TY - JOUR

T1 - About the blowup of quasimodes on Riemannian manifolds

AU - Sogge, Christopher D.

AU - Toth, John A.

AU - Zelditch, Steve

N1 - Funding Information:
The first and third authors were supported by the National Science Foundation, Grants DMS-0555162, DMS-0099642, DMS-0904252. The second author was partially supported by NSERC Grant OGP0170280 and a William Dawson Fellowship.
Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.

PY - 2011/1

Y1 - 2011/1

N2 - On any compact Riemannian manifold (M,g) of dimension n, the L 2-normalized eigenfunctions φλ satisfy ||φλ||∞≤ Cλ1-n/2 where -Δφλ =λ2 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).

AB - On any compact Riemannian manifold (M,g) of dimension n, the L 2-normalized eigenfunctions φλ satisfy ||φλ||∞≤ Cλ1-n/2 where -Δφλ =λ2 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).

KW - Eigenfunctions estimates

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U2 - 10.1007/s12220-010-9168-6

DO - 10.1007/s12220-010-9168-6

M3 - Article

AN - SCOPUS:79951516184

SN - 1050-6926

VL - 21

SP - 150

EP - 173

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

IS - 1

ER -