Quasimode concentration on compact space forms

TL;DR

This paper demonstrates that upper bounds for $L^2$-norms of $L^1$-normalized quasimodes are sharp on compact space forms, enabling characterization of these manifolds via quasimode decay rates and concentration near geodesics.

Contribution

It fully characterizes compact manifolds of constant sectional curvature using decay rates and concentration properties of quasimodes, resolving a longstanding problem.

Findings

Upper bounds for $L^2$-norms are sharp on all compact space forms.

Characterization of constant curvature manifolds via quasimode decay rates.

Manifolds are characterized by quasimode concentration near periodic geodesics.

Abstract

We show that the upper bounds for the $L^2$-norms of $L^1$-normalized quasimodes that we obtained in [9] are always sharp on any compact space form. This allows us to characterize compact manifolds of constant sectional curvature using the decay rates of lower bounds of $L^1$-norms of $L^2$-normalized log-quasimodes fully resolving a problem initiated by the second author and Zelditch [15]. We are also able to characterize such manifolds by the concentration of quasimodes near periodic geodesics as measured by $L^2$-norms over thin geodesic tubes.