the classification of locally conformally flat yamabe solitons

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Advances in Mathematics 240 (2013) 346–369www.elsevier.com/locate/aim

The classification of locally conformally flat Yamabesolitons

Panagiota Daskalopoulosa,∗, Natasa Sesumb

a Department of Mathematics, Columbia University, New York, NY 10027, USAb Department of Mathematics, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA

Received 5 March 2012; accepted 16 March 2013Available online 9 April 2013

Communicated by C. Kenig

Abstract

This paper addresses the classification of locally conformally flat gradient Yamabe solitons. In the firstpart it is shown that locally conformally flat gradient Yamabe solitons with positive sectional curvatureare rotationally symmetric. In the second part the classification of all radially symmetric gradient Yamabesolitons is given and their correspondence to smooth self-similar solutions of the fast diffusion equation onRn is shown. In the last section it is shown that any eternal solution to the Yamabe flow with positive Riccicurvature and with the scalar curvature attaining an interior space–time maximum must be a steady Yamabesoliton.c⃝ 2013 Elsevier Inc. All rights reserved.

Keywords: Yamabe solitons; Fast diffusion; Self-similar

1. Introduction

This paper addresses the classification of complete locally conformally flat Yamabe gradientsolitons.

Definition 1.1. A Riemannian manifold (Mn, gi j ) is called a Yamabe gradient soliton if thereexist a smooth scalar (potential) function f : Mn

→ R and a constant ρ ∈ R such that

(R − ρ) gi j = ∇i∇ j f. (1.1)

∗ Corresponding author.E-mail addresses: [email protected] (P. Daskalopoulos), [email protected] (N. Sesum).

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P. Daskalopoulos, N. Sesum / Advances in Mathematics 240 (2013) 346–369 347

If ρ > 0, ρ < 0 or ρ = 0, then g is called a Yamabe shrinker, Yamabe expander or Yamabesteady soliton respectively. By scaling the metric, we may assume with no loss of generality thatρ = 1,−1, 0 respectively.

When f is a constant function in (1.1) we say that the corresponding Yamabe soliton is atrivial Yamabe soliton. It has been known (see [9,12,18]) that every compact Yamabe soliton isof constant scalar curvature, hence trivial, since in this case the potential function f turns out tobe constant.

Yamabe solitons are special solutions to the Yamabe flow

∂

∂tgi j = −R gi j . (1.2)

This flow was introduced by R. Hamilton [14] as an approach to solve the Yamabe problem onmanifolds of the positive conformal Yamabe invariant. It is the negative L2-gradient flow of thetotal scalar curvature, restricted to a given conformal class. Hamilton [14] showed the existenceof the normalized Yamabe flow (which is the re-parametrization of (1.2) to keep the volume fixed)for all time; moreover, in the case when the scalar curvature of the initial metric is negative, heshowed the exponential convergence of the flow to a metric of constant scalar curvature.

Since then, a number of works have been established on the convergence of the Yamabe flowon a compact manifold to a metric of constant scalar curvature. Chow [8] showed the convergenceof the flow under the conditions that the initial metric is locally conformally flat and of positiveRicci curvature. The convergence of the flow for any locally conformally flat initial metric wasshown by Ye [22]. Inspired by this result, Del Pino and Saez [11] proved the convergence to thesphere of a conformally flat metric on Rn evolving by the Yamabe flow and satisfying a decaycondition at infinity.

More recently, Schwetlick and Struwe [20] obtained the convergence of the Yamabe flow ona general compact manifold under a suitable Kazdan–Warner type of condition that rules outthe formation of bubbles and that is verified (via the positive mass Theorem) in dimensions3 ≤ n ≤ 5. The convergence result for any general compact manifold was established byBrendle [2,3] (up to a technical assumption, in dimensions n ≥ 6, on the rate of vanishing ofthe Weyl tensor at the points at which it vanishes): starting with any smooth metric on a compactmanifold, the normalized Yamabe flow converges to a metric of constant scalar curvature.

Even though the analogue of Perelman’s monotonicity formula is still lacking for the Yamabeflow, one expects that Yamabe soliton solutions model finite time singularities. This expectationhas been justified in Corollary 5.1 of this article which indicates that in certain cases of Type IIsingularities one may expect the steady Yamabe solitons to be the singularity models.

Although the Yamabe flow on compact manifolds is well understood, the complete non-compact case is unsettled. In [10] the authors showed that in the conformally flat case and undercertain conditions on the initial data, which in particular imply that the initial metric admitsthe asymptotic behavior of the cylindrical metric at infinity, complete non-compact solutionsto the Yamabe flow develop a finite time singularity and after re-scaling the metric convergesto the Barenblatt solution (a certain type of a shrinker, corresponding to the Type I singularity,see (1.4) for the definition). The general case even when the solution is conformally equivalentto Rn is not well understood. In fact it is shown in Proposition 1.5 that there exist infinitelymany shrinking solitons which behave as cylinders at infinity and they are all different than theBarenblatt solution given by (1.4). All such solutions are prototypes of Type I singularities of thecomplete non-compact Yamabe flow.

348 P. Daskalopoulos, N. Sesum / Advances in Mathematics 240 (2013) 346–369

Remark 1.2. (i) If gi j defines a Yamabe shrinker according to Definition 1.1, then the (timedependent) metric gi j given by gi j (t) = (T − t)φ∗

t (g), t < T , where φt is a one parameterfamily of diffeomorphisms generated by the vector field X = ∇ f/(T − t), defines an ancientsolution to the Yamabe flow (1.2) (also called a Yamabe shrinker) which vanishes at time T andsatisfies

R −1

T − t

gi j (t) = ∇i∇ j f.

(ii) If gi j defines a Yamabe expander according to Definition 1.1, then the (time dependent)metric gi j defined by gi j (t) = t φ∗

t (gi j ), t > 0, where φt is a one parameter family of diffeomor-phisms generated by X = ∇ f/t , is a solution to the Yamabe flow (1.2) (also called a Yamabeexpander) which is defined on 0 < t < ∞ and satisfies

R −1t

gi j (t) = ∇i∇ j f.

(iii) If gi j defines a Yamabe steady soliton according to Definition 1.1, then the (time dependent)metric gi j defined by gi j (t) = φ∗

t (gi j ), −∞ < t < ∞, where φt is a one parameter family ofdiffeomorphisms generated by ∇ f is an eternal solution to the Yamabe flow (1.2) (also called aYamabe steady soliton) which satisfies

R gi j (t) = ∇i∇ j f.

Our first result establishes the rotational symmetry of locally conformally flat Yamabesolitons.

Theorem 1.3 (Rotational Symmetry of Yamabe Solitons). All locally conformally flat completeYamabe gradient solitons with positive sectional curvature have to be rotationally symmetric.

A number of months after our article first appeared on the arXiv, a related work by Cao,Sun and Zhang [5] was posted. Inspired by our work, it was shown in [5] that every completenontrivial gradient Yamabe soliton admits a special global warped product structure with a one-dimensional base. Consequently, locally conformally flat complete gradient Yamabe solitonswith nonnegative Ricci curvature are rotationally symmetric. A related work by Catino, Man-tegazza and Mazzieri [7] also appeared on the arXiv after article [5] was posted.

We will show at the end of Section 2 that the result in [6] implies that rotationally symmetriccomplete Yamabe solitons with nonnegative sectional curvature are globally conformally flat,

namely gi j = u4

n+2 dx2, where dx2 denotes the standard metric on Rn and u4

n+2 is the conformalfactor. We have the following result.

Proposition 1.4 (PDE Formulation of Yamabe Solitons). Let gi j = u4

n+2 dx2 be a conformallyflat rotationally symmetric Yamabe gradient soliton. Then, u is a smooth solution to the ellipticequation

n − 1m

∆um+ β x · ∇u + γ u = 0, on Rn (1.3)

where β ≥ 0 and

γ =2β + ρ

1 − m, m =

n − 2n + 2

.


