Ricci solitons

What it is

A Ricci soliton is a metric $g$ with a vector field $X$ and a constant $\lambda$ such that $\operatorname{Ric} + \tfrac12\mathcal L_X g = \lambda g$. When $X = \nabla f$ it is a gradient soliton, $\operatorname{Ric} + \nabla^2 f = \lambda g$. The Ricci flow starting at $g$ then evolves by pulling back along the flow of $X$ and rescaling. The soliton is shrinking for $\lambda > 0$, steady for $\lambda = 0$ and expanding for $\lambda < 0$. Hamilton introduced them; the concept card gives the short version.

The basic examples:

• Einstein metrics (with $f$ constant).
• The Gaussian shrinker on $\mathbb R^n$, with $f = |x|^2/4$.
• The round cylinders $S^{k}\times\mathbb R^{n-k}$.
• Hamilton's cigar, a steady soliton on $\mathbb R^2$.
• The Bryant soliton, the rotationally symmetric steady soliton on $\mathbb R^3$.
• The Koiso–Cao and Feldman–Ilmanen–Knopf Kähler solitons.

On closed manifolds Perelman showed that every shrinker is a gradient shrinker, and steady and expanding solitons are Einstein. Earlier, Hamilton (dimension two) and Ivey (dimension three) had shown that closed solitons in those dimensions have constant curvature.

How a soliton moves under the flow

Suppose $X$ is complete and set $\sigma(t) = 1 - 2\lambda t$. Let $\varphi_t$ be the diffeomorphisms generated by $X/\sigma(t)$, with $\varphi_0 = \mathrm{id}$. Then

solves the Ricci flow with $g(0) = g$. To check it, differentiate at time $t$ and use that the Ricci tensor is scale invariant and natural under diffeomorphisms: both $\partial_t g$ and $-2\operatorname{Ric}(g(t))$ equal $\varphi_t^{*}(\mathcal L_X g - 2\lambda g)$.

On a closed manifold $X$ is always complete. Z.-H. Zhang showed that $\nabla f$ is complete on any complete gradient soliton, so no extra hypothesis is needed in that case.

The sign of $\lambda$ decides the time interval:

Type	$\lambda$	$\sigma(t)$	Exists for	Flow type
Shrinking	$\tfrac12$	$1-t$	$t < 1$	Ancient; singular at $t = 1$ unless flat
Steady	$0$	$1$	all $t$	Eternal
Expanding	$-\tfrac12$	$1+t$	$t > -1$	Immortal

So shrinkers and steady solitons are ancient solutions, and a non-flat shrinker is a flow that becomes singular at $t = 1$ and has the same shape at every earlier time. In Perelman's language, solitons and breathers are the fixed points and periodic orbits of the flow on metrics modulo diffeomorphism and scaling.

Identities every gradient soliton satisfies

Tracing the equation, and combining it with the contracted Bianchi identity, gives Hamilton's identities:

Taking the divergence of the middle one gives an equation for the scalar curvature with drift Laplacian $\Delta_f = \Delta - \nabla_{\nabla f}$:

Subtracting the first identity from the third gives $\Delta f - |\nabla f|^2 + 2\lambda f = n\lambda - C$. On a closed manifold with $\lambda \le 0$ the maximum principle forces $f$ to be constant, which is the elementary reason a closed gradient steady or expanding soliton is Einstein. The non-gradient case needs Perelman's monotone quantities.

On a steady soliton the third identity reads $R + |\nabla f|^2 = C$. B.-L. Chen proved that every complete ancient solution has $R \ge 0$, without any curvature bound, and that every complete three-dimensional ancient solution has nonnegative sectional curvature. So $R \ge 0$ on all complete shrinkers and steady solitons, and $|\nabla f|^2 \le C$ on a steady soliton. The three-dimensional statement replaces the bounded-curvature argument through Hamilton–Ivey pinching.

Worked examples

The Gaussian. On flat $\mathbb R^n$, $f = |x|^2/4$ gives $\nabla^2 f = \tfrac12 g$, a shrinker, and $f = -|x|^2/4$ gives an expander.

Round cylinders. With $\lambda = \tfrac12$, the sphere $S^k$ needs $\operatorname{Ric} = \tfrac12 g$, so its radius is $\sqrt{2(k-1)}$, for $k \ge 2$. On $S^k \times \mathbb R^{n-k}$ take $f = |x|^2/4$ on the Euclidean factor. Under the flow the sphere shrinks with radius squared $2(k-1)(1-t)$ while the line stays fixed, which is the model of a neckpinch when $k = n-1$.

