负曲率度量的稀疏性的普遍性
Prevalence of sparsity for negatively curved metrics
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中文总结 AI 辅助
该研究证明，当$k$足够大时，$M$上$C^k$光滑负曲率度量中，长度谱呈指数稀疏的度量普遍存在且稠密，且其界的指数仅随正则性$k$次线性增长，相关结果受前期研究动机驱动。
中文摘要 AI 辅助
设$M$为任意维数的闭流形，$M$上负曲率度量$g$的长度谱是$g$所有闭测地线的长度构成的集合。我们证明，长度谱呈指数稀疏（即不同长度间的间隙被一个随长度指数衰减的量下界约束）的度量，在$M$上$C^k$光滑负曲率度量中是普遍存在的，因此当$k$足够大时是稠密的。此外，与所有已知构造不同，我们的界中的指数仅随正则性$k$次线性增长。本研究的动机源于我们近期关于扩张圆映射的普遍未标记长度谱刚性的结果，以及DeWitt、Durham、Reber和O'Hare的最新结果：具有指数稀疏谱的$C^k$光滑负曲率度量是局部刚性的。
英文摘要
Let $M$ be a closed manifold of arbitrary dimension. The length spectrum of a negatively curved metric $g$ on $M$ is the set of lengths of all possible closed geodesics of $g$. We prove that metrics whose length spectrum is exponentially sparse, i.e., the gaps between distinct lengths are bounded below by a quantity decaying exponentially in the length, are prevalent, and hence dense, among $C^k$-smooth negatively curved metrics on $M$ for sufficiently large $k$. Moreover, unlike in all known constructions, the exponent in our bound grows only sublinearly in the regularity $k$. This work is motivated by our recent result on prevalent unmarked length spectral rigidity for expanding circle maps and a very recent result by DeWitt, Durham, Reber, and O'Hare that $C^k$-smooth negatively curved metrics with exponentially sparse spectra are locally rigid.
发表机构
- Universitat de Barcelona（巴塞罗那大学）
- Centre de Recerca Matemàtica（数学研究中心）
- Institute of Science and Technology Austria（奥地利科学技术研究所）
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