Focal points and supnorms of eigenfunctions
Focal points and supnorms of eigenfunctions

Christopher Sogge, Johns Hopkins University
Fine Hall 110
Please note special location. If (M,g) is a compact real analytic Riemannian manifold, we give a necessary and sufficient condition for there to be a sequence of quasimodes saturating supnorm estimates. The condition is that there exists a selffocal point x_0\in M for the geodesic flow at which the associated PerronFrobenius operator U: L^2(S_{x_0}^*M) \to L^2(S_{x_0}^*M) has a nontrivial invariant function. The proof is based on von Neumann's ergodic theorem and stationary phase. In two dimensions, the condition simplifies and is equivalent to the condition that there be a point through which the geodesic flow is periodic. This is joint work with Steve Zelditch.