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Shortest closed geodesics on three-dimensional isospectral manifolds
Abstract
We prove that the spectrum of a smooth closed three-manifold gives a positive lower bound for the length of its shortest nonconstant closed geodesic. The argument uses three curvature heat coefficients and the first positive singularity of the retarded wave trace. The heat coefficients yield a curvature--length dichotomy: sufficiently large maximum curvature forces a closed geodesic shorter than the curvature scale. We then show that a shortest closed geodesic lying below the conjugate radius produces a singularity of the sine trace. Combining these estimates gives a lower bound depending only on the common spectrum. The compactness theorem of Anderson then implies smooth compactness of the full isospectral family modulo diffeomorphisms.
Related
- Closed geodesics on compact semi-Riemannian manifolds
- Growth of Simple Closed Geodesics in Finite-Volume Hyperbolic Manifolds
- Simple geodesics in closed hyperbolic manifolds
- Elliptic closed geodesics on non-negatively curved Finsler manifolds
- Multiplicity of closed geodesics on odd-dimensional bumpy Finsler compact space form
BibTeX
@misc{li2026shortest,
title = {{Shortest closed geodesics on three-dimensional isospectral manifolds}},
author = {Yuxiang Li and Yunqing Wu and Jie Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28380},
eprint = {2609.28380},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28380},
}
RIS
TY - UNPB
TI - Shortest closed geodesics on three-dimensional isospectral manifolds
AU - Yuxiang Li
AU - Yunqing Wu
AU - Jie Zhou
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28380
UR - https://arxiv.org/abs/2609.28380
AB - We prove that the spectrum of a smooth closed three-manifold gives a positive lower bound for the length of its shortest nonconstant closed geodesic. The argument uses three curvature heat coefficients and the first positive singularity of the retarded wave trace. The heat coefficients yield a curvature--length dichotomy: sufficiently large maximum curvature forces a closed geodesic shorter than the curvature scale. We then show that a shortest closed geodesic lying below the conjugate radius produces a singularity of the sine trace. Combining these estimates gives a lower bound depending only on the common spectrum. The compactness theorem of Anderson then implies smooth compactness of the full isospectral family modulo diffeomorphisms.
ER -