Ricci Flow

Introduction

Ricci flow is the evolution equation

$$ \frac{\partial g_{ij}}{\partial t} = -2R_{ij} $$

for a one-parameter family of Riemannian metrics $g(t)$ on a fixed manifold $M$, the rate of change of the metric being minus twice its Ricci tensor. The equation is intrinsic, it is invariant under the diffeomorphism group of $M$, and it is weakly parabolic: the principal symbol of the linearisation is degenerate in the directions generated by diffeomorphisms, so the standard existence theory of parabolic systems does not apply directly. Hamilton's theorem of short-time existence, obtained by fixing a gauge, is the beginning of the theory, and the reaction-diffusion form of the evolution equations of the curvature is what makes the flow tractable: the Riemann tensor evolves by a nonlinear heat equation, and the maximum principle then controls the curvature, forbids certain singularities and forces the convergence of the flow under curvature hypotheses.

The flow is a geometric evolution equation, and its study is one of the places at which the analysis of this Part meets the Riemannian geometry of Part II. The Riemann curvature tensor, the Ricci and scalar curvatures, the Levi-Civita connection and the associated Laplacians are the notions of Part II, and they are used here without re-derivation; the elliptic operators and the maximum principle for parabolic equations belong to the analysis of this Part, the abstract theory of well-posed evolution equations being that of the preceding article on semigroups, and the reaction-diffusion structure of the curvature equations being the same as in the harmonic map flow discussed earlier. What is developed here is the equation itself: its local well-posedness, the evolution of the curvature, the exact solutions and solitons, the maximum principles, the singularity formation that the flow exhibits, and the convergence theorems under curvature conditions.

The Equation and Short-Time Existence

Definition. A Ricci flow on a smooth manifold $M$ is a smooth family $\{g(t)\}_{t\in[0,T)}$ of complete Riemannian metrics satisfying

$$ \frac{\partial g_{ij}}{\partial t}(x,t) = -2R_{ij}(x,t), $$

where $R_{ij}$ is the Ricci tensor of $g(t)$. The flow is normalised if the volume of $M$ is required to remain constant, in which case an extra term $2\bar Rg_{ij}/n$ is added, $\bar R$ being the average scalar curvature.

Theorem (Hamilton's short-time existence). Let $M$ be a compact manifold and $g_0$ a smooth Riemannian metric on $M$. There is a $T>0$ and a smooth solution $g(t)$ of the Ricci flow on $[0,T)$ with $g(0)=g_0$, and the solution is unique among smooth solutions with the given initial datum; the maximal interval of existence is finite if and only if the curvature becomes unbounded.

Proof. Quoted as standard (Hamilton). The equation is not parabolic because of its diffeomorphism invariance, and the proof introduces the DeTurck trick: one adds to the flow the Lie derivative $\mathcal{L}_{W(g)}g$ of the metric along a vector field $W$ determined by the difference of the connections of $g$ and of a fixed background metric, obtaining the Ricci–DeTurck equation

$$ \frac{\partial g_{ij}}{\partial t} = -2R_{ij} + \nabla_iW_j+\nabla_jW_i , $$

which has a strictly parabolic principal symbol of quasilinear type. The parabolic existence theorem gives a solution for short time, and one undoes the gauge by solving the flow of the time-dependent vector field $W$ and pulling the solution back; the diffeomorphism invariance of the Ricci tensor makes the pulled-back solution a Ricci flow. Uniqueness follows from the uniqueness for the parabolic equation in the fixed gauge.

Remark (the weakly parabolic character). The degeneracy of the symbol is not a defect to be removed but the expression of the invariance: the flow is defined only up to the action of the diffeomorphism group, and to prove a theorem about the metric one must either fix the gauge as above or work with a quantity invariant under the group. The curvature evolution below is of the second kind: the curvature is already invariant, and its equation is parabolic.

Theorem (evolution of the volume). Along a Ricci flow,

$$ \frac{d}{dt}\operatorname{Vol}(M,g(t)) = -\int_M R\,dV , $$

so the volume decreases when the scalar curvature is positive and increases when it is negative.

