Curvature and harmonic analysis on compact manifolds

TL;DR

This paper explores the relationship between curvature and eigenfunction concentration on compact manifolds, providing new sharp estimates for quasimodes that characterize space forms through their $L^q$-norm growth.

Contribution

It introduces novel sharp $L^q$-estimates for log-quasimodes on compact manifolds, characterizing space forms via $L^q$-norm growth for certain exponents.

Findings

Sharp $L^q$-estimates for log-quasimodes are established.

Characterization of space forms based on $L^q$-norm growth.

No such characterization exists for $q > q_c$.

Abstract

We discuss problems that relate curvature and concentration properties of eigenfunctions and quasimodes on compact boundaryless Riemannian manifolds. These include new sharp $L^q$-estimates, $q\in (2,q_c]$, $q_c=2(n+1)/(n-1)$, of log-quasimodes that characterize compact connected space forms in terms of the growth rate of $L^q$-norms of such quasimode for these relatively small Lebesgue exponents $q$. No such characterization is possible for any exponent $q> q_c$.