Curvature and harmonic analysis on compact manifolds

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