Proof. The derivative of the volume form is $\frac{\partial}{\partial t}dV = \frac12g^{ij}\frac{\partial g_{ij}}{\partial t}dV = -g^{ij}R_{ij}dV = -R\,dV$, the middle identity being the identity $\frac{\partial}{\partial t}\log\det g = g^{ij}\partial_tg_{ij}$; integrating gives the formula.

The Evolution of Curvature

Theorem (evolution of the scalar curvature). Along a Ricci flow,

$$ \frac{\partial R}{\partial t} = \Delta R + 2|R_{ij}|^2 , $$

where $\Delta$ is the Laplace–Beltrami operator of $g(t)$ and $|R_{ij}|^2 = R_{ij}R^{ij}$.

Proof. Quoted as standard. The computation differentiates the twice-contracted Bianchi identity and the definition $R=g^{ij}R_{ij}$; the derivative of the inverse metric contributes $-2R^{ij}$ contracting the Ricci tensor and the derivative of the Ricci tensor contributes its own evolution, and the result is the displayed reaction-diffusion equation, the reaction term being the square of the Ricci tensor.

Theorem (evolution of the Ricci tensor). Along a Ricci flow,

$$ \frac{\partial R_{ij}}{\partial t} = \Delta_LR_{ij} := \Delta R_{ij} + 2R_{ikjl}R^{kl} - 2R_{ik}R^k{}_j , $$

where the operator $\Delta_L$ is the Lichnerowicz Laplacian acting on symmetric $2$-tensors, $\Delta$ the rough (connection) Laplacian and $R_{ikjl}$ the Riemann tensor.

Proof. Quoted as standard. The derivative of the Ricci tensor is computed from the derivative of the connection, itself a difference of Christoffel symbols; the result is the displayed operator, in which the zeroth-order terms are quadratic in the curvature, as they must be by scaling.

Theorem (evolution of the Riemann tensor). Along a Ricci flow,

$$ \frac{\partial \mathrm{Rm}}{\partial t} = \Delta \mathrm{Rm} + \mathrm{Rm}*\mathrm{Rm} , $$

where $\mathrm{Rm}*\mathrm{Rm}$ denotes a universal bilinear combination of the curvature tensor with itself.

Proof. Quoted as standard. The computation is the tensor analogue of the preceding one and its result is commonly summarised in the schematic form displayed, the exact coefficients being irrelevant for the maximum-principle arguments that use only the positivity of the quadratic terms in a suitable sense.

Corollary (the maximum principle for the scalar curvature). Along a Ricci flow on a compact manifold,

$$ R(x,t)\ge\min_{M}R(\cdot,0) \qquad \text{for all } (x,t), $$

and if $R(\cdot,0)\ge0$ then $R(\cdot,t)\ge0$ for all $t$; more generally a lower bound $R\ge-c$ is preserved.

Proof. The evolution equation gives $\partial_tR\ge\Delta R$, a differential inequality of parabolic type. Given $\varepsilon>0$, the function $R_\varepsilon = R+\varepsilon t$ satisfies $\partial_tR_\varepsilon\ge\Delta R_\varepsilon+\varepsilon$, so its minimum over $M\times[0,T']$ cannot be attained in the interior: at an interior minimum, $\partial_tR_\varepsilon\le0$ and $\Delta R_\varepsilon\ge0$, whereas the inequality gives $\partial_tR_\varepsilon\ge\varepsilon>0$. Hence the minimum is attained on the parabolic boundary, and letting $\varepsilon\to0$ gives the claim; the preservation of a lower bound $-c$ follows by applying the same argument to $R+c$.

Remark (the maximum principle for systems). The maximum principle for a scalar quantity is immediate; for the full curvature tensor it requires a pinching condition, because the reaction term $\mathrm{Rm}*\mathrm{Rm}$ does not preserve an arbitrary positivity cone. The Hamilton–Ivey estimate, which bounds the curvature of a three-dimensional Ricci flow below by $-C/(t_0-t)$ for a curvature singularity at $t_0$, is the sharp example: the reaction term is controlled by a positivity cone chosen so that the quadratic terms act in the favourable direction.