In the case of expanders β > 0. In addition, any smooth solution to the elliptic equation (1.3) withβ and γ as above defines a gradient Yamabe soliton.

The above proposition reduces the classification of Yamabe solitons to the classification ofglobal smooth solutions of the elliptic equation (1.3).

To simplify the notation, we will assume from now on that ρ = 1 in (1.1) (and hence in Propo-sition 1.4 as well) in the case of the Yamabe shrinkers, and that ρ = −1 in the case of the Yamabeexpanders. This can be easily achieved by scaling our metric g. The following result provides theclassification of radially symmetric and smooth solutions of the elliptic equation (1.3).

Proposition 1.5 (Classification of Radially Symmetric Yamabe Solitons). Let m = (n − 2)/(n +

2). The elliptic equation (1.3) admits non-trivial radially symmetric smooth solutions if and onlyif β ≥ 0 and γ := (2β + ρ)/(1 − m) > 0. More precisely, we have the following.

i. Yamabe shrinkers ρ = 1: For any β > 0 and γ = (2β + 1)/(1 − m), there exists a oneparameter family uλ, λ > 0, of smooth radially symmetric solutions to Eq. (1.3) on Rn ofslow-decay rate at infinity, namely uλ(x) = O(|x |

−2/(1−m)) as |x | → ∞. We will refer tothem as cigar solutions. In the case γ = βn the solutions are given in the closed form

uλ(x) =

Cn

λ2 + |x |2

11−m

, Cn = (n − 2)(n − 1) (1.4)

and will refer to them as the Barenblatt solutions. When β = 0 and γ = 1/(1 − m) Eq. (1.3)admits the explicit solutions of fast-decay rate

uλ(x) =

Cn λ

λ2 + |x |2

21−m

, Cn = (4n(n − 1))12 . (1.5)

We will refer to them as the spheres.ii. Yamabe expanders ρ = −1: For any β > 0 and γ = (2β − 1)/(1 − m) > −1/(1 − m)

there exists a one parameter family uλ, λ > 0, of smooth radially symmetric solutions toEq. (1.3) on Rn .

iii. Yamabe steady solitons ρ = 0: For any β > 0 and γ = 2β/(1 − m) > 0 there exists aone parameter family uλ, λ > 0, of smooth solutions to Eq. (1.3) on Rn which satisfy theasymptotic behavior uλ(x) = O((log |x |/|x |

2)1/(1−m)), as |x | → ∞. We will refer to themas logarithmic cigars. For β = γ = 0, the solution uλ is a constant, defining the Euclideanmetric on Rn .

In all of the above cases the solution uλ is uniquely determined by its value at the origin.

Remark 1.6 (Self-similar Solutions of the Fast-diffusion Equation). There is a clear connectionbetween Yamabe solitons and self-similar solutions to the fast diffusion equation

∂ u

∂t=

n − 1m

∆um . (1.6)

i. Yamabe shrinkers ρ > 0: The function u is a solution to the elliptic equation (1.3) if andonly if u(x, t) = (T − t)γ u(x (T − t)β) is an ancient solution to (1.6) which vanishes at T . Theexistence of such solutions is proven in [21] (Proposition 7.4) and it was also noted in [13].

ii. Yamabe expanders ρ < 0: The function u is a solution to the elliptic equation (1.3) if andonly if u(x, t) = t−γ u(x t−β) is a solution to (1.6) which is defined for all 0 < t < ∞.


iii. Yamabe steady solitons ρ = 0: The function u is a solution to the elliptic equation (1.3)if and only if u(x, t) = e−γ t u(x e−βt ) is an eternal solution to (1.6). The existence of suchsolutions (without a proof) was first noted in [13].

In all of the above cases, g(t) = u4

n+2 (·, t) dx2 defines a solution to the Yamabe flow (1.2).

Combining the above results leads to the following classification of Yamabe solitons.

Theorem 1.7. Let g be a complete locally conformally flat Yamabe gradient soliton with positive

sectional curvature. Then, g is conformally equivalent to Rn , namely g = u4

n+2 dx2 where usatisfies the elliptic equation (1.3), for some β ≥ 0 and γ := (2β + ρ)/(1 − m) > 0. Theclassification of all such metrics is given in Proposition 1.5.

Conversely, if β ≥ 1/(2m) and γ ≥ 0 then the Yamabe soliton whose conformal factor usatisfies elliptic equation (1.3) has positive sectional curvature.

2. Yamabe solitons are rotationally symmetric

In this section we will establish the rotational symmetry of locally conformally flat Yamabesolitons with nonnegative sectional curvature, Theorem 1.3. Our proof is inspired by the proof ofthe analogous theorem for complete gradient steady Ricci solitons in [4] by Cao and Chen.

Proof of Theorem 1.3. We will first deal with the case of steady solitons

R gi j = ∇i∇ j f (2.1)

where we refer to f as a potential function. The other two cases of shrinkers and expanders canbe treated in the same way as it will be explained at the end of the proof.

Since R > 0, the potential function f is strictly convex and therefore it has at most one criticalpoint. Denote G = |∇ f |

2 and observe that in any neighborhood of the level surface

Σc := {x ∈ M : f (x) = c},

for a regular value c of f (meaning that G = 0), we can express the metric g as

g =1

G( f, θ)d f 2

+ gab( f, θ) dθadθb (2.2)

where (θ2, . . . , θn) denote intrinsic coordinates for Σc.We wish to show that G = G( f ), gab = gab( f ) and that (Σc, gab) is a space form of positive

constant curvature. This would mean that g has the form

g = ψ2( f ) d f 2+ φ2( f ) gSn−1 (2.3)

where gSn−1 denotes the standard metric on the unit sphere Sn−1. As in [4] it can be argued thatf has exactly one critical point, leading to the fact that g is a rotationally symmetric metric onRn .

Next we derive some identities on Yamabe solitons that will be used later in this paper.

Lemma 2.1. If G := |∇ f |2, then

∇G = 2R ∇ f. (2.4)

Furthermore,

(n − 1)∇ R = −Ric (∇ f, ·). (2.5)


Proof. Fix p ∈ M and choose normal coordinates around p so that the metric matrix is diagonalat p. Then,

∇i G = 2 ∇i∇ j f ∇ j f = 2Rgi j∇ j f

implying ∇i G = 2R ∇i f . In other words,

∇G = 2R ∇ f.

Moreover, continuing to compute in normal coordinates around p ∈ M , if we apply ∇k to oursoliton equation ∇i∇ j f = R gi j we obtain

∇k∇i∇ j f = ∇k R gi j

implying that

∇i∇k∇ j f + R jkil∇

l f = ∇k R gi j .

Tracing the previous equation in k and j , we obtain

∇i∆ f + Ril∇l f = ∇i R.

On the other hand, after tracing the soliton equation we get

∆ f = n R

and therefore

n∇i R + Ril∇l f = ∇i R.

We conclude that the following identity holds on any Yamabe steady soliton,

(n − 1)∇ R = −Ric (∇ f, ·). �

In the following proposition we will show that the Ricci tensor of our steady soliton metricg has at most two distinct eigenvalues. Cao and Chen proved the same theorem in [4] usingthe properties of the Cotton tensor together with the Ricci soliton equation. Our proof uses theHarnack expression for the Yamabe flow that has been introduced by B. Chow in [8].

Proposition 2.2. At any point p ∈ Σc, the Ricci tensor of g has either a unique eigenvalueλ or two distinct eigenvalues λ and µ of multiplicities 1 and n − 1 respectively. In eithercase, e1 = ∇ f/|∇ f | is an eigenvector with eigenvalue λ. Moreover, for any orthonormal basise2, . . . , en tangent to the level surface Σc at p, we have

i. Ric(e1, e1) = λ

ii. Ric(e1, eb) = R1b = 0, b = 2, . . . , niii. Ric(ea, eb) = Raaδab, a, b = 2, . . . , n,

where either R11 = · · · Rnn = λ or R11 = λ and R22 = · · · = Rnn = µ.

