If is a compact real analytic Riemannian manifold, we give a necessary and sufficient condition for there to be a sequence of quasimodes of order 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 for the geodesic flow at which the associated Perron–Frobenius operator has a nontrivial invariant function. The proof is based on an explicit Duistermaat–Guillemin–Safarov pre-trace formula and von Neumann's ergodic theorem.
Cite this article
Christopher D. Sogge, Steve Zelditch, Focal points and sup-norms of eigenfunctions. Rev. Mat. Iberoam. 32 (2016), no. 3, pp. 971–994DOI 10.4171/RMI/904