4-manifolds

What it is

Four-manifolds are where topology is most exotic. Freedman classified closed simply connected topological 4-manifolds by the intersection form and the Kirby–Siebenmann invariant, and in particular showed that a homotopy 4-sphere is homeomorphic to $S^4$. Smooth structures behave differently. Donaldson's 1983 theorem says that a definite intersection form on a closed, simply connected smooth 4-manifold is diagonalisable, so many forms that Freedman realises topologically have no smooth realisation. Combining the two results yields exotic smooth structures on $\mathbb R^4$. The smooth Poincaré conjecture in dimension four, that every homotopy 4-sphere is diffeomorphic to $S^4$, is the dimension-four counterpart of the conjecture Perelman settled in dimension three.

For Ricci flow, curvature conditions give strong results:

• Positive curvature operator (Hamilton, 1986): the manifold is diffeomorphic to $S^4$ or $\mathbb{RP}^4$, the result shown above.
• Positive isotropic curvature. Hamilton introduced surgery here in 1997. Chen, Tang and Zhu completed the classification: compact 4-manifolds with positive isotropic curvature are, up to diffeomorphism, $S^4$, $\mathbb{RP}^4$, quotients of $S^3\times\mathbb R$, or connected sums of these.

Without such a condition the flow still exists for a short time and, on a closed manifold, stays non-collapsed on finite time intervals. What is missing is the three-dimensional control of the singular regions.

The curvature operator in dimension four

On an oriented 4-manifold the Hodge star splits two-forms as $\Lambda^2 = \Lambda^+ \oplus \Lambda^-$, each of rank three, and the curvature operator becomes a block matrix:

where $W^\pm$ are the self-dual and anti-self-dual Weyl tensors and $B:\Lambda^-\to\Lambda^+$ encodes the trace-free Ricci tensor. So $\operatorname{tr}A = \operatorname{tr}C = R/4$, and $B = 0$ exactly when the metric is Einstein. Write $a_1\le a_2\le a_3$ and $c_1\le c_2\le c_3$ for the eigenvalues of $A$ and $C$, and $b_1\le b_2\le b_3$ for the square roots of the eigenvalues of $BB^T$.

The standard models, in the normalisation where the unit sphere has $A = C = \mathrm{Id}$:

Model	$A$	$C$	$B$	PIC?
Round $S^4$, sectional curvature 1	$\mathrm{Id}$	$\mathrm{Id}$	$0$	yes
$S^3\times\mathbb R$, unit $S^3$	$\tfrac12\mathrm{Id}$	$\tfrac12\mathrm{Id}$	$\tfrac12\mathrm{Id}$	yes
$S^2\times\mathbb R^2$, unit $S^2$	$\operatorname{diag}(0,0,\tfrac12)$	$\operatorname{diag}(0,0,\tfrac12)$	$\operatorname{diag}(0,0,\tfrac12)$	no
$\mathbb{CP}^2$, $\operatorname{Ric} = 6g$	$\operatorname{diag}(0,0,6)$	$2\,\mathrm{Id}$	$0$	no

In an evolving frame the curvature obeys a reaction–diffusion equation whose reaction part splits along these blocks. With $X^\#$ the cofactor matrix of a $3\times3$ matrix, the ODE is

Papers that scale the operator differently drop the overall factor 2. Every row of the table solves this ODE, up to the scaling of its flow, which is a quick check on the convention.

The Gauss–Bonnet and signature formulas tie this curvature to topology. For a closed oriented 4-manifold with Euler characteristic $\chi$ and signature $\tau$, and with $\lvert W^\pm\rvert$ the norm as an operator on $\Lambda^\pm$,

For an Einstein metric the trace-free term vanishes, which gives the Hitchin–Thorpe inequality $2\chi \ge 3\lvert\tau\rvert$ (see Anderson's survey). Naff and Ozuch describe dimension four as critical for the equation: the curvature PDE has a quadratic nonlinearity, and the integral of $\lvert\operatorname{Rm}\rvert^2$ is scale invariant exactly here.

Why it matters

Without curvature assumptions, a 4-dimensional flow can develop singularities whose models are not known. Understanding them is the natural next step after dimension three. Naber classified non-compact 4D shrinkers with bounded, non-negative curvature operator. Bamler's theory shows that tangent flows at the first singular time are shrinking solitons whose singular set has codimension at least four, which in dimension four leaves room only for isolated singular points. A usable classification of 4D shrinkers would open the way to Ricci flow through singularities in dimension four, and perhaps to topological applications.

In dimension three, Hamilton–Ivey pinching forces blow-up limits to have non-negative curvature, so they are $\kappa$-solutions. Naff and Ozuch note that a version of this estimate holds for flows with positive isotropic curvature, and that no general analogue is known. Máximo (arXiv:1204.5967) showed that blow-up limits on closed 4-manifolds need not even have non-negative Ricci curvature. Naff and Ozuch expect an abundance of singularity models in this dimension.