The cigar. On $\mathbb R^2$ take

Its scalar curvature is $R = 4/\cosh^2 s$, positive with exponential decay in distance, and $R + |\nabla f|^2 = 4$. It is asymptotic to a cylinder of finite circumference, so its volume growth is linear. Hamilton showed that, up to scaling, it is the only complete non-flat steady soliton on a surface whose bounded curvature attains its maximum at an origin.

The product $\mathbb R \times$ cigar is a steady soliton in dimension three. Balls of large radius $r$ far from the tip have curvature below $r^{-2}$ but volume only about $r^2$, so it is not $\kappa$-noncollapsed at large scales, and Perelman's no local collapsing theorem excludes it as a blow-up limit of a flow on a closed manifold in finite time.

The Bryant soliton. On $\mathbb R^n$, $n \ge 3$, Bryant found a rotationally symmetric steady gradient soliton with positive sectional curvature. Geodesic spheres of radius $s$ have diameter of order $\sqrt s$, so balls of radius $r$ have volume of order $r^{(n+1)/2}$, and the curvature falls off like $1/s$. Bryant's construction is cited in the literature as unpublished notes.

Kähler examples. Koiso and Cao independently found a closed $U(2)$-invariant shrinker on $\mathbb{CP}^2 \# \overline{\mathbb{CP}}{}^2$, with positive Ricci curvature. Feldman, Ilmanen and Knopf found complete noncompact $U(n)$-invariant shrinkers that are cone-like at infinity, whose Ricci curvature has no fixed sign. Wang and Zhu proved existence on every toric Fano manifold with non-vanishing Futaki invariant, including $\mathbb{CP}^2 \# 2\overline{\mathbb{CP}}{}^2$. Cao also gave steady solitons and expanders on $\mathbb C^n$. See Kähler–Ricci solitons.

Non-gradient examples. Noncompact solitons need not be gradient. Lauret, and Baird and Danielo, constructed non-gradient expanders on Nil and Sol manifolds, in the setting of homogeneous spaces.

Why it matters

Singularities are studied by blowing up, and the blow-up limits are often solitons. A finite-time singularity is Type I if $\sup_t\,(T-t)\sup_M|\operatorname{Rm}| < \infty$. Limits at Type I singularities are gradient shrinkers: Naber showed that a suitable rescaling limit is a gradient shrinker, and Enders, Müller and Topping showed that blow-ups around singular points converge to nontrivial ones.

The other types of limit are tied to the other kinds of soliton through Hamilton's Harnack inequality. Under nonnegative curvature operator and positive Ricci curvature, Hamilton showed that a Type II singularity model is a steady soliton, and Chen and Zhu showed that a Type III model is an expander. The Harnack quantity vanishes on expanding solitons.

Perelman's surgery needs a complete list of the high-curvature regions, and that list is made of solitons and their relatives. By the canonical neighbourhood theorem those regions are modelled on ancient $\kappa$-solutions, and each such solution has an asymptotic gradient shrinker as $\tau \to \infty$. Necks are modelled on $S^2\times\mathbb R$. Brendle later proved that a noncompact ancient $\kappa$-solution in dimension three is a shrinking cylinder, a quotient of one, or the Bryant soliton, so among simply connected noncompact models the Bryant soliton is the only one with a cap. Knowing which solitons exist in a given dimension is essentially knowing what the flow can do there.

In higher dimensions Bamler's structure theory gives the general statement: non-collapsed limits of Ricci flows are smooth away from a set of codimension at least four, and their tangent flows are gradient shrinkers, possibly with a singular set of that size. The codimension-4 card has the terms.

Entropy and stability

Perelman's functionals single out solitons among all metrics. On a closed manifold,

Along Ricci flow, with $f$ solving the conjugate heat equation and $\tau = T - t$ in the second case,

The integrands are the soliton equations themselves. So the first derivative vanishes only on gradient steady solitons, and the second only on gradient shrinkers with $\lambda = 1/(2\tau)$. Taking infima gives the energies $\lambda(g)$ and $\mu(g,\tau)$ and $\nu(g) = \inf_\tau \mu(g,\tau)$, all monotone. The scale-invariant $\lambda(g)\operatorname{Vol}(g)^{2/n}$ handles expanders while it is nonpositive. This is the proof that closed solitons are gradient: a soliton returns to itself up to scale and diffeomorphism, so the monotone quantity is constant along it. Feldman, Ilmanen and Ni built the matching entropy for expanders. See Perelman's $\mathcal W$-entropy for more.

