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The library

Every book, paper, set of notes, course, video and reference page that this site recommends anywhere — in the topic guides, on the Path, in the guidebook, on the People and Applications pages — in one place. Each one shows where it is recommended and which other resources it connects to, so you can see how the reading fits together rather than one link at a time.

792resources

1,283recommendations

8reading trails

The backbone

Where the most-used resources are recommended

The 24 books, papers and notes recommended in the most places, and where along the way you meet them: the twelve Path stages with their guidebook courses, then the rest of the site. A resource that spans many columns is one that ties the subject together. Click a title for its connections.

The 24 resources recommended in the most places, and where each is recommended: Path stages 1 to 12, topic guides, the canon, People and Applications.
Resource	Path stage (and its guidebook course)												Elsewhere
	1	2	3	4	5	6	7	8	9	10	11	12	Guides	Canon	People	Apps
Ricci Flow and the Poincare ConjectureMorgan–Tian 2006									4	4	12	18	4	1	2
Notes on Perelman's papersKleiner–Lott 2006						1				1	1	28	6	1	2
The entropy formula for the Ricci flow and its geometric…Perelman 2002						2					1	15	9	1
Getting a feel for Ricci flow: Two dimensionsTopping					1	2		1	4		15	3
Ricci flow with surgery on three-manifoldsPerelman 2003											1	12	4	1
Recent developments in Ricci flowsBamler 2021												1	7	1	1
Basic ConstructionsHatcher							12			1
Finite extinction time for the solutions to the Ricci…Perelman 2003										1		4	2	1
The Ricci Flow: An IntroductionChow–Knopf 2004											3		3		2
Completion of the Proof of the Geometrization ConjectureMorgan–Tian 2008												3	1	1	2
Extrinsic Geometric FlowsAndrews et al. 2020													6		1
Vector SpacesAxler	9
Generic mean curvature flow I; generic singularitiesColding–Minicozzi 2009						1							4		1
Prologue: The problem of measureTao			8
Three-manifolds with positive Ricci curvatureHamilton 1982													6
Estimates for the extinction time for the Ricci flow on…Colding–Minicozzi 2003										1		1	2		1
Hamilton’s Ricci FlowChow et al. 2006											2		3
Lecture notes on the Kähler-Ricci flowSong–Weinkove 2012													4		1
Partial Differential EquationsJost						5
Singular Ricci flows IKleiner–Lott 2014													3	1	1
Structure theory of non-collapsed limits of Ricci flowsBamler 2020													3	1	1
Chapter 10: Mathematical InductionHammack		6
Deformation of Kähler matrics to Kähler-Einstein metrics…Cao 1985													4
Hamilton-Perelman's Proof of the Poincaré Conjecture and…Cao–Zhu 2006												1	1	1	1

Read in order

Reading trails

Short routes through resources that belong together, in the order to read them. Each step links to its entry below, with everything connected to it.

Perelman's proof and the expositions that checked it

Perelman posted three preprints in 2002–03 and never published them in a journal. Teams of mathematicians then wrote out every step in full, and those expositions are how most people read the proof today. Read the preprints first for the ideas, then one exposition beside them. The Poincaré conjecture and geometrization guides give the background.

Paper
The entropy formula for the Ricci flow and its geometric applications Open The entropy formula for the Ricci flow and its geometric applicationsGrisha Perelman · 2002Perelman I: the entropy, no local collapsing and the canonical neighbourhood picture.
Paper
Ricci flow with surgery on three-manifolds Open Ricci flow with surgery on three-manifoldsGrisha Perelman · 2003Perelman II: Ricci flow with surgery on 3-manifolds, the engine of the geometrization proof.
Paper
Finite extinction time for the solutions to the Ricci flow on certain three-manifolds Open Finite extinction time for the solutions to the Ricci flow on certain three-manifoldsGrisha Perelman · 2003Perelman III: on simply connected manifolds the flow goes extinct in finite time, a shortcut to the Poincaré conjecture.
Notes
Notes on Perelman's papers Open Notes on Perelman's papersBruce Kleiner and John Lott · 2006Kleiner–Lott: line-by-line notes on the preprints, the standard companion. Read them side by side, section by section.
Book
Ricci Flow and the Poincare Conjecture Open Ricci Flow and the Poincare ConjectureJohn W. Morgan and Gang Tian · 2006Morgan–Tian: a complete, book-length proof of the Poincaré conjecture.
Paper
Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture Open Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization ConjectureHuai-Dong Cao and Xi-Ping Zhu · 2006Cao–Zhu: an independent exposition of the proof of geometrization, useful when a step in one account won't click.
Paper
Completion of the Proof of the Geometrization Conjecture Open Completion of the Proof of the Geometrization ConjectureJohn Morgan and Gang Tian · 2008Morgan–Tian: completing the proof of Thurston's geometrization conjecture.
Book
Geometrisation of 3-Manifolds Open Geometrisation of 3-ManifoldsLaurent Bessières, Gérard Besson, Michel Boileau et al. · 2010Bessières, Besson, Boileau, Maillot and Porti: a book-length account of geometrization.