The proof of Proposition 2.2 will make use of the evolution of the Harnack expression for thescalar curvature, that has been introduced by B. Chow in [8]. We will compute its evolution andexpress it in a form that is convenient for our purposes. This computation does not depend onhaving the soliton equation, but only on evolving the metric by the Yamabe flow.

Assume that we have a complete eternal locally conformally flat Yamabe flow (every steadysoliton is an eternal solution),

gt = −R g, (2.6)


where g has positive Ricci curvature. Choose a vector field X to satisfy

∇i R +1

n − 1Ri j X j = 0. (2.7)

The vector field X is well defined since Ric > 0 (and therefore defines an invertible matrix).Following Chow [8] we define Harnack expression for eternal Yamabe flow, namely

Z(g, X) = (n − 1)∆R + ⟨∇ R, X⟩ +1

2(n − 1)Ri j X i X j + R2. (2.8)

Note that in (2.8) we have dropped the term R/t , due to the fact that we have a solution thatcomes all the way from t = −∞. To simplify the notation, we define � = ∂t − (n − 1)∆.

Lemma 2.3. The quantity Z defined by (2.8) evolves by

�Z = RZ + Ai j X i X j + gkl Ri j (Rgik − ∇i Xk)(Rg jl − ∇ j Xl) (2.9)

where Ai j is the same matrix that Chow defines by (3.13) in [8].

Proof. We have the following equation due to Chow [8] after dropping all terms with 1/t :

�Z = 3RZ − R3+

12(Rki jl − R2

i j )X i X j

−1

2(n − 1)R Ri j X i X j −

(n − 1)(n − 2)2

|∇ R|2

− (n − 1)Ri j∇i R X j + ⟨∇ R,�X⟩ +Ri j X i

n − 1�X j

− 2Ri j∇k X i∇k X j − 2∇k Ri j∇k X i X j − 2(n − 1)⟨∇∇ R,∇ X⟩

+ Ri j∇k X i∇k X j . (2.10)

Since the evolution equation for Z is independent of the choice of coordinates, we may choosecoordinates so that gi j = δi j and the Ricci tensor is diagonal at the point of consideration. By(2.7) we have

⟨∇ R,�X⟩ +Ri j X i

n − 1�X j =

∇ j R +

Ri j X i

n − 1

�X j = 0 (2.11)

and

2Ri j∇k X i∇k X j + 2∇k Ri j∇k X i X j + 2(n − 1)⟨∇∇ R,∇ X⟩

= 2(∇k X i · ∇k(Ri j X j )+ (n − 1)∇k X i∇k∇i R)

= 2∇k X i∇k(Ri j X j + (n − 1)∇i R) = 0. (2.12)

Combining (2.10), (2.11) and (2.12) yields

�Z = 3RZ − R3+

12(Rki jl Rkl − R2

i j )X i X j −1

2(n − 1)R Ri j X i X j

−(n − 1)(n − 2)

2|∇ R|

2− (n − 1)Ri j∇i R X j + Ri j∇k X i∇k X j . (2.13)

We recall the following basic identity that holds on locally conformally flat manifolds

Rki jl =1

n − 2(Rkl gi j + Ri j gkl − Rk j gil − Ril gk j )−

R (gkl gi j − gk j gil)

n − 1.


If we contract this identity by Rkl , we obtain at a point where gi j = δi j and Ri j is diagonal, that

Rki jl Rkl =1

n − 2

|Ric|2 δi j + Ri j R δi j − R2

i jδi j − R2i jδi j −

R

n − 1(R δi j − Ri j )

=

1n − 2

|Ric|2 +

n

n − 1R Ri j − 2R2

i j −R2

n − 1

δi j

and therefore

12(Rki jl Rkl − R2

i j ) =1

2(n − 2)

|Ric|2 +

n

n − 1R Ri j − n R2

i j −R2

n − 1

δi j . (2.14)

We also have ∇i R = −Ri j X j/(n − 1), hence

|∇ R|2

= gi j∇i R∇ j R =

1

(n − 1)2Rik Xk Ril Xl

=1

(n − 1)2Rik Ril Xk Xlδikδil =

1

(n − 1)2R2

i j X i X jδi j (2.15)

and

− (n − 1)Ri j∇i R X j = Rik Xk Ri j X j = R2i j X i X jδi j . (2.16)

Combining (2.13), (2.14) and (2.16) yield

�Z = 3RZ − R3+ Ai j X i X jδi j + Ri j∇k X i∇k X j (2.17)

where

Ai j =1

n − 2

|Ric|2

2+

R Ri j

n − 1−

R2

2(n − 1)−

n

2(n − 1)R2

i j

gi j .

A direct computation gives

2RZ − R3= 2R

(n − 1)∆R + ⟨∇ R, X⟩ +

12(n − 1)

Ri j X i X j + R2

− R3

= 2(n − 1)R ∆R + 2R ⟨∇ R, X⟩ +R Ri j

n − 1X i X j + R3. (2.18)

After taking the covariant derivative ∇k of (2.7) we find that

∇k∇i R +1

n − 1∇k Ri j X j +

1n − 1

Ri j∇k X j = 0, (2.19)

which gives

∇k∇i R = −1

n − 1

∇k Ri j X j + Ri j∇k X j

.

If we contract (2.19) by gik , by the contracted Bianchi identity ∇i Ri j =12∇ j R, we get

∆R +1

2(n − 1)∇ j R X j +

1n − 1

Ri j∇i X j = 0.

By (2.7) and the previous identity we have

2(n − 1)R ∆R =R Ri j

n − 1X i X j − 2R Ri j∇i X j


which combined with (2.18) yields

2RZ − R3=

R Ri j

n − 1X i X j − 2R Ri j∇i X j −

2R Ri j

n − 1X i X j +

R Ri j

n − 1X i X j + R3

= R3− 2R Ri j∇i X j .

It follows from (2.17) that

�Z = RZ + Ai j X i X jδi j + (2RZ − R3+ Ri j∇k X i∇k X j ) (2.20)

where by the discussion above

I := 2RZ − R3+ Ri j∇k X i∇k X j = R3

− 2R Ri j∇i X j + Ri j∇k X i∇k X j .

Hence, at the chosen coordinates at a point where gi j = δi j and Ri j is diagonal, we have

I = R3− 2R Ri j∇i X j + Ri j∇k X i∇k X j

=

i

Ri i (R2gi i − 2R∇i X i + |∇i X i |

2)+

i=k

Ri i |∇k X i |2

=

i

Ri i (R gi i − ∇i X i )2+

i=k

Ri i |∇k X i |2

= gkl Ri j (R gik − ∇i Xk)(R g jl − ∇ j Xl). (2.21)

By combining (2.20) with (2.21) we readily conclude (2.9). The matrix Ai j is the same thatChow defines by (3.13) in [8]. In local coordinates {xi }, where gi j = δi j and the Ricci tensor isdiagonal at the considered point, we have

Ri j =

λ1. . .

λn

hence

Ai j =

ν1. . .

νn

(2.22)

where

νi =1

2(n − 1)(n − 2)

k,l=i,k>l

(λk − λl)2. �

We now give the proof of Proposition 2.2.

Proof of Proposition 2.2. Assume that the solution (2.6) is a steady soliton, namely it satisfies(2.1). Taking the divergence of the above equation, tracing and then taking the Laplacian yield(see [8] for details)

(n − 1)∆R +12⟨∇ R,∇ f ⟩ + R2

= 0.


With our choice of X in (2.7) we have that Z(g, X) = 0, if g is a steady Yamabe soliton. Thenfrom (2.9) we find

Ai j∇i f ∇ j f ≡ 0, on M.