Solitons are therefore critical points, which raises the question of linear stability. Cao, Hamilton and Ilmanen computed the second variation of $\nu$. On an Einstein shrinker with $\operatorname{Ric} = g/(2\tau)$, on trace-free divergence-free tensors, it is governed by $\tfrac12(\Delta_L + \tfrac1\tau)$, where $\Delta_L$ is the Lichnerowicz Laplacian. The round sphere is stable and $\mathbb{CP}^m$ is neutrally stable. A product of two Einstein manifolds is unstable, the bad direction being a growing mismatch in the sizes of the factors.

They also introduced the Gaussian density $\Theta = e^{\nu}$ of a closed shrinker, a single number for comparing them. Cao's 2009 survey tabulates the closed four-dimensional cases known then, among them:

Shrinker	Type	$\Theta$	Stability recorded in 2009
$S^4$	Einstein	$0.812$	Stable
$\mathbb{CP}^2$	Einstein	$0.609$	Stable (neutrally)
$S^2\times S^2$	Einstein product	$0.541$	Unstable
$\mathbb{CP}^2\#\overline{\mathbb{CP}}{}^2$	Koiso–Cao soliton	$0.518$	Unknown
$\mathbb{CP}^2\#\overline{\mathbb{CP}}{}^2$	Page, non-Kähler Einstein	$0.517$	Unknown
$\mathbb{CP}^2\#2\overline{\mathbb{CP}}{}^2$	Wang–Zhu soliton	$0.4549$	Unknown

Among positive Einstein manifolds of a given dimension, the sphere has the largest value.

Shrinking solitons

#What holds in every dimension

Normalise $\lambda = \tfrac12$ and $R + |\nabla f|^2 = f$. For a complete noncompact gradient shrinker, Cao and Zhou proved that the potential grows like the Gaussian's, with $r$ the distance from a fixed point:

They also proved that the volume growth is at most Euclidean, $\operatorname{Vol}(B_r) \le C r^n$ for large $r$. No curvature assumption is needed. Other general facts: a complete shrinker has finite fundamental group (Wylie), a complete shrinker with bounded curvature is gradient and $\kappa$-noncollapsed (Naber), and one with positive sectional curvature is compact (Munteanu and Wang).

#Dimensions two and three

A complete gradient shrinker in dimension two is $\mathbb R^2$ or a quotient of $S^2$. In dimension three it is $\mathbb R^3$ or a quotient of $S^3$ or of $S^2\times\mathbb R$. The low-dimensional card has the short form. The hypotheses came off in stages:

• Ivey handled the compact case.
• Perelman assumed $\kappa$-noncollapsing with bounded nonnegative curvature.
• Ni and Wallach, and Naber, dropped the non-collapsing and asked only for $\operatorname{Ric} \ge 0$. Naber keeps bounded curvature, and Ni–Wallach allow growth up to $e^{ar}$ for a small constant $a$.
• Cao, Chen and Zhu removed the remaining curvature assumptions by using Chen's theorem.

The same pattern extends to vanishing Weyl curvature in all dimensions: Z.-H. Zhang proved such a shrinker is a finite quotient of $\mathbb R^n$, $S^n$ or $S^{n-1}\times\mathbb R$.

#Dimension four

Naber classified noncompact shrinkers with bounded nonnegative curvature operator: $\mathbb R^4$ or a finite quotient of $S^3\times\mathbb R$ or $S^2\times\mathbb R^2$. Without a sign on curvature, Munteanu and Wang work with a bound on scalar curvature $S$ alone. On a complete four-dimensional gradient shrinker with bounded $S$, they prove $|\operatorname{Rm}| \le cS$, and then:

• If $S \to 0$ at infinity, the shrinker is smoothly asymptotic to a cone.
• If $S$ is bounded below by a positive constant on an end, that end is smoothly asymptotic to a cylinder $\mathbb R\times S^3/\Gamma$, or it converges along each integral curve of $\nabla f$ to $\mathbb R^2\times S^2$ or its $\mathbb Z_2$ quotient.

Kotschwar and Lu Wang proved the matching rigidity in all dimensions: two gradient shrinkers asymptotic along an end to the same regular cone are isometric near infinity on those ends. One consequence is that the only complete connected gradient shrinker asymptotic to a rotationally symmetric cone is the Gaussian. For shrinkers with $S \to 0$ in dimension four, classification therefore reduces to the limiting cones.

