Paper · Aug 30, 2026

Under Ricci flow, a 3-torus goes flat

John Lott

math.DGarXiv:2608.30027v3
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Abstract

We determine the long-time behavior of the canonical Ricci flow on a closed three-manifold of Thurston type , Nil, or Sol, starting from an arbitrary initial metric. In the Euclidean case, the metric converges exponentially fast to a flat metric. The Gromov–Hausdorff limit of 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, , converge, in the pointed Cheeger–Hamilton sense, to an explicit homogeneous expanding Ricci soliton solution.

Comments: minor changes

Ricci flowRicci solitonsGeometrization3-manifoldsLimit spaces & RCDHomogeneous spacesLong-time existence & convergence

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  1. Long-Time Behavior of the -Ricci Flow on the 2-Torus

    Yifan Guo and Xiaokang Wang PDF

    Topics Limit spaces & RCD, Long-time existence & convergence, Ricci solitons · Shared terms “time behavior”, “universal cover”, “Hausdorff”

  2. Long-time behavior of Ricci flow on some complex surfaces

    John Lott PDF

    Topics Long-time existence & convergence · Same author John Lott · Shared terms “time behavior”

  3. Long-time behavior of awesome homogeneous Ricci flows

    Roberto Araujo PDF

    Topics Homogeneous spaces, Long-time existence & convergence · Shared terms “time behavior”, “universal cover”

  4. Large-scale geometry of the Universe

    Yassir Awwad and Tomislav Prokopec PDF

    Topics Geometrization, 3-manifolds, Homogeneous spaces · Shared terms “Thurston”

  5. On the long time behavior of ancient homogeneous Ricci flows

    Anusha M. Krishnan and Francesco Pediconi PDF

    Topics Limit spaces & RCD, Homogeneous spaces, Long-time existence & convergence · Shared terms “time behavior”

  6. On Ricci Solitons and Harmonic Vector Fields in the Thurston Geometry

    Halima Boukhari, Hadjer Okbani and Ahmed Mohammed Cherif PDF

    Topics Geometrization, Homogeneous spaces, Ricci solitons · Shared terms “Thurston”

  7. Geometrisation of 3-manifolds

    Bruno Martelli PDF

    Topics Geometrization, 3-manifolds · Shared terms “Thurston”

  8. A Lean Formalization of the Hamilton–Perelman Proof of Thurston's Geometrization Conjecture

    Yuan Liao, Ziyang Qin, Jui-Hui Chung et al. PDF

    Topics Geometrization · Shared terms “Thurston”, “Hamilton”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Geometrization Thurston's conjecture: every closed 3-manifold can be cut along spheres and tori into pieces that each carry one of eight model geometries. Start with: Three dimensional manifolds, Kleinian groups and hyperbolic geometry (Thurston, 1982) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  2. 10A.8
    Thurston’s Eight GeometriesBook 10A · Three-Manifolds and Geometrization

    The model geometries, the geometrization conjecture, and why it implies Poincaré.

  3. 10A.9
    Hyperbolic Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    Ideal tetrahedra, Mostow rigidity, and the thick–thin decomposition.