Theorems under curvature assumptions

#Positive curvature operator

The proof finds cones of curvature operators preserved by the ODE above and shows that the solution pinches toward the constant-curvature ray. Böhm and Wilking later carried this to all dimensions under 2-positive curvature operator, and Brendle and Schoen reached the $1/4$-pinched sphere theorem by way of invariant cones built from isotropic curvature.

Not every positivity condition survives. Krishnan and Bettiol found cohomogeneity one metrics on $S^4$ and $\mathbb{CP}^2$ with positive sectional curvature that lose this property under the flow.

#Positive isotropic curvature

A metric has positive isotropic curvature (PIC) when the complexified sectional curvature is positive on every totally isotropic 2-plane in $T_pM\otimes\mathbb C$. In dimension four this is equivalent to $a_1 + a_2 > 0$ and $c_1 + c_2 > 0$. Strict pointwise $1/4$-pinching implies PIC, and PIC implies $R > 0$. A conformally flat metric with $R>0$ has $W^\pm = 0$, hence PIC. The table shows why $S^3\times\mathbb R$ qualifies and $S^2\times\mathbb R^2$ does not.

Hamilton (1997) proved that PIC is preserved in dimension four, and proved a pinching estimate that plays the role of Hamilton–Ivey. There are constants $\rho, \Lambda, P$ depending only on the initial metric with $a_1+\rho>0$, $c_1+\rho>0$, $\max\{a_3,b_3,c_3\}\le\Lambda(a_1+\rho)$ and $\max\{a_3,b_3,c_3\}\le\Lambda(c_1+\rho)$, together with a bound that forces $b_3/\sqrt{(a_1+\rho)(c_1+\rho)}$ to be at most 1 in the limit of large curvature. Any blow-up limit therefore satisfies

This gives a canonical-neighbourhood description near singularities in terms of necks modelled on $S^3\times\mathbb R$, their quotients and caps, and that is what makes surgery possible.

The classification came in two steps:

• No essential incompressible space form. An incompressible space form is an embedded $S^3/\Gamma$ injecting on $\pi_1$. Under the assumption that $M$ has no essential one, Hamilton's theorem says $M$ admits a PIC metric if and only if it is diffeomorphic to $S^4$, $\mathbb{RP}^4$, $S^3\times S^1$, the $\mathbb Z_2$-quotient $S^3\tilde\times S^1$, or a connected sum of these. Chen and Zhu (2006) gave a complete proof, supplementing statements in Hamilton's paper.
• The general case. Chen, Tang and Zhu allow such space forms. Surgery along them creates isolated orbifold points, so they run the flow with surgery on orbifolds. After finitely many surgeries the flow becomes extinct, and $M$ is recovered as an orbifold connected sum of spherical orbifolds $S^4/\Delta$. A separate topological argument converts this into the final list.

Two corollaries: Schoen's conjecture holds for $n = 4$: a finite cover of $M$ is diffeomorphic to $S^4$, $S^3\times S^1$ or a connected sum of them. And a compact 4-manifold admits a PIC metric if and only if it admits a conformally flat metric of positive scalar curvature, which fails for $n>4$.

Singularities without curvature assumptions

Type I. Naber showed that the dilation of a Type I singularity is a shrinking soliton, in any dimension.

The general structure. Bamler's theory applies to any non-collapsed limit, using Nash entropy and $\mathbb F$-convergence. The singular set has parabolic Minkowski dimension at most $n-2$, counting time twice, and this is optimal: blowing down static Eguchi–Hanson gives the orbifold $\mathbb R^4/\mathbb Z_2$. His Theorem 2.46 is the statement for $n=4$: the singular space of a tangent flow is a complete smooth orbifold with isolated singularities, and it either has $R>0$ or is isometric to $\mathbb R^4/\Gamma$ for a finite group $\Gamma\subset O(4)$.

Finer results. Fang and Li prove parabolic rectifiability of each stratum of the singular set in dimension four, with a sharp $\mathscr H^2$-volume bound. Under bounded scalar curvature, Bamler and Zhang obtained $L^2$ curvature bounds in dimension four, and Huang and Zhang extend these up to dimension 7.

Not everything is a soliton. Appleton (arXiv:1903.09936) showed that $U(2)$-invariant, asymptotically cylindrical metrics on $TS^2$ can form a Type II singularity modelled on Eguchi–Hanson, which is Ricci-flat. As Bamler's survey records, the possible blow-up limits there are Eguchi–Hanson, the flat $\mathbb R^4/\mathbb Z_2$, the Bryant soliton mod $\mathbb Z_2$, and $\mathbb{RP}^3\times\mathbb R$. The first two must occur, and at the time of the survey it was unknown whether the last two do.

