{
  "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 $[α] \\in H^1(M,\\mathbb{R})$, which is invariant under the flow. If $[α]$ is rational, then as $t\\to\\infty$, $(M,t^{-1}g(t))$ Gromov--Hausdorff converges to $S^1$ with length $\\ell([α])/\\sqrt 2$ where $\\ell([α])$ is the primitive length of $[α]$. On the universal cover, the rescaled flow $(s^{-1}\\tilde g(st),\\tildeα(st))$ smoothly converges to an expander $(g^*(t),α^*(t))$ on $\\mathbb{R}^2$ as $s\\to \\infty$ 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 $s_i\\to\\infty$ such that $(M,s_i^{-1}g(s_it))$ Gromov--Hausdorff converges to a point. Moreover, on the universal cover, $(s_i^{-1}\\tilde g(s_it),\\tildeα(s_it))$ smoothly converges to $(g^*(t),α^*(t))$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.19273",
  "author_total": 2,
  "authors": [
    {
      "name": "Yifan Guo"
    },
    {
      "name": "Xiaokang Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aefebacxkv86s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.19273",
  "fulltext_url": null,
  "id": "aefebacxkv86s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.19273",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7222272294",
  "pdf_url": "https://arxiv.org/pdf/2610.19273",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aefebacxkv86s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.19273",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.19273"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Long-Time Behavior of the $α$-Ricci Flow on the 2-Torus",
  "title_text": "Long-Time Behavior of the α-Ricci Flow on the 2-Torus",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/aefebacxkv86s",
  "venue": "arXiv",
  "versions": []
}