Focal points and sup-norms of eigenfunctions

Focal points and sup-norms 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 sup-norm estimates.  The condition is that there exists a self-focal point x_0\in M for the geodesic flow at which the associated Perron-Frobenius 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.