A conjectural list. Haslhofer (2023) conjectures that every $\kappa$-solution in 4D is, up to scaling and finite quotients, one of the following:

• shrinkers: $S^4$, $\mathbb{CP}^2$, $S^2\times S^2$, $\mathbb R\times S^3$, $\mathbb R^2\times S^2$;
• steadies: the 4D Bryant soliton, the 3D Bryant soliton times a line, and Lai's one-parameter family;
• ovals: Perelman's, 3D ovals times a line, Buttsworth's, and Haslhofer's new bubble-sheet family on $S^4$.

He records that the Type I and neck cases follow from existing results, leaving the bubble-sheet case as the main problem. Conditional on a stronger form of the conjecture, he proves a canonical neighbourhood theorem at cylindrical singularities. He also conjectures that quotient necks $\mathbb R\times S^3/\mathbb Z_k$ with $k\ge3$ cause non-uniqueness of the flow through singularities, with the flow continuing either by the Bryant soliton mod $\mathbb Z_k$ or by Appleton's soliton on $O(-k)$.

Shrinking solitons in dimension four

A gradient shrinker satisfies $\operatorname{Ric} + \nabla^2 f = \tfrac12 g$. In dimension four the known examples are, according to Bamler, Cifarelli, Conlon and Deruelle:

• $\mathbb R^4$ and quotients of $S^4$, $S^3\times\mathbb R$ and $S^2\times\mathbb R^2$;
• the ten del Pezzo surfaces, carrying Kähler–Einstein metrics or compact Kähler–Ricci solitons;
• the Feldman–Ilmanen–Knopf (FIK) soliton on the blow-up of $\mathbb C^2$ at a point;
• the new soliton of that paper on the blow-up of $\mathbb C\times\mathbb P^1$ at a point.

That paper completes the classification of Kähler shrinkers with bounded scalar curvature in complex dimension two. Kim and Ozuch point out that the general problem contains the classification of positive Einstein 4-manifolds, and they cite recent constructions on infinitely many topologies, so a full list looks out of reach.

Structure at infinity. Munteanu and Wang showed that in dimension four, bounded scalar curvature $S$ implies $\lvert\operatorname{Rm}\rvert\le cS$ and that the curvature operator is asymptotically non-negative. If $S\to0$ at infinity, the soliton is smoothly asymptotic to a cone. If $S$ is bounded below by a positive constant, then along each end the soliton is asymptotic to a quotient of $\mathbb R\times S^3$ or converges to a quotient of $\mathbb R^2\times S^2$ along integral curves of $\nabla f$. Bertellotti and Buzano study blow-ups along ends without global curvature assumptions. Cao's survey compares the curvature estimates for shrinkers, steadies and expanders.

Classification under conditions.

• Bounded non-negative curvature operator, non-compact: $\mathbb R^4$ or a finite quotient of $S^2\times\mathbb R^2$ or $S^3\times\mathbb R$ (Naber).
• PIC: a quotient of $S^4$ or $S^3\times\mathbb R$ (Li, Ni and Wang, as recorded by Xiaodong Cao and Tran).
• Harmonic self-dual Weyl tensor, non-Einstein: a finite quotient of $S^2\times\mathbb R^2$ or $S^3\times\mathbb R$ (Wu, Wu and Wylie, same survey).
• Scalar curvature 1, non-compact: a finite quotient of $\mathbb R^2\times S^2$ (Cheng and Zhou, with another proof by Wang and Wu).

Compact shrinkers. Non-Einstein compact examples exist on $\mathbb{CP}^2\#\overline{\mathbb{CP}}{}^2$ (Koiso, Cao) and $\mathbb{CP}^2\#2\overline{\mathbb{CP}}{}^2$ (Wang, Zhu). Whether every compact gradient shrinker obeys Hitchin–Thorpe is, as far as we know, open: Ribeiro with Cheng and Zhou prove it under an upper bound on the range of the potential (arXiv:2203.14916).

Topology enters directly. Chow, Freedman, Shin and Yongjia Zhang use topological lemmas to estimate the curvature of 4D soliton singularity models; a later correction repairs one step with a statement about linking pairings.

Steady and expanding solitons. Steadies appear among the $\kappa$-solutions above. Recent results include $O(3)$-symmetry of non-collapsed steadies with positive curvature operator, the asymptotic geometry of steadies such as Lai's, and classification under half-harmonic Weyl curvature of asymptotically cylindrical steadies and asymptotically conical expanders, by Huai-Dong Cao and Xie.

Blowdowns and generic flows

The FIK soliton lives on $O(-1)\to\mathbb{CP}^1$. It is Kähler and $U(2)$-invariant, and its zero section is a sphere of self-intersection $-1$ that shrinks to a point at the singular time. Its asymptotic cone is over a Berger sphere.

