Browse by idea
Topics
Every paper gets tagged automatically from its title and abstract, using a list of topics I built around Hamilton's and Perelman's program. My own notes and the concept cards use the same tags, so everything about one idea sits together.
Flows
Ricci flow
Hamilton's evolution of a Riemannian metric by its Ricci curvature: ∂g/∂t = −2 Ric.
324 papers · 1 note
Kähler–Ricci flow
Ricci flow restricted to Kähler metrics; reduces to a scalar parabolic Monge–Ampère equation.
56 papers
Generalized & G₂ flows
Generalized Ricci flow, pluriclosed flow, Laplacian/G₂ flows and other non-Kähler relatives.
43 papers
Mean curvature flow
The extrinsic cousin of Ricci flow: hypersurfaces moving by their mean curvature.
12 papers
Yamabe flow
Conformal flow driven by scalar curvature.
2 papers
Curve shortening
The 1-dimensional model: curves moving by curvature become round.
1 paper
Special solutions
Ricci solitons
Self-similar solutions — metrics that evolve only by scaling and diffeomorphism. They model singularities.
236 papers
Einstein metrics
Fixed points (up to scaling) of the Ricci flow: Ric = λg.
53 papers
Kähler–Ricci solitons
Solitons of the Kähler–Ricci flow.
48 papers
Ancient solutions
Flows that exist for all negative time — blow-up limits of singularities.
26 papers
Perelman's toolkit
Singularities
Where curvature blows up — and how to understand or continue through it.
82 papers
(Non)collapsing
κ-noncollapsing and collapsing: volume bounds at scales where curvature is controlled.
48 papers
Perelman entropy
Monotone quantities (𝓕, 𝓦, Nash entropy) that tame the flow.
20 papers
Harnack inequalities
Hamilton's differential Harnack and Li–Yau type estimates.
16 papers
Heat kernel
Heat kernels on evolving backgrounds, conjugate heat equation.
12 papers
Surgery
Cutting out high-curvature necks and gluing in caps to continue the flow.
11 papers
Curvature pinching
Curvature conditions preserved (and improved) by the flow.
9 papers
Reduced volume
Perelman's ℒ-length and reduced volume — monotone along the flow.
6 papers
Topology
4-manifolds
Ricci flow in dimension four — the next frontier.
53 papers
3-manifolds
The dimension where Ricci flow solved the classification problem.
44 papers
Surfaces
Ricci flow in 2D and the uniformization theorem.
34 papers
Sphere theorems
Curvature conditions that force a manifold to be a sphere (Brendle–Schoen …).
29 papers
Geometrization
Thurston's picture: every 3-manifold splits into pieces carrying one of eight geometries.
9 papers
Poincaré conjecture
Every simply connected closed 3-manifold is the 3-sphere.
4 papers · 1 note
Smale conjecture & diffeomorphisms
Using Ricci flow to study spaces of metrics and diffeomorphism groups.
2 papers
Geometry & analysis
Kähler geometry
Complex geometry: Fano manifolds, Monge–Ampère equations, canonical metrics.
147 papers
Stability & uniqueness
When is a flow or soliton stable, rigid, or unique?
125 papers
Boundaries & non-compact
Flows on manifolds with boundary or infinite extent.
88 papers
Scalar curvature
The simplest curvature invariant — and a key monotone quantity along the flow.
82 papers
Homogeneous spaces
Symmetric settings where the flow becomes an ODE.
54 papers
Limit spaces & RCD
Compactness, Gromov–Hausdorff limits, optimal transport and synthetic Ricci bounds.
53 papers
Weak & singular flows
Flows from rough initial data, through singularities, or in synthetic settings.
24 papers
Connections
From the journal
Real analysis
A topic from my own notes. No arXiv papers are tagged with it — it belongs to the groundwork rather than the research.
0 papers · 3 notes
Proofs
A topic from my own notes. No arXiv papers are tagged with it — it belongs to the groundwork rather than the research.
0 papers · 12 notes
Linear algebra
A topic from my own notes. No arXiv papers are tagged with it — it belongs to the groundwork rather than the research.
0 papers · 7 notes
Calculus
A topic from my own notes. No arXiv papers are tagged with it — it belongs to the groundwork rather than the research.
0 papers · 9 notes
Quadratic forms
A topic from my own notes. No arXiv papers are tagged with it — it belongs to the groundwork rather than the research.
0 papers · 1 note
Eigenvalues
A topic from my own notes. No arXiv papers are tagged with it — it belongs to the groundwork rather than the research.
0 papers · 2 notes
Spectral theorem
A topic from my own notes. No arXiv papers are tagged with it — it belongs to the groundwork rather than the research.
0 papers · 1 note
Inner products
A topic from my own notes. No arXiv papers are tagged with it — it belongs to the groundwork rather than the research.
0 papers · 1 note
Integration
A topic from my own notes. No arXiv papers are tagged with it — it belongs to the groundwork rather than the research.
0 papers · 1 note
Derivatives
A topic from my own notes. No arXiv papers are tagged with it — it belongs to the groundwork rather than the research.
0 papers · 2 notes
Limits
A topic from my own notes. No arXiv papers are tagged with it — it belongs to the groundwork rather than the research.
0 papers · 2 notes