It follows from (2.22) that at every point p ∈ M either all eigenvalues of Ricci tensor arethe same, namely λ1 = · · · = λn = λ, or there are two distinct eigenvalues λ and µ withmultiplicities 1 and n − 1 respectively. In the latter case, say ∇1 f = 0 and ∇i f = 0 fori = 2, . . . , n, then ∇ f = |∇ f | e1, with e1 = ∇ f/|∇ f | an eigenvector of the Ricci tensorand λ2 = · · · = λn . In either case, we conclude that ∇ f/|∇ f | is an eigenvector of Ric. The otherproperties of Ricci curvature listed in the statement of Proposition 2.2 now easily follow. �

To conclude the proof of Theorem 1.3 we need the following lemma.

Lemma 2.4. Let c be a regular value of f and Σc = { f = c}. Then, we have the following.

i. The function G = |∇ f |2 and the scalar curvature R are constant on Σc, that is, they are

functions of f only.ii. The mean curvature H of Σc is constant.

iii. The sectional curvature of the induced metric on Σc is constant.

Proof. Let {e1, e2, . . . , en} be an orthonormal frame with e1 = ∇ f/|∇ f | and e2, . . . , en tangentto Σc. By (2.4) we have

∇aG = 2R ∇a f = 0, a = 2, . . . , n (2.23)

since ea, a = 2, . . . , n are tangential directions to the level surfaces Σc on which f is constant.Furthermore, using (2.5) and Proposition 2.2 we get

(n − 1)∇a R = −Ric(∇ f, ea) = 0, a = 2, . . . , n. (2.24)

Observe that (2.23) and (2.24) prove part (i) of our lemma.The second fundamental form of the level surface Σc is given by

hab =fab

√G

=R gab√

G=

H gab

n − 1

where H = (n − 1)R/√

G is the mean curvature of hypersurface Σc. By part (i), both G and Hare constant on Σc and therefore the mean curvature H of Σc is constant. This proves (ii).

It remains to show that (iii) holds. By the Gauss equation, the sectional curvatures of (Σc, gab)

are given by

RΣcabab = Rabab + haahbb − h2

ab = Rabab +H2

(n − 1)2. (2.25)

Since Wi jkl = 0, we get

Ri jkl =1

n − 2(gik R jl − gil R jk − g jk Ril + g jl Rik)

−R

(n − 1)(n − 2)(gik g jl − gil g jk). (2.26)

Using (2.26) and Proposition 2.2 we obtain

Rabab =2

n − 2Raa −

R

(n − 1)(n − 2)=

R − 2R11

(n − 1)(n − 2). (2.27)


Our goal is to show that ∇a R11 = 0, namely R11 is constant on the level surface Σc. This togetherwith R and H being constant on Σc will imply that the sectional curvatures of Σc are constant.

Recall that our metric g can be expressed as g = G( f )−1d f 2+ hab( f, θ)dθa dθb where

we have denoted by ( f, θ2, . . . , θn) the local coordinates on our soliton and by (θ2, . . . , θn) theintrinsic coordinates for Σc. Performing the computation in local coordinates we find

∇a∇1 R =∂2 R

∂θa∂ f−

l

Γ la1∇l R =

∂2 R

∂θa∂ f− Γ 1

a1∇1 R

since R = R( f ). Furthermore setting θ1 := f and using that g1a = G( f )−1δ1a we obtain

Γ 1a1 =

g1k

2

∂gak

∂ f+∂g1k

∂θa−∂ga1

∂θk

=

g11

2

∂ga1

∂ f+∂g11

∂θa−∂ga1

∂ f

= 0

since ga1 ≡ 0 and ∇ag11 = −∇aG/G2= 0. This implies that

∇a∇1 R =∂

∂ f

∂R

∂θa

= 0. (2.28)

On the other hand by (2.5) we have

(n − 1)|∇ f |

∇1 R = −R11.

Differentiating this equality in the direction of the vector ea and using that ∇aG = 0, whereG = |∇ f |

2, yield

(n − 1)|∇ f |

∇a∇1 R = −∇a R11.

Using (2.28) we conclude that

∇a R11 = 0

that is, R11 = λ is constant on Σc. Since R and R11 are constant on Σc, by (2.27) it follows thatRabab is constant on Σc. Since H is also constant on Σc by part (ii), (2.25) immediately impliesthat the sectional curvatures of Σc are constant, which proves (iii). �

Yamabe Shrinkers and Expanders: We will indicate how one argues in the case of shrinkersand expanders that satisfy (1.1) for ρ = 1 and ρ = −1 respectively. First, the same arguments asbefore yield

∇G = 2R ∇ f, (n − 1)∇ R = −Ric (∇ f, ·)

with G = |∇ f |2. To prove Proposition 2.2 for shrinkers and expanders one may proceed with

exactly the same reasoning and calculation. We still define

Z(g, X) = (n − 1)∆R + ⟨∇ R, X⟩ +1

2(n − 1)Ri j X i X j

+ R2

and choose X to be the vector field such that

∇i R +1

n − 1Ri j X j

= 0.


In the case of Yamabe shrinkers (ρ = 1) and Yamabe expanders (ρ = −1) satisfying (R −ρ) gi j= ∇i∇ j f , putting X = ∇ f and assuming that the Yamabe shrinkers become extinct at T = 0,we get

Z(g, X) =ρ

(−t)R

and therefore

∂Z

∂t=ρ

t2 R −ρ

t((n − 1)∆R + R2).

If we plug all of the above in (2.9), using (1.1) with ρ = 1 we obtain

ρ

t2 R −ρ

tR2

= −ρ

tR2

+ Ai j∇i f ∇ j f +ρ2

t2 R

implying that

Ai j∇i f ∇ j f = 0. (2.29)

In the case of expanders (ρ = −1) we argue similarly as above. Note that in this case we have(R + 1)gi j = ∇i∇ j f and if we consider expanders with positive sectional curvature, f is stillstrictly convex and has at most one critical point.

In the case of Yamabe shrinkers (ρ = 1) we have (R − 1)gi j = ∇i∇ j f and even thoughR > 0, f may not be convex so we need to argue slightly differently, as in [4]. Note that the set{q|∇ f (q) = 0} is of measure zero. The same argument that was used before for steady solitons,implies that locally our soliton is rotationally symmetric. In other words whenever |∇ f |(p) = 0we have rotational symmetry in a neighborhood of the level surface Σ f (p). This means thatlocally our soliton has a warped product structure

g = ds2+ ψ2(s)gSn−1 . (2.30)

Look at a cross section Sn−1 of our manifold at point p, in whose neighborhood the manifoldis rotationally symmetric and we have the warped product structure. Let this cross sectioncorrespond to s = 0. Then s measures the distance from that cross section on both sides fromit and our metric is of the form (2.30) for s ∈ (−a, b) for a, b > 0. As long as the warpingfunction is not zero we can extend the warping product structure. In other words, if ψ(s0) = 0,then by continuity the metric will have the warping product structure ds2

+ψ2(s, θ) gSn−1 a littlebit past s0. Since the set of critical points of f is of measure zero, by using the same argumentsthat were used before to prove the rotational symmetry in a neighborhood of the level surfacescorresponding to regular values, we obtain thatψ(s, θ) is almost everywhere a function of s only.Therefore, by the smoothness of our metric!, g has to be of the form (2.30) everywhere as longas ψ does not vanish. We can have three possible scenarios.

i. g has the form (2.30) for all s ∈ (−∞,∞) in which case our soliton splits off a line and thatcontradicts the positivity of the sectional curvature.

ii. g has the form (2.30) for all s ∈ (−∞, a) and ψ(a) = 0 or for all s ∈ (−b,∞) andψ(−b) = 0 which corresponds to a soliton having only one end and f having exactly onecritical point.

iii. g has the form (2.30) for all s ∈ (−a, b) and ψ(−a) = ψ(b) = 0 which corresponds tohaving a compact Yamabe soliton which is known to be trivial [9,12,18]. �


Proposition 2.5. All complete noncompact rotationally symmetric Yamabe solitons with positiveRicci curvature are either nonflat and globally conformally equivalent to Rn , or flat.

