About the blowup of quasimodes on Riemannian manifolds

Abstract

On any compact Riemannian manifold $(M,g)$ of dimension $n$, the $L^2$-normalized eigenfunctions ${\varphi }_{\lambda }$ satisfy $||{\varphi }_{\lambda }|{|}_{\infty }\le C{\lambda }^{\frac{n-1}{2}}$ where $-\Delta {\varphi }_{\lambda }={\lambda }^2{\varphi }_{\lambda }.$ 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 ${\mathbb{R}}^n/\Gamma$. We say that $S^n$, but not ${\mathbb{R}}^n/\Gamma$, 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 ${\mathcal{L}}_x$ of geodesic loops at $x$ has positive measure in $S_x^{\ast }M$. We strengthen this result here by showing that such a manifold must have a point where the set ${\mathcal{R}}_x$ of recurrent directions for the geodesic flow through x satisfies $|{\mathcal{R}}_x|>0$. 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 quasi-modes with $L^{\infty }$-norms that are $\Omega ({\lambda }^{(n-1)/2})$.

20 pages

References in corpus (1)

• L^p norms of eigenfunctions in the completely integrable case

Cited by in corpus (1)

• An improvement on eigenfunction restriction estimates for compact boundaryless Riemannian manifolds with nonpositive sectional curvature