Asymptotically conical shrinkers do occur as singularity models. A 2026 preprint of Kim and Ozuch shows that the Feldman–Ilmanen–Knopf shrinker is strictly dynamically stable: the set of metrics on a closed four-manifold whose first singularity is modelled on it is $C^{14}$-open, and nonempty exactly when the manifold contains a sphere of self-intersection $\pm 1$. Li, Yan and Zhang prove strong uniqueness of tangent flows when one tangent flow is a smooth asymptotically conical shrinker.

#Rigidity and the Kähler case

A soliton is rigid if it is a quotient of a product of an Einstein manifold with a Euclidean factor. For constant scalar curvature, recent preprints prove rigidity when the focal set of $f$ is totally geodesic, for five-dimensional shrinkers and for Kähler shrinkers. In a different direction, Brendle and Tsiamis extend the classification of manifolds with positive isotropic curvature to all dimensions $n\ge 5$, which they say completes the classification of complete noncompact gradient shrinkers with uniformly positive isotropic curvature.

In complex dimension two the picture is complete under bounded scalar curvature. Bamler, Cifarelli, Conlon and Deruelle constructed the last candidate, on the blow-up of $\mathbb C\times\mathbb P^1$ at one point, as a blow-up limit of a Kähler–Ricci flow. Their list is: a Kähler–Einstein metric or the Wang–Zhu soliton on a Fano surface, the Gaussian on $\mathbb C^2$, the Feldman–Ilmanen–Knopf soliton on the blow-up of $\mathbb C^2$ at the origin, the cylinder $\mathbb C\times\mathbb P^1$, and the new soliton. A 2026 preprint of Li and Zhang proves a scalar curvature bound for Kähler–Ricci shrinkers depending only on dimension, which would make that hypothesis automatic. Xu and Zhou use the classification for volume-collapsing singularities of Kähler–Ricci flow on surfaces.

Steady solitons

In dimension two the answer is the cigar. In dimension three the dividing line is non-collapsing.

Brendle proved that a non-flat $\kappa$-noncollapsed three-dimensional steady gradient soliton is, up to scaling, the Bryant soliton. Perelman had written in 2002 that he could prove uniqueness in this class, without giving the argument, and Brendle's paper describes itself as solving that problem. Roughly, the proof first shows that the soliton is asymptotically cylindrical, then upgrades approximate rotational Killing fields near infinity to exact ones.

Without non-collapsing the statement fails. Lai constructed a family of $\mathbb Z_2\times O(2)$-symmetric three-dimensional steady gradient solitons with positive curvature operator, confirming a conjecture of Hamilton. Each is a flying wing, asymptotic to a sector of angle $\alpha \in (0,\pi)$. All are collapsed, and along the geodesic fixed by the $O(2)$-action the scalar curvature tends to $R(p)\sin^2(\alpha/2)$, where $p$ is the critical point of $f$. So it does not vanish at infinity. The soliton converges to $\mathbb R\times$ cigar along the edges.

In dimension $n\ge 4$, non-collapsing is no longer enough. Lai's construction also gives non-collapsed $\mathbb Z_2\times O(n-1)$-symmetric solitons with positive curvature operator that are not rotationally symmetric. Brendle's higher-dimensional theorem says what does suffice: a steady gradient soliton of dimension $n\ge 4$ with positive sectional curvature that is asymptotically cylindrical is rotationally symmetric. In dimension four this covers $\kappa$-noncollapsed solitons with positive isotropic curvature. A 2026 preprint of Fang shows that every complete four-dimensional $\kappa$-noncollapsed gradient steady soliton with positive curvature operator has an effective isometric $O(3)$ action with principal orbits $S^2$. Symmetric ansätze of this kind are collected under rotational symmetry and warped products.

Expanding solitons

An expander's flow starts at $t = -1$, and in many examples the initial object is a cone, so an expander is a way for a flow to emerge from a conical singularity. Deruelle proved that for a simply connected closed manifold $(X, g_X)$ with $\operatorname{Rm}(g_X) \ge 1$, the cone over $X$ has a unique expanding gradient soliton with nonnegative curvature operator asymptotic to it. The same paper shows that the space of positively curved asymptotically conical expanders is connected, joining each one to a rotationally symmetric expander.

Expanders also describe long-time behaviour. For example, Wood shows that invariant flows on certain twisted principal bundles over $S^1$ are immortal and Type III, and after normalising they converge to a homogeneous expanding soliton. On stability, Lavoyer and Thompson prove linear stability for all steady and expanding gradient Kähler–Ricci solitons, and strict linear stability for asymptotically conical expanders. Flows through singularities in general are the subject of weak and singular flows.