Hamilton's programme, 1982–1999

Before Perelman, Hamilton spent two decades building the theory and laying out the plan that Perelman completed. Read in order, these papers show the subject being invented: the flow, its curvature estimates, its singularities, and the first surgeries. See the Ricci flow guide for the context.

Paper
Three-manifolds with positive Ricci curvature Open Three-manifolds with positive Ricci curvatureRichard S. Hamilton · 1982The paper that started the subject: the flow is introduced, and 3-manifolds with positive Ricci curvature become round.
Paper
Four-manifolds with positive curvature operator Open Four-manifolds with positive curvature operatorRichard S. Hamilton · 1986The same in dimension four, for positive curvature operator.
Paper
The Ricci flow on surfaces Open The Ricci flow on surfacesRichard S. Hamilton · 1988Ricci flow on surfaces, where the cigar soliton first appears.
Paper
The Harnack estimate for the Ricci flow Open The Harnack estimate for the Ricci flowRichard S. Hamilton · 1993The Harnack estimate, which controls how curvature grows.
Paper
The formations of singularities in the Ricci Flow Open The formations of singularities in the Ricci FlowRichard Hamilton · 1993Hamilton's own survey of the singularity problem, and the plan Perelman completed.
Paper
A Compactness Property for Solutions of the Ricci Flow Open A Compactness Property for Solutions of the Ricci FlowRichard S. Hamilton · 1995The compactness theorem that makes blow-up limits possible.
Paper
Four-manifolds with positive isotropic curvature Open Four-manifolds with positive isotropic curvatureRichard S. Hamilton · 1997Four-manifolds with positive isotropic curvature, where Ricci flow with surgery first appears.
Paper
Non-singular solutions of the Ricci flow on three-manifolds Open Non-singular solutions of the Ricci flow on three-manifoldsRichard S. Hamilton · 1999Non-singular solutions on 3-manifolds, and how the geometric pieces appear in the long run.

Learning Ricci flow, from first notes to the frontier

The order the Path recommends for stage 11 and beyond: notes written for newcomers, then the textbook, then Hamilton himself, then the modern courses and surveys that lead to current research.

Notes
Lectures on the Ricci Flow (free notes) Open Lectures on the Ricci Flow (free notes)Peter ToppingTopping: start here. Short, careful notes, written for readers arriving from a first course in Riemannian geometry.
Article
An Illustrated Introduction to the Ricci Flow Open An Illustrated Introduction to the Ricci FlowGabriel Khan · 2022Khan: an illustrated introduction. Pictures first, to build intuition beside the analysis.
Book
The Ricci Flow: An Introduction Open The Ricci Flow: An IntroductionBennett Chow and Dan Knopf · 2004Chow–Knopf: the standard textbook treatment, with full computations.
Paper
Three-manifolds with positive Ricci curvature Open Three-manifolds with positive Ricci curvatureRichard S. Hamilton · 1982Hamilton's first paper. Read it once you can follow Chow–Knopf.
Notes
Ricci flow lecture notes (free PDF) Open Ricci flow lecture notes (free PDF)Richard BamlerBamler: a modern course, from short-time existence up to Perelman's entropy.
Book
Hamilton’s Ricci Flow Open Hamilton’s Ricci FlowBennett Chow, Peng Lu and Lei Ni · 2006Chow–Lu–Ni: the reference to keep at hand, with a full treatment of Harnack estimates.
Notes
285G: Poincaré's conjecture (free lecture notes) Open 285G: Poincaré's conjecture (free lecture notes)Terence TaoTao: a graduate course on Perelman's proof, written as blog posts.
Notes
Recent developments in Ricci flows Open Recent developments in Ricci flowsRichard H Bamler · 2021Bamler: a survey of recent developments, a map of the modern field.

Through singularities: from surgery to canonical flows

Perelman continued the flow past singularities by surgery, which depends on choices. The work that followed replaced surgery by a canonical flow, proved it unique, classified the singularity models in dimension three, and used the result to study spaces of metrics and diffeomorphism groups. The weak and singular flows and Smale conjecture guides tell the story.

