Long-Time Behavior of the α-Ricci Flow on the 2-Torus
Abstract
We study the α-Ricci flow $(g(t),α(t))$, which is a Ricci flow coupled with a harmonic 1-form heat flow, on the 2-torus M. We prove global existence for this flow as well as a Type III curvature estimate. The long-time behavior of the α-Ricci flow depends on the cohomology class [α] ∈ H1(M,ℝ), which is invariant under the flow. If [α] is rational, then as t→∞, (M,t−1g(t)) Gromov--Hausdorff converges to S1 with length ℓ([α])/√2 where ℓ([α]) is the primitive length of [α]. On the universal cover, the rescaled flow (s−1g̃(st),α̃(st)) smoothly converges to an expander (g*(t),α*(t)) on ℝ2 as s→ ∞ where α*(t) is nonzero and constant in spacetime and g*(t) is flat and expanding in the α* direction. If [α] is irrational, then there is a sequence si→∞ such that (M,si−1g(sit)) Gromov--Hausdorff converges to a point. Moreover, on the universal cover, (si−1g̃(sit),α̃(sit)) smoothly converges to (g*(t),α*(t)).
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BibTeX
@misc{guo2026longtime,
title = {{Long-Time Behavior of the $α$-Ricci Flow on the 2-Torus}},
author = {Yifan Guo and Xiaokang Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19273},
eprint = {2610.19273},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19273},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Long-Time Behavior of the α-Ricci Flow on the 2-Torus
AU - Yifan Guo
AU - Xiaokang Wang
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.19273
UR - https://arxiv.org/abs/2610.19273
AB - We study the α-Ricci flow $(g(t),α(t))$, which is a Ricci flow coupled with a harmonic 1-form heat flow, on the 2-torus M. We prove global existence for this flow as well as a Type III curvature estimate. The long-time behavior of the α-Ricci flow depends on the cohomology class [α] ∈ H^1(M,ℝ), which is invariant under the flow. If [α] is rational, then as t→∞, (M,t^−1g(t)) Gromov--Hausdorff converges to S^1 with length ℓ([α])/√2 where ℓ([α]) is the primitive length of [α]. On the universal cover, the rescaled flow (s^−1g̃(st),α̃(st)) smoothly converges to an expander (g^*(t),α^*(t)) on ℝ^2 as s→ ∞ where α^*(t) is nonzero and constant in spacetime and g^*(t) is flat and expanding in the α^* direction. If [α] is irrational, then there is a sequence s_i→∞ such that (M,s_i^−1g(s_it)) Gromov--Hausdorff converges to a point. Moreover, on the universal cover, (s_i^−1g̃(s_it),α̃(s_it)) smoothly converges to (g^*(t),α^*(t)).
ER -