Paper detail

About the blowup of quasimodes on Riemannian manifolds

On any compact Riemannian manifold $(M, g)$ of dimension $n$, the $L^2$-normalized eigenfunctions ${ϕ_λ}$ satisfy $||ϕ_λ||_{\infty} \leq C λ^{\frac{n-1}{2}}$ where $-Δϕ_λ = λ^2 ϕ_λ.$ 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 $\R^n/Γ$. We say that $S^n$, but not $\R^n/Γ$, 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 set ${\mathcal L}_x$ 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 ${\mathcal R}_x$ 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 $o(λ^{n-1)/2})$, 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 quasi-modes with $L^\infty$-norms that are $Ω(λ^{(n-1)/2})$.