Resumen de Focal points and sup-norms of eigenfunctions

Christopher D. Sogge, Steve Zelditch

• 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 of order o(λ) saturating sup-norm estimates. In particular, it gives optimal conditions for existence of eigenfunctions satisfying maximal sup norm bounds. The condition is that there exists a self-focal point x0∈M for the geodesic flow at which the associated Perron–Frobenius operator Ux0:L2(S∗x0M)→L2(S∗x0M) has a nontrivial invariant L2 function. The proof is based on an explicit Duistermaat–Guillemin–Safarov pre-trace formula and von Neumann's ergodic theorem.