Proof. It is known that every rotationally symmetric metric is locally conformally flat. In [6]it has been shown that all complete locally conformally flat manifolds of dimension n ≥ 3with nonnegative Ricci curvature enjoy nice rigidity properties: they are either flat or locallyisometric to a product of a sphere and a line or they are globally conformally equivalent to Rn

or to a spherical spaceform Sn/Γ . The second case contradicts our assumption on positivity ofthe curvature. Hence, the only possibility for complete nonflat locally conformally flat Yamabesolitons is being globally conformally equivalent to the Euclidean space. �

Proposition 2.5 and Theorem 1.3 reduce the classification of gradient Yamabe solitons withpositive sectional curvature to the classification of radially symmetric Yamabe solitons that are

conformally equivalent to Rn , namely they have the form g = u4

n+2 dx2. We will classify suchsolitons and study their geometric properties in the next two sections.

3. PDE formulation of Yamabe solitons

Our aim in this section is to prove Proposition 1.4. We will assume that the metric g is globallyconformally equivalent to Rn (we will call it conformally flat) and rotationally symmetric andthat satisfies (1.1). We may express g as

g = u(r)4

n+2 (dr2+ r2gSn−1)

where (r, θ2, . . . , θn) denote spherical coordinates. We choose next cylindrical coordinates onRn and define v(s) by

v(s)4

n+2 = r2u(r)4

n+2 , r = es . (3.1)

Then g = v(s)4

n+2 ds2c , where dsc = ds2

+ gSn−1 is the cylindrical metric. Denote by w the

conformal factor in cylindrical coordinates, namely w(s) = v(s)4

n+2 and g = v(s)4

n+2 ds2c =

w(s) ds2c .

We will use index s to refer to the s direction and indices 2, 3, . . . , n to refer to sphericaldirections. By (1.1) we have

(R − ρ) gi j = ∇i∇ j f (3.2)

for a potential function f which is radially symmetric. Using the formulas

∇i∇ j f = fi j − Γ li j fl

and

Γ li j =

gkl

2

∂gik

∂x j+∂g jk

∂xi−∂gi j

∂xk

for a function f = f (s) that only depends on s we have

∇s∇s f = fss − Γ sss fs and ∇i∇i f = −Γ s

ii fs, i ≥ 2.

Since

Γ sss =

ws

2w, Γ s

ii = −ws

2w, i ≥ 2


we conclude that

∇s∇s f = fss −ws fs

2wand ∇i∇i f =

ws fs

2w, i ≥ 2.

The last two relations and the soliton equation (3.2) imply that

fss −ws fs

2w= (R − ρ)w and

ws fs

2w= (R − ρ)w. (3.3)

If we subtract the second equation from the first we get

fss −fsws

w= 0.

This is equivalent to ( fs/w)s = 0 (since w > 0) which implies that

fs

w= C. (3.4)

The scalar curvature R of the metric g = w(s) (ds2+ gSn−1) is given by

R = −4(n − 1)

n − 2w−

n+24

(w

n−24 )ss −

(n − 2)2

4w

n−24

. (3.5)

The second equation in (3.3) and (3.4) imply that

ws =2C(R − ρ)w. (3.6)

Combining (3.5) and (3.6) gives

ws = −8(n − 1)C(n − 2)

w−n+2

4 +1(w

n−24 )ss −

(n − 2)2

4w

n−24 + ρ

(n − 2)4(n − 1)

wn+2

4

.

Setting θ := Cm/(2(n − 1)) we conclude that w satisfies the equation

(wn−2

4 )ss + θ (wn+2

4 )s −(n − 2)2

4w

n−24 + ρ

(n − 2)4(n − 1)

wn+2

4 = 0. (3.7)

To facilitate future references we also remark that (3.7) can be re-written as

wss =(α − 1)α

w2s

w− (α + 1) θ wsw +

4αw −

α + 4α

ρ w2, α =4

n − 2. (3.8)

We conclude from (3.7) that g = v(s)4

n+2 ds2c = w(s) ds2

c is a Yamabe soliton if and only if vsatisfies the equation

(vn−2n+2 )ss + θ vs −

(n − 2)2

4v

n−2n+2 + ρ

(n − 2)4(n − 1)

v = 0. (3.9)

If we go back to Euclidean coordinates, i.e. we set

u4

n+2 (r) = e−2 sv4

n+2 (s), s = log r (3.10)


then, after a direct calculation, we conclude that u satisfies the elliptic equation

∆un−2n+2 + θ x · ∇u +

11 − m

(2θ +m

n − 1ρ) u = 0. (3.11)

Recalling the notation m := (n − 2)/(n + 2), Eq. (3.11) takes the form

n − 1m

∆um+ β x · ∇u + γ u = 0 (3.12)

with

β =n − 1

mθ and γ =

2β + ρ

1 − m.

Observe also that if u is a radially symmetric smooth solution of Eq. (3.12) then the above dis-

cussion (done backwards) implies that g = u4

n+2 dx2 satisfies the Yamabe soliton equation (1.1)with potential function f defined in terms of w by (3.4). To finish the proof of Proposition 1.4we need to show the following.

Claim 3.1. If g = u4

n+2 dx2 defines a complete Yamabe gradient soliton, then β ≥ 0. In the caseof a noncompact Yamabe shrinker or expander, β > 0.

Proof of Claim. We have seen in Remark 1.6 that a Yamabe soliton g = u4

n+2 dx2 defines a

solution g = u4

n+2 dx2 of the Yamabe flow (2.6), or equivalently, of the fast diffusion equation(1.6). Hence, if g is a Yamabe shrinker or a steady soliton, then the scalar curvature R of gsatisfies R ≥ 0 (this can be seen by using the Aronson–Benilan inequality). It follows that thescalar curvature R = −

4 (n−1)n−2 u−1∆um of the metric g satisfies R ≥ 0 as well. Eq. (3.12) implies

that

R = (1 − m) (γ + βr (log u)r ) = (2β + ρ)+ (1 − m)βr (log u)r (3.13)

since 1 − m = 4/(n + 2) and γ (1 − m) = 2β + ρ. Hence, R(0) = (1 − m) γ = 2β + ρ. Weconclude that γ ≥ 0 on a Yamabe shrinker or a steady soliton (since R(0) ≥ 0).

In the case of a Yamabe shrinker, it is shown in Proposition 7.4 in [21] that γ > 1/(1−m), orequivalently β > 0, (otherwise (3.12) does not admit a smooth global solution). This in particularimplies that on a Yamabe shrinker R(0) > 1.

In the case of a Yamabe steady soliton, γ ≥ 0 implies that β ≥ 0 as well. If γ = 0 then β = 0and it follows from (3.4) that w and hence u are constant (remember that C = 2(n − 1)θ/m in(3.4) and θ = mβ/(n − 1)). In this case u defines the flat metric.

It remains to prove the claim for Yamabe expanders. To this end, we observe first that if therewere a smooth solution of (3.12) with β ≤ 0 (which implies that γ ≤ −ρ/2 < 0 as well) then,u(x, t) := t−γ u(x t−β) would be a solution of (1.6) with initial data identically equal to zero.The uniqueness result in [17] would imply that u ≡ 0. Hence, β > 0. This concludes the proofof the claim and Proposition 1.4. �

4. Classification of radially symmetric Yamabe solitons

In this section we discuss the existence of radially symmetric and conformally flat Yamabesolitons. As we have seen in the previous section, this is equivalent to having a smooth positiveglobal solution of Eq. (1.3). The existence of such solutions is stated in Proposition 1.5. Before


we proceed with the proof of this proposition we will establish some preliminary results. Webegin by recalling the following result of S.-Y. Hsu (Lemma 1.1 in [19]), whose sketch of theproof will be included here due to the completeness of the argument.