Exact Solutions, Solitons and the Flow in Model Cases

Definition. A Ricci soliton is a Riemannian metric $g$ satisfying

$$ R_{ij} + \nabla_iX_j + \nabla_jX_i = \lambda g_{ij} $$

for a complete vector field $X$ and a constant $\lambda$. The soliton is steady if $\lambda=0$, shrinking if $\lambda>0$ and expanding if $\lambda<0$. A solution of the Ricci flow that is self-similar — $g(t)=c(t)\phi_t^*g_0$ for a scaling $c(t)>0$ and a family of diffeomorphisms $\phi_t$ — is generated by a soliton, with $\lambda$ determined by the scaling.

Example (Einstein metrics). If $g_0$ is Einstein, $R_{ij}=\lambda g_{ij}$, then the homothetic family $g(t)=(1-2\lambda t)g_0$ solves the Ricci flow, because both sides scale proportionally; the metric is a fixed point of the normalised flow, and the unnormalised flow is a pure rescaling. The sign of $\lambda$ determines the behaviour: the flow shrinks the metric if $\lambda>0$, expands it if $\lambda<0$ and is stationary if $\lambda=0$.

Example (the round sphere, the flat torus, the hyperbolic manifold). On the round $n$-sphere of radius one, $R_{ij}=(n-1)g_{ij}$, so $g(t)=(1-2(n-1)t)g_0$ shrinks homothetically and the curvature blows up as $t\to\frac{1}{2(n-1)}$, where the metric degenerates to a point. On the flat torus $R_{ij}=0$ and every flat metric is a fixed point. On a compact quotient of hyperbolic $n$-space, $R_{ij}=-(n-1)g_{ij}$, so $g(t)=(1+2(n-1)t)g_0$ expands forever and the curvature decays as $1/t$; the volume grows as $t^{n/2}$ by the evolution-of-volume formula. The three cases are the three Einstein signs and the three asymptotic behaviours of the flow.

Example (the cigar and the Bryant soliton). On $\mathbb{R}^2$ the metric $g=\frac{dx^2+dy^2}{1+r^2}$, $r^2=x^2+y^2$, is a complete steady Ricci soliton, the cigar soliton, rotationally symmetric and asymptotic to a cylinder at infinity; its curvature decays but its injectivity radius does not grow, and it is the model of a steady singularity. In dimension three the analogous complete rotationally symmetric steady soliton is the Bryant soliton, unique up to scaling, asymptotic to a cylinder at infinity and with curvature decaying as $1/s$ in the distance $s$. The two examples are the standard steady solitons, and they show that a complete soliton need not be compact.

Example (the Gaussian soliton). On $\mathbb{R}^n$ the flat metric with the vector field $X=\frac12x\cdot\nabla$ satisfies $R_{ij}+\nabla_iX_j+\nabla_jX_i=g_{ij}$, so the Gaussian soliton is a shrinking soliton; the corresponding self-similar solution of the flow is the rescaled flat metric. The example shows that a soliton need not have positive curvature and that the vector field, not the curvature, may carry the self-similarity; with the opposite sign of the vector field the same flat metric is an expanding soliton, which exhibits the dependence of the soliton type on the sign of $\lambda$.

Singularities and Monotonicity

Definition. A solution $g(t)$ on a maximal interval $[0,T)$ is singular at $T$ if $T<\infty$ and the curvature is unbounded as $t\to T^-$. The singularity is Type I if $\sup_M|\mathrm{Rm}|(x,t)\le C/(T-t)$ and Type II otherwise. A singularity is necklike if, after parabolic rescaling about a sequence of points, the solution converges to a cylinder $\mathbb{R}\times S^{n-1}$.

