The Ergodicity of Geodesic Flows on Rank One Manifolds of Nonpositive Curvature
Abstract
In this paper, we study the ergodicity of geodesic flows on closed rank-one manifolds of nonpositive sectional curvature. We introduce the infinite-order vanishing set of the Gaussian curvature in dimension two and of the fiberwise second moment of the reduced Jacobi determinant in higher dimensions. We bound the Liouville measure of the singular set in terms of the volume of this subset and, for surfaces, obtain a corresponding Hausdorff dimension bound. In particular, if this subset has zero volume, then the singular set has zero Liouville measure, and the geodesic flow is ergodic. We also give a geometric criterion for finitely many exceptional regions whose fundamental groups have virtually Abelian images of rank smaller than the dimension of the manifold. Under strict negativity of sectional curvature outside these regions, we obtain a Hausdorff dimension bound for the singular set and prove that the geodesic flow is ergodic with respect to Liouville measure. By extending this method to real-analytic metrics, we characterize the singular set as the zero set of a nontrivial real-analytic determinant constructed from curvature operators, thereby establishing ergodicity in this setting without further assumptions.
Related
- Measure of Maximal Entropy for Geodesic Flows on Noncompact Manifolds of Nonpositive Curvature
- Ergodicity of the geodesic flow on surfaces of nonpositive curvature
- Ergodicity of geodesic flows on Aravinda-Farrell twisted doubles
- Inhomogeneous Strichartz estimates on manifolds with nonpositive curvature and applications
- Strong laws, random monotone vector fields and gradient flows on metric spaces of nonpositive curvature
BibTeX
@misc{liu2026ergodicity,
title = {{The Ergodicity of Geodesic Flows on Rank One Manifolds of Nonpositive Curvature}},
author = {Fei Liu and Xiaokai Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.12381},
eprint = {2609.12381},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12381},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Ergodicity of Geodesic Flows on Rank One Manifolds of Nonpositive Curvature
AU - Fei Liu
AU - Xiaokai Liu
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2609.12381
UR - https://arxiv.org/abs/2609.12381
AB - In this paper, we study the ergodicity of geodesic flows on closed rank-one manifolds of nonpositive sectional curvature. We introduce the infinite-order vanishing set of the Gaussian curvature in dimension two and of the fiberwise second moment of the reduced Jacobi determinant in higher dimensions. We bound the Liouville measure of the singular set in terms of the volume of this subset and, for surfaces, obtain a corresponding Hausdorff dimension bound. In particular, if this subset has zero volume, then the singular set has zero Liouville measure, and the geodesic flow is ergodic. We also give a geometric criterion for finitely many exceptional regions whose fundamental groups have virtually Abelian images of rank smaller than the dimension of the manifold. Under strict negativity of sectional curvature outside these regions, we obtain a Hausdorff dimension bound for the singular set and prove that the geodesic flow is ergodic with respect to Liouville measure. By extending this method to real-analytic metrics, we characterize the singular set as the zero set of a nontrivial real-analytic determinant constructed from curvature operators, thereby establishing ergodicity in this setting without further assumptions.
ER -