Lemma 4.1 (S.-Y. Hsu [19]). Let u > 0 be a smooth radially symmetric solution to Eq. (3.12) onBR0(0), for some R0 > 0. Assume that β > 0, γ = 0 and mγ < β (n − 2). Then, for allr ∈ [0, R0), we have

u + k rur > 0, k :=β

γ

and

ur < 0 if γ > 0 and ur > 0 if γ < 0.

Proof. Define h(r) := u(r)+ k rur (r), r ∈ [0, R0). A direct computation shows that

h′+

n − 2 − (m/k)

r− (1 − m)

ur

u+

β

n − 1ru1−m

h =

n − 2 − (m/k)

ru > 0

since m/k < n − 2 by assumption. If H(r) := u(r)m−1 exp β

n−1

r0 u(ρ)1−m ρ dρ

> 0, then

the previous inequality implies that

d

dr

rn−2−m/k H(r) h(r)

> 0

hence h(r) > 0 for all r ∈ [0, R0). Furthermore, by Eq. (3.12) we have

n − 1m

r1−n rn−1(um)r

r = −γ h(r)

implying that ur < 0 if γ > 0 and ur > 0 if γ < 0. �

The following lemma concerns the Yamabe shrinkers only.

Lemma 4.2. Assume that g = v(s)4

n+2 ds2c is a rotationally symmetric Yamabe shrinker ex-

pressed in cylindrical coordinates, namely v satisfies the equation

(vn−2n+2 )ss +

m

n − 1β vs −

(n − 2)2

4v

n−2n+2 +

(n − 2)4(n − 1)

v = 0. (4.1)

If β ≥ 1/(2m) then

v(s) ≤(n − 1)(n − 2)

n+24 for all s ∈ R.

Proof. If β = 1/(2m) our solution is given by (1.4) and the desired inequality follows immedi-ately. In the case β > 1/(2m), if the desired inequality fails, there exists a local maximum point

s0 ∈ R of v such that v(s0) >(n − 1)(n − 2)

n+24 and v(s0) ≥ v(s) for all s ≤ s0. Multiply

Eq. (4.1) by eλs , λ := (n − 2)/2 and integrate on (−∞, s0]. Using that v′(s0) = 0 we obtain:

λ2 s0

−∞

vn−2n+2 eλs ds − λ v

n−2n+2 (s0) eλs0 +

mβ

n − 1v(s0) eλs0 −

λmβ

n − 1

s0

−∞

veλs ds

=

s0

−∞

(n − 2)2

4v

n−2n+2 −

n − 24(n − 1)

v

eλs ds. (4.2)


By our choice of λ we conclude

vn−2n+2

n − 2

mβ v

4n+2 (s0)−

12(n − 1)(n − 2)

eλs0 =

m

2(β −

12m)

s0

−∞

v eλs ds.

Since v(s0)4

n+2 ≥ (n − 1)(n − 2) and v(s0) = vn−2n+2 (s0) v

4n+2 (s0), we get

v(s0) (β −1

2m) eλs0 ≤

n − 22

(β −1

2m)

s0

−∞

v eλs ds.

Since β > 1/(2m), we may divide the previous inequality by β − 1/(2m) and use that λ =

(n − 2)/2 to conclude that

v(s0) eλs0 ≤ λ

s0

−∞

v eλs ds ≤ λv(s0)

s0

−∞

eλs ds = v(s0) eλs0 .

It follows that v(s) ≡ v(s0) for all s ≤ s0, which is impossible since lims→−∞ v(s) = 0. �

Next, using Lemmas 4.1 and 4.2 we prove the following proposition concerning properties ofthe scalar curvature R.

Proposition 4.3. Let g = u(r)4

n+2 dx2 be a rotationally symmetric and conformally flat Yamabesoliton with scalar curvature R. Assume β > 0.

(i) If g is a Yamabe expanding soliton with γ > 0, or a Yamabe shrinking, or a steady soliton,then R ≥ 0. In the case of a Yamabe expanding soliton with γ < 0, we have R ≤ 0.

(ii) If g is a Yamabe shrinking soliton with β ≥ 1/(2m), then R ≥ 1.(iii) In the case of a steady soliton, an expander with γ > 0 and a shrinker with β ≥ 1/(2m),

the scalar curvature R is a decreasing function in r .(iv) In all the cases of Yamabe solitons considered in (iii) we have R > ρ, where ρ = −1, 0, 1

for expanders, steady solitons and shrinkers, respectively.

Proof. As observed in the previous section, in the case of Yamabe shrinkers and steady solitonsan immediate consequence of the Aronson–Benilan inequality for the fast diffusion equation isthat R ≥ 0.

In the case of a Yamabe expander (ρ = −1) the condition mγ < β (n−2) in Hsu’s Lemma 4.1is satisfied for any β > 0, hence u + k rur > 0. On the other hand, by (3.13) we have

R = (1 − m)γ1 + k r (log u

r )

which immediately implies that R > 0 if γ > 0 and R < 0 if γ < 0, proving part (i).To show (ii) we first observe that since

R − 1 = β2 + (1 − m) r (log u)r

and β > 0, we have R ≥ 1 near r = 0. To show that this inequality is preserved for all r > 0,we pass to cylindrical coordinates recalling that

vs =n + 2

2C(R − 1) v, C = 2β > 0

which follows from (3.6) with ρ = 1 because w = v4

n+2 . From our previous observation we havelims→−∞ R − 1 > 0, yielding that vs > 0 for s close to −∞. By Lemma 4.2 and (4.1) we have

vn−2n+2

ss +

mβ

n − 1vs > 0.


If vs(s0) = 0 at some s0, it follows thatv

n−2n+2

ss |s=s0 ≥ 0, implying vss |s=s0 ≥ 0. We conclude

that vs ≥ 0 for all s, which readily implies that R ≥ 1 for all s. This proves part (ii).It remains to show that in certain cases R is decreasing in r . It follows (see in [8]) that the

scalar curvature R of a Yamabe soliton satisfies the elliptic equation

(n − 1)∆g R +12

⟨∇ R,∇ f ⟩g + R (R − ρ) = 0, ρ ∈ {0,+1,−1}.

In the case of a rotationally symmetric soliton we have shown in (3.4) that in cylindricalcoordinates we have fs = C w, where C = 2β. It follows that (in radial plane coordinates)R satisfies the equation

(n − 1)∆R + 2(n − 1)m ∇ R · ∇ log u + β x · ∇ R u1−m+ R (R − ρ) u1−m

= 0 (4.3)

where the Laplacian and the gradient are taken with respect to the usual Euclidean metric. Byparts (i) and (ii), R (R − ρ) ≥ 0 everywhere, in the considered cases. Since u is strictly positive,Eq. (4.3) implies that R cannot achieve local minimum at a point x ∈ Rn , unless R ≡ 0 or

R ≡ ρ. The latter possibility and (3.6) would imply that v = wn+2

4 is constant in s, whichcorresponds to a singular solution to (3.11) and not our solution. Since R is a radial function andR(0) > 0 in the considered cases, it follows that R must be a decreasing function of r and thisproves part (iii).