Paper
Ricci flow with surgery on three-manifolds Open Ricci flow with surgery on three-manifoldsGrisha Perelman · 2003Perelman II: Ricci flow with surgery, the starting point.
Paper
Singular Ricci flows I Open Singular Ricci flows IBruce Kleiner and John Lott · 2014Kleiner–Lott: singular Ricci flows, a canonical flow through singularities with no surgery parameters.
Paper
Uniqueness and stability of Ricci flow through singularities Open Uniqueness and stability of Ricci flow through singularitiesRichard H. Bamler and Bruce Kleiner · 2017Bamler–Kleiner: singular Ricci flows are unique and stable.
Paper
Ancient solutions to the Ricci flow in dimension 3 Open Ancient solutions to the Ricci flow in dimension 3S. Brendle · 2018Brendle: the classification of ancient κ-solutions in dimension three.
Paper
Ricci flow and diffeomorphism groups of 3-manifolds Open Ricci flow and diffeomorphism groups of 3-manifoldsRichard H. Bamler and Bruce Kleiner · 2017Bamler–Kleiner: Ricci flow and diffeomorphism groups of 3-manifolds.
Paper
Ricci flow and contractibility of spaces of metrics Open Ricci flow and contractibility of spaces of metricsRichard H. Bamler and Bruce Kleiner · 2019Bamler–Kleiner: Ricci flow and the contractibility of spaces of metrics.

Ricci flow in every dimension

Perelman's picture of singularities was special to dimension three. Bamler's theory of 2020 extends much of it to all dimensions, using ideas first developed for Einstein manifolds. See the singularities and limit spaces guides.

Paper
Regularity of Einstein Manifolds and the Codimension 4 Conjecture Open Regularity of Einstein Manifolds and the Codimension 4 ConjectureJeff Cheeger and Aaron Naber · 2014Cheeger–Naber: non-collapsed limits of manifolds with bounded Ricci curvature, Einstein manifolds among them, are singular only in codimension four. The elliptic model for what came next.
Paper
Entropy and heat kernel bounds on a Ricci flow background Open Entropy and heat kernel bounds on a Ricci flow backgroundRichard H Bamler · 2020Bamler: entropy and heat kernel bounds on a Ricci flow background, the start of the theory.
Paper
Compactness theory of the space of Super Ricci flows Open Compactness theory of the space of Super Ricci flowsRichard H Bamler · 2020Bamler: compactness theory for super Ricci flows and metric flows.
Paper
Structure theory of non-collapsed limits of Ricci flows Open Structure theory of non-collapsed limits of Ricci flowsRichard H Bamler · 2020Bamler: structure theory of non-collapsed limits of Ricci flows in all dimensions.
Notes
Recent developments in Ricci flows Open Recent developments in Ricci flowsRichard H Bamler · 2021Bamler's survey, for the whole picture in one place.

The sphere theorems

Which curvature conditions force a manifold to be a sphere? Ricci flow answered the classical question of the 1950s, the differentiable quarter-pinched sphere theorem, in 2007. The sphere theorems guide has the details.

Paper
Three-manifolds with positive Ricci curvature Open Three-manifolds with positive Ricci curvatureRichard S. Hamilton · 1982Hamilton: in dimension three, positive Ricci curvature makes a closed manifold a spherical space form.
Paper
Four-manifolds with positive curvature operator Open Four-manifolds with positive curvature operatorRichard S. Hamilton · 1986Hamilton: in dimension four, positive curvature operator does the same.
Paper
Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes Open Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-PlanesMario J. Micallef and John Douglas Moore · 1988Micallef–Moore: positive isotropic curvature, studied with minimal 2-spheres.
Paper
Manifolds with positive curvature operators are space forms Open Manifolds with positive curvature operators are space formsChristoph Boehm and Burkhard Wilking · 2006Böhm–Wilking: positive curvature operator in every dimension.
Paper
Manifolds with 1/4-pinched Curvature are Space Forms Open Manifolds with 1/4-pinched Curvature are Space FormsS. Brendle and R. M. Schoen · 2007Brendle–Schoen: quarter-pinched manifolds are space forms, the differentiable sphere theorem.
Paper
Classification of manifolds with weakly 1/4-pinched curvatures Open Classification of manifolds with weakly 1/4-pinched curvaturesS. Brendle and R. M. Schoen · 2007Brendle–Schoen: the weakly quarter-pinched case.
Paper
Sphere Theorems in Geometry Open Sphere Theorems in GeometryS. Brendle and R. M. Schoen · 2009Brendle–Schoen: a survey of sphere theorems in geometry.
Book
Ricci Flow and the Sphere Theorem Open Ricci Flow and the Sphere TheoremSimon Brendle · 2010Brendle: *Ricci Flow and the Sphere Theorem*, the whole proof as a book.