Theorem (Hamilton–Ivey). Along a Ricci flow on a compact three-manifold that becomes singular at $T<\infty$, the minimum of the sectional curvature satisfies

$$ \lambda_{\min}(x,t)\ge -\frac{C}{T-t} $$

for a constant $C$ depending only on the initial metric; consequently the curvature is bounded below in the blow-up scale, and a blow-up limit of the singularity has nonnegative curvature operator or is a steady soliton.

Proof. Quoted as standard (Hamilton–Ivey). The proof applies the maximum principle to the evolution of the minimum sectional curvature, whose evolution is governed by a reaction term that is favourable once the curvature is sufficiently negative; the resulting bound on the reaction term controls the possible blow-up.

Theorem (Perelman's monotonicity). The entropy functionals

$$ \mathcal{F}(g,f) = \int_M\bigl(R+|\nabla f|^2\bigr)e^{-f}dV, \qquad \mathcal{W}(g,f,\tau) = \int_M\Bigl[\tau\bigl(R+|\nabla f|^2\bigr)+f-n\Bigr](4\pi\tau)^{-n/2}e^{-f}dV $$

are monotone along the coupled flow $\partial_tg=-2R_{ij}$, $\partial_tf=-\Delta f+|\nabla f|^2-R$ (and $\partial_t\tau=-1$ for $\mathcal{W}$): $\frac{d}{dt}\mathcal{F}\ge0$ and $\frac{d}{dt}\mathcal{W}\ge0$, with equality only for the corresponding solitons.

Proof. Quoted as standard (Perelman). The computation is a long integration by parts in which the evolution equations for $g$ and $f$ are substituted into the derivative of the functional, and the result is a sum of squares: $\frac{d}{dt}\mathcal{W} = 2\tau\int\bigl|R_{ij}+\nabla_i\nabla_jf-\frac{1}{2\tau}g_{ij}\bigr|^2(4\pi\tau)^{-n/2}e^{-f}dV$, which is nonnegative and vanishes exactly on a shrinking soliton.

Theorem (no local collapsing). For a Ricci flow on a compact manifold with a curvature bound on a parabolic ball and a lower bound on the injectivity radius at the base point, the solution is $\kappa$-noncollapsed at scale $r$ for a constant $\kappa>0$ depending on the bounds; consequently a region of bounded curvature cannot collapse to lower dimension, and the blow-up limits of a singularity are nonflat.

Proof. Quoted as standard (Perelman). The monotonicity of the reduced volume — a localised version of the entropy, defined by a path integral over backwards geodesics — supplies a lower bound for the volume of a parabolic ball in terms of its radius and the curvature bound; the noncollapsing is what rules out the formation of lower-dimensional singular sets and makes the blow-up analysis possible.

Remark (surgery and long-time behaviour). The local structure of a three-dimensional singularity is that of a neck, a cap or a compact positively curved component, and the Ricci flow with surgery cuts the necks and caps the resulting boundaries, continuing the flow on the modified manifold; the monotonicity and noncollapsing theorems supply the estimates that make the surgery possible finitely many times in each compact time interval. This is the analysis that underlies the classification of three-manifolds by the flow; the statement of the classification and the topology it produces belong to the surrounding theory, and the part of it that is pure geometry of three-manifolds is the concern of Part II. What belongs to this article is the analytic content: the parabolic well-posedness, the curvature evolution, the maximum principles, the monotone quantities and the compactness derived from them.

Convergence Theorems

Theorem (Hamilton's convergence theorem). Let $M$ be a compact three-manifold with a Riemannian metric $g_0$ of positive Ricci curvature. Then the normalised Ricci flow starting at $g_0$ exists for all time and converges, after normalisation, to a metric of constant positive sectional curvature; consequently $M$ admits a spherical space form structure.

Proof. Quoted as standard (Hamilton 1982). The curvature pinching improves along the flow: the maximum principle applied to the evolution of the curvature and of the traceless Ricci tensor shows that the eigenvalues of the curvature operator become close to one another, the flow does not develop a singularity before the normalised time, and the limit is Einstein; in dimension three an Einstein metric of positive scalar curvature is of constant sectional curvature.

