Concepts

The models for every 3-dimensional singularity. An ancient κ-solution is a complete, non-flat ancient solution with bounded, nonnegative curvature operator that is κ-noncollapsed at all scales. Perelman proved that blow-up limits of finite-time singularities in dimension 3 are exactly these.

The 3-dimensional list is now complete: shrinking round spherical space forms, the round cylinder $S^2\times\mathbb{R}$ and its $\mathbb{Z}_2$-quotient, the Bryant soliton, and Perelman's ancient oval. Classifying the noncompact case was Brendle's theorem; the compact case is Brendle–Daskalopoulos–Šešum.

The structural heart of Perelman's argument. For a 3-dimensional flow and each $\varepsilon > 0$ there is a scale $r > 0$ such that every point with

has a neighborhood which, rescaled to unit curvature, is $\varepsilon$-close in $C^{[1/\varepsilon]}$ to a corresponding piece of an ancient κ-solution: an $\varepsilon$-neck, an $\varepsilon$-cap, or a closed manifold of positive curvature.

In plain terms: high curvature leaves you no choices. Anywhere the flow is about to fail, the geometry is one of a short list of shapes you already understand — which is exactly what makes surgery possible.

Ricci flow is not a local equation — curvature far away can influence you instantly. Perelman's pseudolocality theorem says it is almost local anyway: a region that starts out nearly Euclidean stays regular for a definite amount of time, no matter how wild the rest of the manifold is.

Quantitatively, if a ball $B(x_0, r_0)$ has almost-Euclidean isoperimetric constant and $R \ge -r_0^{-2}$ at time zero, then

near $x_0$ for $0 < t \le (\varepsilon r_0)^2$. It is the tool that lets you do surgery in one place without the repair being destroyed by geometry somewhere else.

The engine behind every blow-up argument. Given a sequence of pointed Ricci flows with uniformly bounded curvature and a uniform lower bound on the injectivity radius at the basepoints, a subsequence converges in the pointed Cheeger–Gromov $C^\infty$ sense to a limit Ricci flow.

Rescale a sequence approaching a singularity, apply this, and the limit is an ancient solution — the singularity model. The injectivity radius hypothesis is the hard one, and it is precisely what κ-noncollapsing delivers. Before Perelman, the possibility of collapsing was the gap that stopped Hamilton's program.

Bamler's reworking of Perelman's entropy into a compactness theory. Given a conjugate heat kernel measure $\nu_{\tau} = (4\pi\tau)^{-n/2}e^{-f}\,dg$ based at a spacetime point, the pointed Nash entropy is

which is monotone in $\tau$ and bounded by Perelman's μ-functional. Uniform Nash-entropy bounds give compactness in Bamler's $\mathbb{F}$-topology, whose limits are metric flows: objects that are no longer smooth manifolds but still carry a heat flow and a conjugate heat flow.

Bamler's structure theory in all dimensions. Any noncollapsed limit of Ricci flows decomposes into a regular part, which is an honest smooth Ricci flow, and a singular set of parabolic codimension at least 4.

Four is the sharp number: Ricci-flat cones such as the Eguchi–Hanson space show that a 4-dimensional singular stratum really occurs. The result generalises the Cheeger–Colding–Naber theory for static Einstein manifolds to flows, and turns "the singular set is small" from a hope into a theorem.

Surgery works, but it depends on arbitrary parameters: how thin a neck must be before you cut. Kleiner and Lott removed the choice. A singular Ricci flow is a 4-dimensional spacetime $\mathcal{M}$ with a time function, a time vector field and a metric on the time slices, satisfying the canonical neighborhood assumption below any scale.

For every compact 3-manifold, such a flow exists and continues through its singularities with no parameters at all. Bamler–Kleiner then proved it is unique and depends continuously on the initial metric: Ricci flow through singularities is a canonical, deterministic process.

A topological payoff of flowing through singularities. For a spherical space form $M = S^3/\Gamma$, the inclusion of the isometry group into the diffeomorphism group

is a homotopy equivalence. Hatcher proved the case $M = S^3$ by hand in 1983; Bamler–Kleiner proved the general case by running Ricci flow on families of metrics and using uniqueness of singular flows to contract the space of metrics onto the round one.

The curvature condition that makes higher-dimensional Ricci flow work. A manifold has PIC if for every orthonormal 4-frame,

It looks technical, and it is exactly the condition preserved by Ricci flow (Hamilton in dimension 4, Brendle–Schoen in general). Brendle and Schoen proved that pointwise $1/4$-pinched manifolds satisfy a version of it, which gave the differentiable sphere theorem: such a manifold is diffeomorphic — not merely homeomorphic — to a spherical space form.

A machine for inventing preserved curvature conditions. Under Ricci flow the curvature operator satisfies

so by Hamilton's maximum principle for tensors, any closed convex $O(n)$-invariant cone preserved by the ODE $\tfrac{d}{dt}\operatorname{Rm} = \operatorname{Rm}^2 + \operatorname{Rm}^{\#}$ is preserved by the flow. Böhm and Wilking built a continuous family of such cones pinching down to the constant-curvature ray, proving that manifolds with positive curvature operator are space forms. Wilking later gave a Lie-algebraic recipe producing most known invariant conditions in one stroke.

On a Kähler manifold the flow preserves the Kähler condition, and the class evolves linearly:

Everything therefore reduces to a scalar parabolic complex Monge–Ampère equation for a potential $\varphi$. The maximal existence time is determined by cohomology alone — the flow runs until the class leaves the Kähler cone. Song and Tian showed the singularities correspond to the operations of the minimal model program: divisorial contractions and flips, performed analytically by the flow.

Couple the metric to a closed 3-form $H$ (the torsion, or B-field):

This is the renormalization group flow of the two-dimensional sigma model at one loop, which is where Friedan met Ricci flow in physics before geometers took it up. It is also the natural flow on a Courant algebroid in generalized geometry, and it specialises to pluriclosed flow on complex manifolds.

Shrinking gradient solitons are the singularity models, so classifying them classifies singularities. In dimension 3 the list is complete and short: every complete gradient shrinker with bounded curvature is a quotient of $\mathbb{R}^3$, $S^3$ or $S^2\times\mathbb{R}$ (Perelman, with Ni–Wallach and Cao–Chen–Zhu).

Dimension 4 is open and active. Known examples include the Gaussian shrinker, $S^4$, $S^3 \times \mathbb{R}$, $S^2\times\mathbb{R}^2$ and the Kähler shrinker on $\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$ found by Koiso and Cao. Whether that list is everything is one of the field's central questions.

What could "Ricci flow" mean on a space with no derivatives at all? The static answer came first: Lott–Villani and Sturm define $\operatorname{Ric} \ge K$ on a metric measure space by convexity of an entropy along Wasserstein geodesics, giving the RCD spaces.

For flows, Sturm and McCann–Topping characterise a super Ricci flow by a monotonicity: solutions of the heat equation, run along the flow, must not spread apart faster than the Wasserstein distance allows,

Smooth super Ricci flows are exactly those with $\partial_t g \le -2\operatorname{Ric}$, so the definition is the right one — and it makes sense verbatim on spaces where curvature cannot be written down.