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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 $C^\infty$ is finite.
Colin Guillarmou 1 ; Thibault Lefeuvre 1
@article{10_4007_annals_2019_190_1_6,
author = {Colin Guillarmou and Thibault Lefeuvre},
title = {The marked length spectrum of {Anosov} manifolds},
journal = {Annals of mathematics},
pages = {321--344},
publisher = {mathdoc},
volume = {190},
number = {1},
year = {2019},
doi = {10.4007/annals.2019.190.1.6},
mrnumber = {3990606},
zbl = {07097501},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2019.190.1.6/}
}
TY - JOUR AU - Colin Guillarmou AU - Thibault Lefeuvre TI - The marked length spectrum of Anosov manifolds JO - Annals of mathematics PY - 2019 SP - 321 EP - 344 VL - 190 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2019.190.1.6/ DO - 10.4007/annals.2019.190.1.6 LA - en ID - 10_4007_annals_2019_190_1_6 ER -
%0 Journal Article %A Colin Guillarmou %A Thibault Lefeuvre %T The marked length spectrum of Anosov manifolds %J Annals of mathematics %D 2019 %P 321-344 %V 190 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2019.190.1.6/ %R 10.4007/annals.2019.190.1.6 %G en %F 10_4007_annals_2019_190_1_6
Colin Guillarmou; Thibault Lefeuvre. The marked length spectrum of Anosov manifolds. Annals of mathematics, Tome 190 (2019) no. 1, pp. 321-344. doi: 10.4007/annals.2019.190.1.6
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