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Ricci flow and the Poincaré conjecture

a home for everyone drawn to the Poincaré conjecture, Ricci flow and the geometric flows around them

Papers checked every weekday · newest added November 6, 2026 · kept by N. Pinch

∂g∂t=−2 Ric⁡(g)(1)\frac{\partial g}{\partial t} = -2\,\operatorname{Ric}(g) \tag{1}

Abstract

Neckpinch is a meeting point for everyone interested in Hamilton's Ricci flow (1), Perelman's proof of the Poincaré conjecture and geometrization, and the geometric flows that share their methods: researchers keeping up with the arXiv, students working toward the subject, teachers, and enthusiasts who want to understand what Perelman did. It follows new research as it appears (1,480 papers so far, checked every weekday and tagged by topic), lists the meetings where the community gathers, and collects 55 topic guides with checked references, the people working in the field, and a reading path from first-year calculus to Perelman's preprints with a guidebook written for it. Corrections and suggestions are welcome and are acted on: write in.

Keywords: Ricci flow, Poincaré conjecture, Perelman, geometrization, Ricci solitons, singularities and surgery, mean curvature flow, curve-shortening flow, Yamabe flow.

MSC2020: 53E20 (Ricci flows), 53E10 (flows related to mean curvature), 57K30 (general topology of 3-manifolds).

Figure 1. Curve-shortening flow, the one-dimensional cousin of (1), rescaled to keep the enclosed area A fixed. Every embedded closed curve becomes round (Gage–Hamilton 1986, Grayson 1987), so the isoperimetric ratio 4πA/L² tends to 1. Ricci flow does the analogous thing to a metric: on a closed surface the normalised flow converges to constant curvature (Hamilton 1988, Chow 1991). Drag on the figure to draw a curve of your own.

New Ricci flow papers on arXiv

All 1,480 papers

The newest papers in the index, cited the way you would cite them. 17 appeared in the last seven days. Titles open the PDF, names open that author's papers; the full list has abstracts, topic filters and BibTeX.

  1. Yuxing Deng and Guohao Wang, Classification of cohomogeneity one steady gradient Ricci solitons in dimension four and five, arXiv:2611.08206 [math.DG], Nov 5, 2026.
  2. Di Zhang, On the anisotropic mean curvature flow in the Gaussian space, arXiv:2611.07791 [math.DG], Nov 5, 2026. Wider flows
  3. Jianyu Ou and Guoqiang Wu, Scalar curvature bounds and rigidity of Ricci shrinkers, arXiv:2611.07381 [math.DG], Nov 5, 2026.
  4. Laura Flower, Man-Chun Lee and Felix Schulze, Uniqueness of Ricci flows from singular initial data, arXiv:2611.06814 [math.DG], Nov 4, 2026.
  5. Reto Buzano and Shengwen Wang, Perelman's entropy and Gromov-Hausdorff versus dynamical stability of round spheres, arXiv:2611.06678 [math.DG], Nov 4, 2026.
  6. Yasemin Karababa, Rigidity of Ricci solitons for -related metallic semi-Riemannian structures on para-Kenmotsu manifolds, arXiv:2611.06535 [math.DG], Nov 4, 2026.
  7. Maxwell Stolarski and Wei-Bo Su, Existence of Lagrangian Mean Curvature Flow Singularities with Bounded Mean Curvature, arXiv:2611.06248 [math.DG], Nov 4, 2026. Wider flows
  8. Chandan Kumar Mondal, Conjugate radius and parabolicity along the Ricci flow, arXiv:2611.03823 [math.DG], Nov 2, 2026.

Meetings and seminars

Calendar, seminars and recordings

Conferences, workshops and schools on Ricci flow and the wider geometric flows, checked against their official pages.

DatesMeetingWhereFocus
Nov 7–8, 2026AMS Special Session on Geometric AnalysisArizona, USARelated
Jan 4–8, 2027Geometric Analysis Days (JAG 2027)Luminy (Marseille), FranceRelated
Jan 12, 2027AMS Special Session on Recent Trends in Metric Geometry and TopologyIllinois, USA (JMM 2027)Related
Jan 13, 2027AMS Special Session on Geometric Inequalities, Convergence, and Scalar CurvatureIllinois, USA (JMM 2027)Related
Jan 14, 2027AMS Special Session on Recent Advances on Geometric Variational Problems and CurvatureIllinois, USA (JMM 2027)Related
Feb 22 – Mar 5, 2027Local and Nonlocal Minimal Surfaces and Geometric FlowsMilan, ItalyRelated

9 more meetings on the calendar.

Topic guides

All 55 guides

Short guides to the ideas behind the proof. Each states the idea in a few lines, then lists the foundational papers in reading order and the newest work. Start with these six, in this order.

Ricci flow 508 papers
Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads…
Ricci solitons 418 papers
Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons.
Singularities 309 papers
On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off.
Perelman entropy 34 papers
Perelman found that Ricci flow is a gradient flow once a measure is fixed. His 𝓕-functional and 𝓦-entropy increase along the flow; a lower bound…
Surgery 26 papers
Stop the flow just before a neck pinches, cut along the thin neck, glue standard caps onto the two ends, and restart.
Poincaré conjecture 8 papers
Every closed, simply connected 3-manifold is homeomorphic to the 3-sphere. Poincaré posed the question in 1904, and Perelman proved it in 2002–03…

The wider geometric flows

Papers, with a Ricci-flow-only switch

Ricci flow has relatives: flows that deform a curve, a hypersurface or a conformal metric by its curvature, and share its tools (maximum principles, monotonicity formulas, blow-up analysis of singularities). Their papers are included and marked Wider flows; 633 of the 1,480 papers are theirs.