Kähler–Einstein metrics, from Calabi–Yau to K-stability

When does a complex manifold carry a Kähler–Einstein metric? Aubin and Yau answered for $c_1 < 0$ , and Yau for $c_1 = 0$ , in the 1970s; for Fano manifolds the answer, K-stability, took until 2012. The Kähler–Ricci flow runs through the whole story. See Kähler–Einstein and K-stability.

Paper
On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I Open On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, IShing‐Tung Yau · 1978Yau: the Calabi conjecture and the complex Monge–Ampère equation.
Paper
Deformation of Kähler matrics to Kähler-Einstein metrics on compact Kähler manifolds Open Deformation of Kähler matrics to Kähler-Einstein metrics on compact Kähler manifoldsHuai-Dong Cao · 1985Cao: the Kähler–Ricci flow, and a parabolic proof of the Calabi–Yau theorem.
Paper
Kähler-Einstein metrics with positive scalar curvature Open Kähler-Einstein metrics with positive scalar curvatureGang Tian · 1997Tian: Kähler–Einstein metrics with positive scalar curvature, where K-stability is introduced.
Paper
Scalar Curvature and Stability of Toric Varieties Open Scalar Curvature and Stability of Toric VarietiesS.K. Donaldson · 2002Donaldson: the algebraic definition of K-stability.
Paper
Kahler-Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities Open Kahler-Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularitiesXiu-Xiong Chen, Simon Donaldson and Song Sun · 2012Chen–Donaldson–Sun: K-stable Fano manifolds carry Kähler–Einstein metrics (first of three papers).
Paper
Kähler-Ricci flow, Kähler-Einstein metric, and K-stability Open Kähler-Ricci flow, Kähler-Einstein metric, and K-stabilityXiuxiong Chen, Song Sun and Bing Wang · 2015Chen–Sun–Wang: the same theorem by the Kähler–Ricci flow.
Book
An Introduction to Extremal Kähler Metrics Open An Introduction to Extremal Kähler MetricsGábor Székelyhidi · 2014Székelyhidi: *An Introduction to Extremal Kähler Metrics*, the textbook.

Mean curvature flow, the extrinsic mirror

Mean curvature flow has the same singularities, monotone quantities and surgeries as Ricci flow, in a setting where you can draw the pictures. Ideas have crossed between the two flows in both directions. See the mean curvature flow guide.

Paper
Flow by mean curvature of convex surfaces into spheres Open Flow by mean curvature of convex surfaces into spheresGerhard Huisken · 1984Huisken: convex hypersurfaces shrink to round points, the analogue of Hamilton's first theorem.
Paper
Asymptotic behavior for singularities of the mean curvature flow Open Asymptotic behavior for singularities of the mean curvature flowGerhard Huisken · 1990Huisken: the monotonicity formula, and why singularities look like self-shrinkers.
Paper
Mean curvature flow with surgeries of two–convex hypersurfaces Open Mean curvature flow with surgeries of two–convex hypersurfacesGerhard Huisken and Carlo Sinestrari · 2008Huisken–Sinestrari: surgery for 2-convex hypersurfaces, developed in parallel with Perelman's.
Paper
Generic mean curvature flow I; generic singularities Open Generic mean curvature flow I; generic singularitiesTobias H. Colding and William P. Minicozzi · 2009Colding–Minicozzi: entropy, and why generic singularities are spheres and cylinders.
Paper
On the Multiplicity One Conjecture for Mean Curvature Flows of surfaces Open On the Multiplicity One Conjecture for Mean Curvature Flows of surfacesRichard H Bamler and Bruce Kleiner · 2023Bamler–Kleiner: the multiplicity-one conjecture for surfaces in $R^{3}$ .
Book
Lecture Notes on Mean Curvature Flow Open Lecture Notes on Mean Curvature FlowCarlo Mantegazza · 2011Mantegazza's lecture notes: a careful course from first principles to singularities.

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Every resource

792 resources, collected automatically from every page that recommends one. Open an entry to see where it is recommended and what it connects to.

Connections come from where resources are recommended together, from the reading trails, and from shared authors or other editions of the same work; they are not citations. How the library is built. Newest addition: Oct 31, 2026.