论文标题
定量标记的长度光谱刚度
Quantitative marked length spectrum rigidity
论文作者
论文摘要
我们考虑了一个封闭的Riemannian歧管$ M $负曲率和尺寸至少3个，其长度光谱与本地对称空间$ n $的长度足够接近（多重）。使用Hamenstädt的方法，我们表明$ M $和$ n $的量大约相等。然后，我们显示Besson-Courtois-Gallot映射$ f：m \ to n $是一种差异性，衍生界限接近1，并取决于两个标记的长度光谱函数的比率。因此，我们完善了Hamenstädt和Besson-Courtois-Gallot的结果，如果它们的标记长度相等，它们显示出$ m $和$ n $是等值的。我们还使用耳al的方法以及由于pugh引起的gromov紧凑型定理的版本，证明了紧凑型负弯曲的表面的相似结果。
We consider a closed Riemannian manifold $M$ of negative curvature and dimension at least 3 with marked length spectrum sufficiently close (multiplicatively) to that of a locally symmetric space $N$. Using the methods of Hamenstädt, we show the volumes of $M$ and $N$ are approximately equal. We then show the Besson-Courtois-Gallot map $F: M \to N$ is a diffeomorphism with derivative bounds close to 1 and depending on the ratio of the two marked length spectrum functions. Thus, we refine the results of Hamenstädt and Besson-Courtois-Gallot, which show $M$ and $N$ are isometric if their marked length spectra are equal. We also prove a similar result for compact negatively curved surfaces using the methods of Otal together with a version of the Gromov compactness theorem due to Pugh.
