关于黎曼流形上拟模的爆破
About the blowup of quasimodes on Riemannian manifolds
摘要：
在任何维数为$n$的紧致黎曼流形$(M, g)$上，$L^2$归一化的本征函数${ϕ_λ}$满足$||ϕ_λ||_{\infty} \leq C λ^{\frac{n-1}{2}}$其中$-Δϕ_λ = λ^2 ϕ_λ.$由于它是由标准$n$ -球面$S^n$上的带球面调和函数得到的，所以在所有$(M, g)$的类中边界是尖锐的.当然，对于许多黎曼流形，例如平坦环面$\R^n/Γ$，它不是尖锐的。我们说$S^n$，而不是$\R^n/Γ$，是一个具有最大本征函数增长的黎曼流形。本文的问题是确定具有最大特征函数增长的$(M, g)$。在较早的工作中，我们两个人表明，这样的$(M, g)$必须有一个点$x$的集合${\mathcal L}_x$的测地线循环在$x$有积极的措施，在$S^*_x M$。我们加强了这一结果，在这里，这样的流形必须有一个点，其中集合${\mathcal R}_x$的回归方向的测地线流通过x满足$|{\mathcal R}_x|>0$。我们还表明，如果没有这样的点，$L^2$规范化的准模有sup-norms是$o(λ^{n-1)/2})$，并在另一个极端，我们表明，如果有一个点吹$x$在第一个返回映射的流量是身份，那么有一个序列的准模与$L^\infty$ -规范是$Ω(λ^{(n-1)/2})$。
英文摘要：
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 $Ω(λ^{(n-1)/2})$.
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