Stability here means the sense of Cao, Hamilton and Ilmanen, based on the second variation of Perelman's $\nu$-entropy. Kim and Ozuch report that it is rare. Apart from $S^4$ and $\mathbb{CP}^2$, the positive Einstein metrics known before recent constructions are unstable, as are the compact Kähler shrinkers and the Bamler–Cifarelli–Conlon–Deruelle soliton. $\mathbb{CP}^2$ itself is dynamically unstable, and its perturbations develop blowdown singularities, at least numerically.

Naff and Ozuch (2025) proved that the FIK soliton is strictly linearly stable. The proposal is that a generic flow has only a few stable models, each matched to a surgery:

Soliton	Topological operation
$S^4$	a component vanishes in a round point
$S^3\times\mathbb R$	undoes a connected sum
$S^2\times\mathbb R^2$	believed to be a handle surgery
FIK on $O(-1)$, or $O(1)$ in the opposite orientation	blows down a sphere of self-intersection $\pm1$

Kim and Ozuch (2026, preprint) upgrade this to dynamical stability:

• A flow containing a region close to a rescaled large compact part of the soliton, with controlled geometry elsewhere, develops a singularity with unique tangent flow FIK.
• The singular time slice has an isolated conical point, and by Gianniotis and Schulze the flow continues by blowing down the sphere.
• On a closed 4-manifold, the set of metrics whose first singularity is modelled on FIK is $C^{14}$-open, and it is non-empty if and only if the manifold contains a sphere of self-intersection $\pm1$.

They deduce that every closed simply connected smooth 4-manifold, including exotic ones, can be reached by Ricci flow from a standard $k\mathbb{CP}^2\#\ell\overline{\mathbb{CP}}{}^2$ through finitely many blowdowns, and that on $2\mathbb{CP}^2\#9\overline{\mathbb{CP}}{}^2$ infinitely many open sets of metrics reach pairwise non-diffeomorphic manifolds after one blowdown. These are constructions of chosen initial data, not a way to read the smooth structure off an arbitrary metric. Shin proves a related open formation result in $C^{2,\alpha}$ with sharp asymptotics of the exceptional sphere.

Key results

Years are those of the paper or first preprint.

Year	Result	Who
1986	Positive curvature operator implies $S^4$ or $\mathbb{RP}^4$	Hamilton
1997	PIC preserved; pinching estimate; surgery	Hamilton
2003	$U(n)$-invariant shrinkers, including the blowdown soliton	Feldman, Ilmanen, Knopf
2006	Complete proof of the PIC classification without essential space forms	Chen, Zhu
2007	Non-compact shrinkers with bounded non-negative curvature operator	Naber
2008	Complete PIC classification, by orbifold surgery	Chen, Tang, Zhu
2020	4D tangent flows live on orbifolds with isolated points	Bamler
2022	Kähler shrinkers in complex dimension two classified	Bamler, Cifarelli, Conlon, Deruelle, with earlier work
2023	Classification conjecture for 4D $\kappa$-solutions	Haslhofer
2025	Blowdown soliton strictly linearly stable	Naff, Ozuch
2026	Blowdown soliton dynamically stable; applications	Kim, Ozuch

Neighbouring flows

Kähler–Ricci flow. On complex surfaces the flow is far better understood. Song and Weinkove contract exceptional curves by Kähler–Ricci flow, the Kähler counterpart of the blowdown above (arXiv:1003.0718), and on a projective surface this runs the analytic minimal model program.

Mean curvature flow. Colding, Minicozzi and Pedersen propose studying a homotopy 4-sphere by embedding it in $\mathbb R^5$ and running mean curvature flow, where generic singularities are understood. Kim and Ozuch stress a difference: for hypersurfaces only spheres and cylinders are stable, while Ricci flow in dimension four has a Kähler model as well.

Where it stands

Open problems include:

• Classifying 4D gradient shrinkers, at least under geometric assumptions such as bounded curvature or asymptotically conical ends.
• Understanding which singularities form generically. Naff and Ozuch's conjecture says that for an open dense set of metrics every high-curvature region is close to a stable shrinking or steady soliton.
• Constructing the flow. Bamler's Conjecture 5.8 asks for a Ricci flow through singularities in which topology changes along cylinders or cones and time slices may have isolated orbifold points. In dimension three the analogous flow is unique.
• Using Ricci flow to study Einstein metrics on 4-manifolds.

Whether Ricci flow can say anything about the smooth Poincaré conjecture is an open, long-range question. Akbulut posted an outline of a proof in 2022 by other means (arXiv:2209.09968); as far as we know the conjecture is still generally treated as open. Watanabe's 2018 result that $\operatorname{Diff}(S^4)$ is not homotopy equivalent to $O(5)$ shows how different dimension four is from dimension three, where the analogous statement holds and Ricci flow gave a proof of the generalized Smale conjecture.