Spherical harmonics with maximal norm growth

Abstract

Sogges $L^p$-estimates give upper bounds to the $L^p$-norms of $L^2$-normalized eigenfunctions on smooth and compact manifolds. They are also sharp on the sphere, where Gaussian beams saturate the maximal $L^p$-norm growth for small $p$ and zonal harmonics for large $p$. But these maximisers are very sparse in the orthonormal basis of eigenfunctions (ONBE); their densities are both zero. Sogge and Zelditch proposed the question of whether there exists a manifold supporting an ONBE that contains a positive density subsequence of maximisers. In this talk, I will give the positive answer to this question and construct such example for all small $p$.