About the blowup of quasimodes on Riemannian manifolds

Christopher D. Sogge, John A. Toth, Steve Zelditch

Research output: Contribution to journalArticlepeer-review

28 Scopus citations

Abstract

On any compact Riemannian manifold (M,g) of dimension n, the L 2-normalized eigenfunctions φλ satisfy ||φλ||≤ Cλ1-n/2 where -Δφλ2 φλ. The bound is sharp in the class of all (M,g) since it is obtained by zonal spherical harmonics on the standard n-sphere Sn . But of course, it is not sharp for many Riemannian manifolds, e.g., flat toriℝn/Γ. We say that Sn, but not ℝn/Γ, is a Riemannian manifold with maximal eigenfunction growth. The problem which motivates this paper is to determine the (M,g) with maximal eigenfunction growth. In an earlier work, two of us showed that such an (M,g) must have a point x where the se ℒx of geodesic loops at x has positive measure in S*xM. We strengthen this result here by showing that such a manifold must have a point where the set ℛx of recurrent directions for the geodesic flow through x satisfies |ℛ|x >0. We also show that if there are no such points, L2-normalized quasimodes have sup-norms that are o(λ(n-1)/2), and, in the other extreme, we show that if there is a point blow-down x at which the first return map for the flow is the identity, then there is a sequence of quasimodes with L-norms that are Ω(λ(n-1)/2).

Original languageEnglish (US)
Pages (from-to)150-173
Number of pages24
JournalJournal of Geometric Analysis
Volume21
Issue number1
DOIs
StatePublished - Jan 1 2011

Keywords

  • Eigenfunctions estimates

ASJC Scopus subject areas

  • Geometry and Topology

Fingerprint Dive into the research topics of 'About the blowup of quasimodes on Riemannian manifolds'. Together they form a unique fingerprint.

Cite this