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Under Ricci flow, a 3-torus goes flat
Abstract
We determine the long-time behavior of the canonical Ricci flow on a closed three-manifold of Thurston type ℝ3, Nil, or Sol, starting from an arbitrary initial metric. In the Euclidean case, the metric g(t) converges exponentially fast to a flat metric. The Gromov--Hausdorff limit of (M,t−1g(t)) is a point in the Euclidean and Nil cases, and a metric circle or a compact interval of positive length in the Sol case. In all three cases, the blowdown flows lifted to the universal cover, s−1g̃(sτ), converge, in the pointed Cheeger--Hamilton sense, to an explicit homogeneous expanding Ricci soliton solution.
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BibTeX
@misc{lott2026under,
title = {{Under Ricci flow, a 3-torus goes flat}},
author = {John Lott},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.30027},
eprint = {2608.30027},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30027},
}
RIS
TY - UNPB
TI - Under Ricci flow, a 3-torus goes flat
AU - John Lott
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2608.30027
UR - https://arxiv.org/abs/2608.30027
AB - We determine the long-time behavior of the canonical Ricci flow on a closed three-manifold of Thurston type ℝ^3, Nil, or Sol, starting from an arbitrary initial metric. In the Euclidean case, the metric g(t) converges exponentially fast to a flat metric. The Gromov--Hausdorff limit of (M,t^−1g(t)) is a point in the Euclidean and Nil cases, and a metric circle or a compact interval of positive length in the Sol case. In all three cases, the blowdown flows lifted to the universal cover, s^−1g̃(sτ), converge, in the pointed Cheeger--Hamilton sense, to an explicit homogeneous expanding Ricci soliton solution.
ER -