Mean curvature flow 449 papers
A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense.
Curve shortening 68 papers
Plane curves moving by their curvature. Convex curves shrink to round points (Gage–Hamilton), and every embedded curve becomes convex first (Grayson).
Yamabe flow 28 papers
The Yamabe flow deforms a metric inside its conformal class at a rate set by scalar curvature. It is a parabolic approach to the Yamabe problem…
Other curvature flows 236 papers
Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose…
Inverse mean curvature flow 66 papers
A hypersurface expands with normal speed 1/H, the reciprocal of its mean curvature. Huisken and Ilmanen's weak version keeps the Hawking mass…
Gauss curvature flow 46 papers
Convex hypersurfaces shrink with normal speed a power K^α of their Gauss curvature. Firey proposed it as a model of stones worn by waves; for α>…
Lagrangian mean curvature flow 38 papers
In a Calabi–Yau manifold, mean curvature flow keeps Lagrangian submanifolds Lagrangian. Its hoped-for limits are special Lagrangians, the volume…
Volume-preserving & constrained flows 78 papers
Curvature flows with a constraint: subtract the average of the mean curvature and the enclosed volume stays fixed while area decreases, so spheres…
Anisotropic & nonlocal flows 44 papers
When surface energy depends on direction, as in crystals, mean curvature flow becomes anisotropic, and the Wulff shape replaces the round sphere.
Self-shrinkers & translators 122 papers
Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding.

The Guidebook

See every stage

How to learn Ricci flow from scratch: 12 stages, from first-year calculus to Perelman's preprints, each with the books, notes and courses that teach it, and milestones you can tick off in your own browser.

  1. 1Calculus & Linear Algebra
  2. 2Proofs & Real Analysis
  3. 3Measure & Integration
  4. 4Functional Analysis
  5. 5Complex Analysis
  6. 6PDE & the Heat Equation
  7. 7Topology & the Fundamental Group
  8. 8Smooth Manifolds
  9. 9Riemannian Geometry
  10. 103-Manifolds & Geometrization
  11. 11Hamilton's Ricci Flow
  12. 12Perelman & the Proof

Written here: 11 courses and 19 books, one for each stage, built from the axioms with full proofs. All 203 planned chapters are written.

From Poincaré to Perelman

A short history of the Poincaré conjecture: a question asked in 1904, answered with a heat equation for geometry.

1904Poincaré asks the question. In the fifth supplement to Analysis Situs, Henri Poincaré asks whether a closed 3-manifold with trivial fundamental group must be the 3-sphere.
1961Higher dimensions fall first. Stephen Smale proves the analogue in dimensions ≥ 5 using the h-cobordism theorem. Michael Freedman settles dimension 4 in 1982.
1982Thurston and Hamilton. Thurston announces the geometrization conjecture. Hamilton introduces Ricci flow and proves that 3-manifolds with positive Ricci curvature are spherical space forms.
1990sHamilton's program. Harnack inequalities, singularity analysis and the idea of surgery. Hamilton lays out a plan to prove geometrization with Ricci flow, but the singularities remain an obstacle.
2000A Millennium Problem. The Clay Mathematics Institute lists the Poincaré conjecture among its seven Millennium Prize Problems.
2002–03Perelman's three preprints. Grisha Perelman posts three papers to arXiv: the entropy formula, Ricci flow with surgery, and finite extinction time. They complete Hamilton's program.
2006Verification and the Fields Medal. Kleiner–Lott, Morgan–Tian and Cao–Zhu publish detailed expositions. Perelman is awarded the Fields Medal at ICM Madrid and declines it.
2009Beyond dimension three. Brendle and Schoen use Ricci flow to prove the differentiable sphere theorem, showing the method reaches well beyond 3-manifolds.
2010The Millennium Prize. Clay awards Perelman the Millennium Prize. He declines it as well.
2014–20Flows through singularities. Kleiner–Lott construct canonical singular Ricci flows. Bamler–Kleiner prove uniqueness and stability, Brendle classifies 3D ancient solutions, and Bamler develops a structure theory in all dimensions.
2024Richard Hamilton (1943–2024). The inventor of the Ricci flow dies. His flow is now a standard tool of geometric analysis.
TodayYour turn. Hundreds of new papers each year push the flow into higher dimensions, Kähler geometry, physics and beyond. The newest ones are on the Papers page.

Applications

All 27, with their papers

Where this mathematics is used outside pure mathematics, and the unusual work it leads to, each labelled by how established it is and how directly it descends from Ricci flow.

Flattening 3D shapes: game textures, shoe patterns, face matching Demonstrated · From Ricci flow itself
The [uniformization theorem](/topics/surfaces/) says every surface can be deformed, without distorting angles, to have constant curvature.
Image segmentation, film effects and moving interfaces In use · Same mathematics
[Curve shortening](/topics/curve-shortening/) and [mean curvature flow](/topics/mean-curvature-flow/), the extrinsic cousins of Ricci flow, have…
Geodesics on rocket casings and hydrogen tanks In use · From the prerequisites
Pressure vessels such as hydrogen tanks and rocket motor casings are made by **filament winding**, in which a machine lays resin-soaked fibre onto a…

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