Theorem (compactness and convergence of noncollapsed flows). The class of pointed solutions of the Ricci flow with a uniform curvature bound on a parabolic ball and a uniform lower bound on the injectivity radius is compact in the pointed Cheeger–Gromov sense; consequently every blow-up limit of a singularity exists and is itself a complete ancient solution of the flow.

Proof. Quoted as standard (Hamilton's compactness theorem, in the form given by Perelman). The curvature bounds give local derivative bounds by the parabolic regularity theory, the injectivity-radius bound prevents collapse, and the Arzelà–Ascoli diagonal argument in a sequence of balls gives a smooth limit after passing to a subsequence; the limit solves the flow because the equation is closed under smooth convergence.

Remark (solitons as singularity models). The blow-up limits of the preceding theorem are ancient solutions — solutions defined for all negative time with a curvature bound — and the classification of ancient solutions in low dimension yields solitons as models. In dimension three the shrinking solitons are the round sphere, the shrinking cylinder and their quotients, and the singularity models are their blow-ups; the monotonicity of the entropy identifies the solitons as the equality cases, so that the analytic criterion and the geometric classification agree.

Summary

Ricci flow is the evolution $\partial_tg_{ij}=-2R_{ij}$ of a Riemannian metric, a weakly parabolic quasilinear equation whose degeneracy is the diffeomorphism invariance and whose short-time existence is obtained by the DeTurck gauge trick; the solution is unique and maximal until the curvature blows up. The volume evolves by $\partial_t\mathrm{Vol}=-\int R\,dV$, the scalar curvature by $\partial_tR=\Delta R+2|R_{ij}|^2$, the Ricci tensor by the Lichnerowicz Laplacian $\partial_tR_{ij}=\Delta R_{ij}+2R_{ikjl}R^{kl}-2R_{ik}R^k{}_j$, and the Riemann tensor by $\partial_t\mathrm{Rm}=\Delta\mathrm{Rm}+\mathrm{Rm}*\mathrm{Rm}$; the scalar equation is a reaction-diffusion equation, and the maximum principle preserves lower bounds on $R$ and, under pinching hypotheses, on the full curvature. Einstein metrics solve the flow homothetically, with the round sphere shrinking to extinction, the flat torus stationary and the hyperbolic metric expanding; the cigar, Bryant and Gaussian solitons are the complete self-similar solutions and the shrinking solitons are the equality cases of the monotone entropy functionals $\mathcal{F}$ and $\mathcal{W}$, whose monotonicity, together with the no-local-collapsing theorem, controls the blow-up of singularities. Positively Ricci curved compact three-manifolds converge under the normalised flow to constant curvature, and the compactness of noncollapsed flows makes the singular limits ancient solutions classified by solitons.

Summary of Notation

Symbol	Meaning
$g(t)$, $g_{ij}$	Evolving Riemannian metric
$R_{ij}$, $R$, $\mathrm{Rm}$	Ricci, scalar and Riemann curvature tensors
$\partial_tg=-2R_{ij}$	Ricci flow equation
$\bar R$	Average scalar curvature of the normalised flow
$\mathrm{Vol}(M,g)$	Volume, with $\partial_t\mathrm{Vol}=-\int R\,dV$
$\Delta$, $\Delta_L$	Laplace–Beltrami operator; Lichnerowicz Laplacian
$\mathrm{Rm}*\mathrm{Rm}$	Quadratic curvature combination in the curvature evolution
$X$, $\lambda$	Soliton vector field and constant; steady, shrinking, expanding
$T$, $\|\mathrm{Rm}\|$	Singular time and curvature norm; Type I condition
$\mathcal{F}$, $\mathcal{W}$, $\tau$	Perelman entropy functionals and the scale parameter
$\kappa$-noncollapsed	No local collapsing at scale $r$ with constant $\kappa$
Cheeger–Gromov	Pointed convergence of metric spaces and of flows