Key results

Year	Result	Who
1988	The cigar soliton; closed surface solitons have constant curvature	Hamilton
1990, 1996	Closed non-Einstein Kähler shrinkers on $\mathbb{CP}^2\#\overline{\mathbb{CP}}{}^2$	Koiso; Cao
1993	Closed three-dimensional solitons have constant curvature	Ivey
2002	Closed solitons are gradient; entropy monotonicity; three-dimensional shrinkers with bounded nonnegative curvature	Perelman
2003	Complete noncompact Kähler shrinkers, cone-like at infinity	Feldman, Ilmanen, Knopf
2004	Shrinkers on toric Fano manifolds with non-vanishing Futaki invariant	Wang, Zhu
2007	Complete three-dimensional gradient shrinkers classified, in stages	Ni, Wallach; Naber; Cao, Chen, Zhu
2007	Type I blow-up limits are gradient shrinkers; bounded-curvature shrinkers are gradient	Naber
2009	Quadratic growth of $f$; at most Euclidean volume growth	Cao, Zhou
2010	Blow-ups at Type I singular points are nontrivial shrinkers	Enders, Müller, Topping
2012	Non-collapsed three-dimensional steady solitons are Bryant	Brendle
2012	Higher-dimensional rotational symmetry under asymptotically cylindrical geometry	Brendle
2015	Rigidity of asymptotically conical shrinkers	Kotschwar, Lu Wang
2015	Expanders smoothing positively curved cones	Deruelle
2016	Structure at infinity of four-dimensional shrinkers	Munteanu, Jiaping Wang
2020	Tangent flows of non-collapsed limits are possibly singular shrinkers	Bamler
2020	Flying wings; non-collapsed non-symmetric steady solitons for $n\ge 4$	Lai
2022	New shrinker on the blow-up of $\mathbb C\times\mathbb P^1$; complex surface classification	Bamler, Cifarelli, Conlon, Deruelle

Years are those of the papers cited in this guide, using the arXiv year where one exists.

Where it stands

In dimension three the picture is clear for shrinkers and for non-collapsed steady solitons. Collapsed steady solitons are richer. Lai's corollary says that a $\mathbb Z_2\times O(2)$-symmetric, non-rotationally symmetric steady soliton with positive curvature operator is a flying wing, but it does not cover solitons without that symmetry.

In dimension four and above, classifying shrinkers is a central open problem. There is progress on asymptotically conical shrinkers, on shrinkers with positive isotropic curvature, on four-manifolds with bounded scalar curvature, and on Kähler shrinkers. Many papers in the feed below are soliton classification and rigidity results. Ancient flows with a smooth compact tangent shrinker have been parametrised in a 2026 preprint. For the self-similar solutions of mean curvature flow, see self-shrinkers and translators; for almost, η- and Yamabe solitons, which relax the soliton equation, see their own topic.

#Open problems

Cao's 2009 survey lists problems that still frame the subject. Lai's solitons answer the one about steady solitons: non-rotationally symmetric ones with positive curvature exist for $n\ge 4$, and in dimension three positive curvature alone does not force Bryant. We have not confirmed the current status of each item below, so check before relying on them:

• Find a non-product, non-Einstein shrinking soliton that is not Kähler. In 2009 every known nontrivial shrinker was Kähler.
• For $n\ge 4$, is there a complete noncompact non-product shrinker with nonnegative Ricci curvature? Positive sectional curvature is excluded by Munteanu and Wang. In the Kähler case, Xu and Zhang now prove a splitting into a Gaussian factor and a compact one.
• Are linearly stable closed four-dimensional shrinkers necessarily Einstein, and are the only linearly stable positive Einstein four-manifolds $S^4$ and $\mathbb{CP}^2$?
• Does the Hitchin–Thorpe inequality $2\chi \ge 3|\tau|$ hold on closed four-dimensional shrinkers?

Beyond the list, Bamler's theory leaves open which singular shrinkers actually occur as tangent flows, and how to continue a flow through them in general.

Related flows

Mean curvature flow has the same three kinds of self-similar solution: self-shrinkers, translators in the role of steady solitons, and expanders, with round cylinders as models in both settings. Hamilton also uses the term flying wing for translating solutions, and Lai compares the non-collapsed solitons for $n\ge 4$ with non-collapsed translators there. The analogy is not automatic: Lai notes that translating flying wings were classified in $\mathbb R^3$ while the Ricci flow conjecture stayed open until that paper.

In the non-Kähler setting, generalized Ricci flow has its own solitons, including algebraic ones on Lie groups. Ricci-flat metrics are the stationary case of the soliton equation.