To show part (iv) note that in the case of expanders for γ > 0 (ρ = −1), since R ≥ 0, weautomatically have R > ρ. In the case of steady solitons (ρ = 0), scalar curvature R satisfies

∂

∂tR = (n − 1)∆R + R2,

so by the strong maximum principle either R > 0 = ρ, or R ≡ 0. The latter is impossible sinceR(0) = 2β > 0, unless (Rn, g) defines a flat Euclidean metric. In the case of shrinkers (ρ = 1),by part (ii) we already have R ≥ 1. If we denote by R := R − 1, we have

∂

∂tR ≥ (n − 1)∆R

so by the strong maximum principle either R > 0 (or equivalently R > 1) or R ≡ 0. In the lattercase we have R ≡ 1, contradicting lims→−∞(R − 1) > 0. �

Proof of Proposition 1.5. We will separate the cases ρ = 0 (steady solitons), ρ = 1 (shrinkers)and ρ = −1 (expanders).

Yamabe shrinkers ρ = 1: In this case the result is proven in Proposition 7.4 in [21] (seealso [13]). We only need to remark that u solves (1.3) if and only if u = (T − t)γ u(x (T − t)β)is an ancient self-similar solution of the fast diffusion equation (1.6).

Yamabe expanders ρ = −1: We look for a smooth global radially symmetric solution to theelliptic equation (1.3) on Rn with β > 0 and γ = (2β − 1)/(1 − m).

It is well known (c.f. in [21], Section 3.2.2) that for any λ > 0, Eq. (1.3) admits a uniquesmooth radial solution u = uλ(r), with uλ(0) = λ, defined in a neighborhood of the origin. Thisfollows via the change of variables

r = es, X (s) =r ur

u, Y (r) = r2 u1−m

which (in the radial case) transforms equation (1.3) to an autonomous system for X and Y .


Hence we only need to show that such solution is globally defined. To this end it suffices toprove that u remains positive and bounded. We will first show that u remains positive for allr > 0. This simply follows from expressing (1.3) as

div

n − 1m

∇um+ β x · u

= (nβ − γ ) u

and observing that (nβ− γ ) u ≥ 0 in our case, as long as u ≥ 0. Integrating in a ball Br (0) givesthe differential inequality

n − 1m

(um)r + βr u ≥ 0

which easily implies the lower bound

u(r) ≥

µ+

β (1 − m)

2(n − 1)r2

−1

1−m

with µ = u(0)m−1. To establish the bound from above we argue as follows. If γ > 0,Lemma 4.1 implies that ur ≤ 0 which yields that u(r) ≤ u(0). If γ < 0, Lemma 4.1 impliesr (log u)r ≤ −γ /β which yields that u(r) ≤ C r−γ /β , r ≥ r0 for some r0 > 0.

Yamabe steady solitons ρ = 0: For any given β > 0 we will establish the existence of a oneparameter family of radial solutions uλ, λ > 0 of Eq. (1.3) with γ = 2β/(1 − m). Notice thatu solves (1.3) if and only if u = e−γ t u(x e−βt ) is an eternal self-similar solution of the fastdiffusion equation (1.6). The existence of such solutions u (without a proof) is noted in [13]. Weonly outline the proof, avoiding the details of standard well known arguments.

It follows from standard ODE arguments that for any λ > 0, Eq. (1.3) admits a unique smoothradial solution uλ with uλ(0) = λ, which is defined in a neighborhood of the origin. Hence weonly need to show that this solution is globally defined and satisfies the asymptotic behavioru(r) ≈ (log r/r2)1/1−m , as r → ∞.

To this end it is more convenient to work in cylindrical coordinates, where v(s) =

r2/(1−m) u(r) with s = log r satisfies Eq. (3.9) with ρ = 0, θ > 0, namely

(vm)ss + θ vs −(n − 2)2

4vm

= 0. (4.4)

Assuming that v is defined on −∞ < s ≤ s0, for some s0 ∈ R, we first observe that vs ≥ 0 forall s ≤ s0. Indeed, since v(−∞) = 0 and v > 0, it follows that vs > 0 near −∞. To show thatthe inequality is preserved we argue that near a point s1 < s0 at which vs(s1) = 0, we still havev(s1) > 0, hence by the above equation (vm)ss > 0, which implies that (vm)s has to increaseand hence it cannot vanish.

Once we know that vs ≥ 0, Eq. (4.4) implies that (vm)ss ≤ Cvm , C = (n − 2)2/4. Settingh = (vm)s and considering h as a function of z = vm we find that h satisfies hh′

≤ Cz, orequivalently f := h2 satisfies f ′

≤ 2Cz. Since f (0) = 0 (this corresponds to s = −∞) weconclude that f (z) ≤ Cz2, or (vm)s ≤ C vm , which readily implies that vm will remain finite forall s ∈ R. This proves that for each λ the solution uλ is globally defined on Rn .

It remains to show that u(r)1−m≈ r−2 log r , as r → ∞. This will be shown separately in

what follows. �


Proposition 4.4. If g = u4

n+2 dx2 is a radially symmetric steady soliton as in Proposition 1.5,then

u4

n+2 (r) = O

log r

r2

, as r → ∞. (4.5)

Proof. We will use cylindrical coordinates and show that if g = w(s) ds2c is a non-trivial steady

Yamabe soliton, then w(s) = O(s) as s → ∞. This is equivalent to (4.5) via the cylindrical

transformation w(s) = r2u(r)4

n+2 and r = es . More precisely, we will show there exist constantsc,C > 0 so that

c s ≤ w(s) ≤ C s, as s → ∞. (4.6)

Recall that w satisfies Eq. (3.8) with ρ = 0, namely

wss =(α − 1)α

w2s

w− (α + 1) θ wsw +

4αw, α =

4n − 2

. (4.7)

Assume first α := 4/(n − 2) < 1. We will first show the bound from above. By (4.7),

wss ≤ (α + 1)θ w

4α(α + 1)θ

− ws

.

Since lims→−∞ws = 0 and w > 0, the above inequality then implies that

ws ≤4

α(α + 1)θ

giving us the upper bound in (4.6). For the lower bound first observe that lims→∞w(s) = +∞

(since solutions of the Yamabe flow with w ≤ C , or equivalently u ≤ C r−2/(1−m), vanish infinite time which is impossible for a steady soliton). If for some large s the right hand side in(4.7) is negative, that is,

w2s +

θα(α + 1)1 − α

wsw2−

41 − α

w2≥ 0

then setting A = θα(α + 1)/(1 − α) and B = 4/(1 − α), we have (since ws ≥ 0) that

2ws ≥ −Aw2+ w2

A2 +

4B

w2 ≥ B w

when w is very large (which is true always when s → ∞). This is impossible since we havejust shown that ws remains bounded, as s → ∞. We conclude that the right hand side in (4.7) ispositive when s is sufficiently large, hence ws is increasing. The bound from below in (4.6) nowreadily follows.

The case α > 1 can be treated similarly as above. The case α = 1 is simpler. �

We conclude this section by showing the positivity of the sectional curvatures of the Yamabesolitons found in Proposition 1.5 in most of the cases.

Proposition 4.5. The logarithmic cigars, the Yamabe expanders, and the Yamabe shrinkers withβ ≥ 1/(2m), that are shown to exist in Proposition 1.5 have strictly positive sectional curvaturesas long as γ := (2β + ρ)/(1 − m) > 0.


Proof. Recall that if w(s) is the conformal factor in cylindrical coordinates, then by (3.6) wehave

ws =1β(R − ρ)w

where ρ = 0 for steady solitons, ρ = −1 for expanders and ρ = 1 for shrinkers. The aboveidentity and Proposition 4.3 imply that ws > 0 since we are assuming γ > 0 and β ≥ 1/(2m).

To express the sectional curvatures of the metric g = w ds2c in terms of w we consider the

geodesic distance s from the origin, that is,

ds =

w(s) ds or s(s) =

s

−∞

w(u) du.

Then our metric reads as g = ds2+ ψ2(s) gSn−1 , with ψ2(s) = w(s) and s ∈ [0,∞). Note that

we have ψ(0) = 0. Differentiating ψ(s)2 = w(s) in s yields

2ψψs√w = ws .

