Riemannian manifolds with maximal eigenfunction growth

Christopher D. Sogge*, Steve Zelditch

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

73 Scopus citations

Abstract

On any compact Riemannian manifold (M, g) of dimension n, the L2-normalized eigenfunctions {φλ} satisfy ∥φλ∥∞ ≤ Cλ(n-1)/2, where -Δφλ = λ2φλ. 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, for example, for flat tori ℝn/Γ. We say that Sn, but not ℝn/Γ, is a Riemannian manifold with maximal eigenfunction growth. The problem that motivates this paper is to determine the (M, g) with maximal eigenfunction growth. Our main result is that such an (M, g) must have a point x where the set ℒx of geodesic loops at x has positive measure in Sx*M. We show that if (M, g) is real analytic, this puts topological restrictions on M; for example, only M = S2 or M = ℝP2 (topologically) in dimension 2 can possess a real analytic metric of maximal eigenfunction growth. We further show that generic metrics on any M fail to have maximal eigenfunction growth. In addition, we construct an example of (M, g) for which ℒx has positive measure for an open set of x but which does not have maximal eigenfunction growth; thus, it disproves a naive converse to the main result.

Original languageEnglish (US)
Pages (from-to)387-437
Number of pages51
JournalDuke Mathematical Journal
Volume114
Issue number3
DOIs
StatePublished - Sep 15 2002

ASJC Scopus subject areas

  • Mathematics(all)

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