Topics · guides to every idea

Ricci flow topics

Every research topic around Ricci flow and the Poincaré conjecture, and the wider geometric flows beside them, has a short guide: a refresher, then the idea in depth, then the foundational papers and surveys to read, then the newest papers from arXiv.

The 31 key concepts are filed under the topic they belong to, from the flow equation and its solitons to the 𝓕 and 𝓦 functionals, κ-noncollapsing and surgery. Each states one idea in a few lines with the equation that carries it, and links to the stage of the Guidebook where it is learned. Concepts marked advanced are the notions recent papers on the Papers page work with every day; knowing them makes those abstracts readable.

New to the subject? Read these in order:
  1. 1 Ricci flow
  2. 2 Ricci solitons
  3. 3 Singularities
  4. 4 Perelman entropy
  5. 5 Surgery
  6. 6 Poincaré conjecture

Flows

12 topics
Ricci flow Hamilton's evolution of a Riemannian metric by its Ricci curvature: ∂g/∂t = −2 Ric. Guide · 16 key references · 508 papers
Mean curvature flow Wider flows The extrinsic cousin of Ricci flow: hypersurfaces moving by their mean curvature. Guide · 16 key references · 449 papers
Other curvature flows Wider flows Gauss curvature, inverse mean curvature, Calabi and other flows that deform a shape or metric by its curvature. Guide · 7 key references · 236 papers
Kähler–Ricci flow Ricci flow restricted to Kähler metrics; reduces to a scalar parabolic Monge–Ampère equation. Guide · 7 key references · 83 papers
Generalized & G₂ flows Generalized Ricci flow, pluriclosed flow, Laplacian/G₂ flows and other non-Kähler relatives. Guide · 6 key references · 79 papers
Volume-preserving & constrained flows Wider flows Curvature flows with a global or local constraint, such as fixed volume or area, that make spheres attractive and prove inequalities. Guide · 7 key references · 78 papers
Curve shortening Wider flows The 1-dimensional model: curves moving by curvature become round. Guide · 6 key references · 68 papers
Inverse mean curvature flow Wider flows Hypersurfaces expanding at speed 1/H, and other inverse curvature flows: the flow behind the Riemannian Penrose inequality. Guide · 8 key references · 66 papers
Gauss curvature flow Wider flows Convex hypersurfaces moving by powers of their Gauss curvature: worn stones and Minkowski problems. Guide · 10 key references · 46 papers
Anisotropic & nonlocal flows Wider flows Curvature flows whose speed depends on direction or on the whole surface: anisotropic, crystalline and fractional mean curvature flow. Guide · 6 key references · 44 papers
Lagrangian mean curvature flow Wider flows Mean curvature flow of Lagrangian submanifolds in Calabi–Yau manifolds, and the hunt for special Lagrangians. Guide · 8 key references · 38 papers
Yamabe flow Wider flows Conformal flow driven by scalar curvature. Guide · 14 key references · 28 papers

Geometry & analysis

17 topics
Stability & uniqueness When is a flow or soliton stable, rigid, or unique? Guide · 7 key references · 350 papers
Kähler geometry Complex geometry: Fano manifolds, Monge–Ampère equations, canonical metrics. Guide · 16 key references · 276 papers
Boundaries & non-compact Flows on manifolds with boundary or infinite extent. Guide · 7 key references · 225 papers
Long-time existence & convergence When a flow exists for all time, how to extend it, and what it converges to. Guide · 11 key references · 214 papers
Scalar curvature The simplest curvature invariant — and a key monotone quantity along the flow. Guide · 7 key references · 173 papers
Weak & singular flows Flows from rough initial data, through singularities, or in synthetic settings: singular Ricci flows, Brakke and level-set flows. Guide · 11 key references · 135 papers
Homogeneous spaces Symmetric settings where the flow becomes an ODE. Guide · 8 key references · 131 papers
Limit spaces & RCD Compactness, Gromov–Hausdorff limits, optimal transport and synthetic Ricci bounds. Guide · 8 key references · 116 papers
Rotational symmetry & warped products Symmetric settings where the flow reduces to a PDE in one space variable: the testing ground for neckpinches. Guide · 8 key references · 80 papers
Kähler–Einstein & K-stability When a Fano manifold carries a Kähler–Einstein metric (the Yau–Tian–Donaldson answer), cscK metrics and the Calabi flow. Guide · 13 key references · 78 papers
Eigenvalues & spectrum Eigenvalues of the Laplacian and of Perelman’s operator −4Δ + R along the flow, and the spectra that decide stability. Guide · 8 key references · 66 papers
Free boundary & capillary Flows whose boundary slides on a support surface, meeting it orthogonally or at a prescribed angle, and other free-boundary problems. Guide · 7 key references · 64 papers
Geometric inequalities Isoperimetric, Alexandrov–Fenchel, Minkowski and Penrose inequalities, many of them proved by running a flow. Guide · 8 key references · 61 papers
Harmonic maps & DeTurck trick Short-time existence through the DeTurck trick, and the harmonic map heat flow behind it. Guide · 8 key references · 55 papers
Sasakian & contact geometry The odd-dimensional cousin of Kähler geometry: Sasaki–Ricci flow, Sasaki–Einstein metrics and solitons on contact manifolds. Guide · 7 key references · 47 papers
Minimal & CMC surfaces Critical points of area: the fixed points of mean curvature flow, and a tool for finite extinction in Ricci flow. Guide · 7 key references · 47 papers
Analytic minimal model program Kähler–Ricci flow as a route through birational geometry: contractions, flips and canonical models. Guide · 9 key references · 41 papers

Where the concept cards came from: they were drafted by an AI at the site owner's request. The owner, who is still learning this material, chose which concepts to include but cannot vouch for every detail, so treat them as a well-drawn map and check the cited papers. Corrections are welcome: see how this site works.