The marked length spectrum of Anosov manifolds

• 成果类型: Article
• 署名作者: Guillarmou, Colin; Lefeuvre, Thibault
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2019.190.1.6
• 发表日期: 2019
• 页码: 321-344
• 关键词: invariant distributions; RIGIDITY; SURFACES; geometry; FLOWS

摘要

In all dimensions, we prove that the marked length spectrum of a Riemannian manifold (M, g) with Anosov geodesic flow and non-positive curvature locally determines the metric in the sense that two close enough metrics with the same marked length spectrum are isometric. In addition, we provide a new stability estimate quantifying how the marked length spectrum controls the distance between the isometry classes of metrics. In dimension 2 we obtain similar results for general metrics with Anosov geodesic flows. We also solve locally a rigidity conjecture of Croke relating volume and marked length spectrum for the same category of metrics. Finally, by a compactness argument, we show that the set of negatively curved metrics (up to isometry) with the same marked length spectrum and with curvature in a bounded set of C-infinity is finite.