of any sequence of eigenfunctions with maximal $L^\infty$ growth, showing that they must be neither more concentrated nor more diffuse than the zonal harmonics. As a consequence, we obtain new proofs of results on the geometry manifolds with maximal eigenfunction growth obtained by Sogge--Zelditch, and Sogge--Toth--Zelditch.