Prevalence of sparsity for negatively curved metrics
Abstract
Let M be a closed manifold of arbitrary dimension. The length spectrum of a negatively curved metric g on M is the set of lengths of all possible closed geodesics of g. We prove that metrics whose length spectrum is exponentially sparse, i.e., the gaps between distinct lengths are bounded below by a quantity decaying exponentially in the length, are prevalent, and hence dense, among Ck-smooth negatively curved metrics on M for sufficiently large k. Moreover, unlike in all known constructions, the exponent in our bound grows only sublinearly in the regularity k. This work is motivated by our recent result on prevalent unmarked length spectral rigidity for expanding circle maps and a very recent result by DeWitt, Durham, Reber, and O'Hare that Ck-smooth negatively curved metrics with exponentially sparse spectra are locally rigid.
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BibTeX
@misc{drach2026prevalence,
title = {{Prevalence of sparsity for negatively curved metrics}},
author = {Kostiantyn Drach and Vadim Kaloshin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12351},
eprint = {2610.12351},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12351},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Prevalence of sparsity for negatively curved metrics
AU - Kostiantyn Drach
AU - Vadim Kaloshin
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12351
UR - https://arxiv.org/abs/2610.12351
AB - Let M be a closed manifold of arbitrary dimension. The length spectrum of a negatively curved metric g on M is the set of lengths of all possible closed geodesics of g. We prove that metrics whose length spectrum is exponentially sparse, i.e., the gaps between distinct lengths are bounded below by a quantity decaying exponentially in the length, are prevalent, and hence dense, among C^k-smooth negatively curved metrics on M for sufficiently large k. Moreover, unlike in all known constructions, the exponent in our bound grows only sublinearly in the regularity k. This work is motivated by our recent result on prevalent unmarked length spectral rigidity for expanding circle maps and a very recent result by DeWitt, Durham, Reber, and O'Hare that C^k-smooth negatively curved metrics with exponentially sparse spectra are locally rigid.
ER -