Since ws > 0, we have ψs > 0.Denote by K0 and K1 the sectional curvatures of the 2-planes perpendicular to the spheres

{s} × Sn−1 and the 2-planes tangential to these spheres, respectively. They are given by

K0 = −ψs s

ψand K1 =

1 − ψ2s

ψ2 .

We will next show that

K0 > 0 and K1 > 0. (4.8)

We will first observe that K0 ≥ 0, namely that −ψs s ≥ 0. Indeed, by direct calculation this isequivalent to −(logw)ss > 0. By (3.6) the last inequality is equivalent to Rs ≤ 0 which followsfrom Proposition 4.3. To show that K1 ≥ 0 we first observe that the inequality K0 ≥ 0 impliesthat ψs s ≤ 0. By Proposition 4.1 in [1] and ψs > 0 we have

lims→0

ψs = 1.

Since ψs s ≤ 0 we obtain that

0 < ψs ≤ 1, for all s ∈ [0,∞).

This shows that K1 ≥ 0.We will next prove that K0 and K1 are strictly positive. We have observed above that ws > 0.

Recall that w satisfies Eq. (3.8) which can be rewritten as

wwss +n − 6

4w2

s +θ

mw2ws +

ρ

n − 1w3

− (n − 2)w2= 0, (4.9)

where θ = mβ/(n − 1) > 0 and ρ = 0,−1, 1, in the case of the logarithmic cigars, expandersand shrinkers, respectively. We have just shown that K0 ≥ 0 and K1 ≥ 0, which are equivalentto w2

s − wwss ≥ 0 and 4w2− w2

s ≥ 0, respectively. Assume that K0 = 0 at an interior point s.Then, at that particular point, which is an interior minimum point for K0, we have (K0)s = 0, orequivalently,

wwss = w2s , wwsss = wswss =

w3s

w. (4.10)


Combining the first identity with Eq. (4.9) yields

n − 24

w2s +

θ

mw2ws +

ρ

n − 1w3

− (n − 2)w2= 0 (4.11)

satisfied at the interior minimum of K0. If we differentiate (4.9) in s and use (4.10) to eliminatewsss and wss we obtain

n − 24

w2s +

3θ2m

w2ws +3ρ

2(n − 1)w3

− (n − 2)w2ws = 0 (4.12)

holding at the interior minimum point for K0. After dividing (4.12) by ws and subtracting (4.11)from (4.12) we obtain

ρ

n − 1w3

+θ

mw2ws = 0. (4.13)

Since ws > 0 we see that this is impossible for ρ ≥ 0. That shows the logarithmic cigarsand the shrinkers with β ≥ 1/(2m) have K0 > 0. In the case of the expanders (ρ = −1)when γ > 0, subtracting (4.13) from (4.11) yields ws = 2w, which combined with (4.13) givesθ = m/(2(n−1)), which is equivalent to β = 1/2 in (3.12) and therefore γ = 0. This contradictsγ > 0.

Similarly, assume K1 = 0 at some interior point s. Then (K1)s = 0 at that point, implying

ws = 2w, wss = 4w.

If we plug these back in (4.9) we obtain

2θmw3

+ρ

n − 1w3

= 0 (4.14)

which is impossible for ρ ≥ 0, covering the logarithmic cigars and the shrinkers. In the case ofthe expanders (ρ = −1), when γ > 0, identity (4.14) implies again that θ = m/(2(n − 1)), orβ = 1/2 and therefore γ = 0 which again contradicts that γ > 0. This finishes the proof of (4.8)and the proposition. �

5. Eternal solutions to the Yamabe flow

As a corollary of the proof of Theorem 1.3 we have the following rigidity result for eternalsolutions to the Yamabe flow that can be viewed as the analogue of Hamilton’s theorem foreternal solutions to the Ricci flow.

Corollary 5.1. Let g(x, t) be a complete eternal solution to the locally conformally flat Yamabeflow on a simply connected manifold M with uniformly bounded sectional curvature and strictlypositive Ricci curvature. If the scalar curvature R assumes its maximum at an interior space–timepoint P0, then g(x, t) is necessarily a gradient steady soliton.

From (2.7) we have that X j = −(n − 1)R−1i j ∇ Ri . Since

Rt = (n − 1)∆R + R2

and since at the point P0 = (x0, t0) where R assumes its maximum, we have ∂R/∂t = 0 and∇i R = 0, we conclude that

Z(g, X) = 0 at P0.


The idea is to apply the strong maximum principle to get that Z ≡ 0, which implies that∇i X j = R gi j (this will follow from the evolution equation for Z ).

To finish the proof of Corollary 5.1 we need the following version of the strong maximumprinciple.

Lemma 5.2. If Z(g, X) = 0 at some point at t = t0, then Z(g, X) ≡ 0 for all t < t0.

Proof. The proof is similar to the proof of Lemma 4.1 in [15]. For the convenience of a readerwe will include the main steps of the proof. In what follows we denote by ∆ the Laplacian withrespect to the metric gi j (·, t). Our lemma will be a consequence of the usual strong maximumprinciple, which assures that if we have a function h ≥ 0 which solves

ht = ∆h

for t ≥ 0 and in addition h > 0 at some point when t = 0, then h > 0 everywhere for t > 0.Assume there is a t1 < t0 such that Z(g, X) = 0 at some point at time t1. We may assume,

without loss of generality, that t1 = 0. Define F0 := Z(0) and allow F0 to evolve by the equation

Ft = (n − 1)∆F.

By the result of Chow we know that F0 = F(·, 0) ≥ 0 and therefore by the maximum principleit will remain so for t ≥ 0. By our assumption there is a point at t = 0 at which F(·, 0) > 0. Bythe strong maximum principle we conclude that F > 0 everywhere as soon as t > 0.

Take φ = δeAt f (x) where f (x) is the function constructed in [16] and satisfies f (x) → ∞

as x → ∞, f (x) ≥ 1 everywhere and has all covariant derivatives bounded. We may choose Asufficiently large (depending on δ) so that

φt > (n − 1)∆φ.

Observe next that since R, Z ≥ 0, Ai j X i X j ≥ 0 and Ric ≥ 0, all terms on the right hand side of(2.9) are nonnegative and therefore

Z t ≥ (n − 1)∆Z .

Hence, Z := Z − F + φ satisfies the differential inequality

Z t ≥ (n − 1)∆Z − Ft + (n − 1)∆F + φt − (n − 1)∆φ

and by the choice of φ and f ,

Z t > (n − 1)∆Z .

Since φ(x) → ∞ as x → ∞, Z attains the minimum inside a bounded set and by the maximumprinciple we have

(Zmin)t > 0

which implies that

Zmin(t) ≥ Zmin(0) = φ(0) > 0.

We conclude that Z ≥ F − φ everywhere for t ≥ 0. We now let δ → 0 in the definition of φ.This yields

Z ≥ F > 0 as soon as t > 0. (5.1)


On the other hand, Z(g, X) = 0 at time t0 > 0 at the point where R attains its maximum, whichcontradicts (5.1). This implies Z(g, X) ≡ 0 everywhere for t < t0 and this finishes the proof ofLemma 5.2. �

Proof of Corollary 5.1. The result readily follows from Lemmas 2.3 and 5.2. Since Z ≡ 0 andsince all terms on the right hand side of (2.9) are nonnegative, we obtain from (2.9) the identity

∇i X j = R gi j

that is, g is a steady soliton. Since ∇i X j = ∇ j X i and since our manifold is simply connected,the vector field X is a gradient of a function, which means that the metric g is a gradient steadysoliton. �

Acknowledgments

The authors are grateful to Robert Bryant and Richard Hamilton for many fruitful discussions.They are also indebted to the referee of the paper for many useful comments. The first authorwas partially supported by NSF grant 0604657. The second author was partially supported byNSF grant 0905749.

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http://arxiv.org/1108.6316

the classification of locally conformally flat yamabe solitons

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