About the Blowup of Quasimodes on Riemannian Manifolds

作者：Sogge Christopher D^*; Toth John A; Zelditch Steve

来源：Journal of Geometric Analysis, 2011, 21(1): 150-173.

DOI：10.1007/s12220-010-9168-6

• Eigenfunctions estimates

On any compact Riemannian manifold (M, g) of dimension n, the L(2)-normalized eigenfunctions phi(lambda) satisfy parallel to phi(lambda)parallel to = C lambda(n-1)/2 where -Delta phi(lambda) =lambda(2)phi(lambda). The bound is sharp in the class of all (M, g) since it is obtained by zonal spherical harmonics on the standard n-sphere S(n). But of course, it is not sharp for many Riemannian manifolds, e. g., flat tori R(n)/Gamma. We say that Sn, but not R(n)/Gamma, 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 set L(x) of geodesic loops at x has positive measure in S(x)*M. We strengthen this result here by showing that such a manifold must have a point where the set R(x) of recurrent directions for the geodesic flow through x satisfies vertical bar R(x)vertical bar > 0. We also show that if there are no such points, L(2)-normalized quasimodes have sup-norms that are o(lambda((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(infinity)-norms that are Omega (lambda((n-1)/2)).

• 出版日期2011-1