Type II Singularities on complete non-compact Yamabe flow

This work concerns with the existence and detailed asymptotic analysis of Type II singularities for solutions to complete non-compact conformally flat Yamabe flow with cylindrical behavior at infinity. We provide the specific blow-up rate of the maximum curvature and show that the solution converges, after blowing-up around the curvature maximum points, to a rotationally symmetric steady soliton. It is the first time that the steady soliton is shown to be a finite time singularity model of the Yamabe flow.


Introduction and preliminaries
Let (M, g 0 ) be a Riemannian manifold without boundary of dimension n ≥ 3. For a mertric g = u 4 n+2 g 0 which is conformal to g 0 , the scalar curvature R of g is given in terms of the scalar curvature R 0 of g 0 by where ∆ g0 denotes the Laplace Beltrami operator with respect to g 0 andc n = 4(n − 1)/(n − 2).
In 1989 R. Hamilton introduced the Yamabe flow as an approach to solve the Yamabe problem on manifolds of positive conformal Yamabe invariant. In the case where M is compact the long time existence and convergence of Yamabe flow is well understood. Hamilton [H] himself showed the existence of the normalized Yamabe flow (which is the re-parametrization of (1.1) to keep the volume fixed) for all time; moreover, in the case when the scalar curvature of the initial metric is negative, he showed the exponential convergence of the flow to a metric of constant scalar curvature. Chow [Ch] showed the convergence of the flow, under the conditions that the initial metric is locally conformally flat and of positive Ricci curvature. The convergence of the flow for any locally conformally flat initially metric was shown by Ye [Ye]. Schwetlick and Struwe [SS] obtained the convergence of the Yamabe flow on a general compact manifold under a suitable Kazdan-Warner type of condition that rules out the formation of bubbles and that is verified (via the positive mass Theorem) in dimensions 3 ≤ n ≤ 5. The convergence result, in its full generality, was established by Brendle [B1] and [B2] (up to a technical assumption, in dimensions n ≥ 6, on the rate of vanishing of Weyl tensor at the points at which it vanishes): starting with any smooth metric on a compact manifold, the normalized Yamabe flow converges to a metric of constant scalar curvature.
Although the Yamabe flow on compact manifolds is well understood, the complete non-compact case is unsettled. In this case one expects to have more types of singularities which could be either of type I or type II according to the definition below.
Definition 1.1. Assume that a solution g(t) of the Yamabe flow (1.1) on a Riemannian manifold has a singularity at time T . This singularity is called type I if lim sup t→T − (T − t) sup M |Rm|(·, t) < +∞.
A singularity which is not of type I, is called type II.
The results mentioned above show that in the generic compact case the only singularities of the Yamabe flow are type I. Moreover the works [DS1,DKS] address the singularity formation of complete non-compact solutions to the conformally flat Yamabe flow whose conformal factors have cylindrical behavior at infinity. These singularities are all of type I.
A natural question to ask is whether the Yamabe flow admits any singularities which are of type II in the non-compact case. The authors in [CD] presented, for the first time in the Yamabe flow, examples of complete solutions which develop a type II singularity, either at finite time T < +∞ or at infinite time T = +∞. Such solutions are conformally equivalent to R n and their initial data has cylindrical behavior at infinity if T < ∞. What distinguishes our type II solutions from the type I solutions which are modeled on shrinkers, is that their initial metric has slower second order decay rate to the cylindrical metric than that of any other Yamabe shrinkers. In this work we study complete non-compact and conformally flat solutions of the Yamabe flow (1.1) on R n which develop type II singularity and provide their detailed asymptotic behavior near the singularity.
Let us briefly discuss next the known results on the singularity formation of non-compact Yamabe flow. Even though the analogue of Perelman's monotonicity formula is still lacking for the Yamabe flow, one expects that Yamabe soliton solutions model finite and infinite time singularities. These are special solutions of the Yamabe flow (1.1) which is characterized by a metric g = g ij and a potential function P so that (R − ρ)g ij = ∇ i ∇ j P, ρ ∈ {1, −1, 0}.
Depending on the sign of the constant ρ, a Yamabe soliton is called a Yamabe shrinker, a Yamabe expander or a Yamabe steady soliton if ρ = 1, −1 or 0 respectively.
The classification of locally conformally flat Yamabe solitons with positive sectional curvature was established in [DS2] (c.f. also [CSZ] and [CMM]). It is shown in [DS2] that such solitons are globally conformally equivalent to R n and correspond to radially symmetric self-similar solutions of the fast-diffusion equation satisfied by the conformal factor defined by g ij = u 4 n+2 δ ij . Here and in the sequel δ ij denotes the standard metric on R n and we set m := (n − 2)/(n + 2). A complete description of those solutions is given in [DS2]. In [CSZ] the assumption of positive sectional curvature was relaxed to that of nonnegative Ricci curvature.
As mentioned above, in [DS1,DKS] the singularity formation of complete noncompact solutions to the conformally flat Yamabe flow with cylindrical behavior at infinity was studied. The singularity profiles are Yamabe shrinking solitons which are determined by the second order asymptotics at infinity of the initial data, which is matched with that of the corresponding soliton. The solutions may become extinct at the extinction time T of the cylindrical tail or may live longer than T . In the first case, the singularity profile is described by a Yamabe shrinker that becomes extinct at time T . This result can be seen as a stability result around the Yamabe shrinkers with cylindrical behavior at infinity. In the second case, the flow develops a singularity at time T which is described by a singular Yamabe shrinker slightly before T and by a matching Yamabe expander slightly after T .
Recently the authors [CD] studied long time behavior of the complete noncompact conformally flat Yamabe flow and in particular showed the stability around the steady solitons. Such solitons are conformally equivalent to R n and rotationally symmetric. They are the analogue to the bowl translating soliton of MCF or the Bryant soliton of the Ricci flow.
In this work we study the asymptotic behavior of type II singularities in the conformally flat non-compact case. More precisely, for a sufficiently small T < +∞, we provide a condition, in terms of the second order decay rate of the initial metric g γ,κ (·, 0) at spatial infinity, which guarantees that the Yamabe flow g γ,κ (·, t) develops a type II singularity at time T with specified blow up rate (1.3) Moreover, we prove that after rescaling the solution g γ,κ (·, t) around highest curvature point by (T − t) −(1+γ) , it converges to a radial steady gradient soliton. Our main result states as follows: Theorem 1.2. Let g 0 = u 1−m 0 (x) δ ij be a conformally flat metric with positive Ricci curvature. For any given γ > 0 and A > 0, there is T 1 > 0 with the following property: for any T < T 1 , if , as |x| → +∞ then the solution of Yamabe flow (1.1) with initial data g 0 will develop a type II singularity at time t = T with specified blow up rate given by Moreover, after rescaling the metric around the highest curvature point by (T − t) −(1+γ) , it converges to the unique radial steady gradient soliton of maximum scalar curvature 2γA.
Theorem 1.2 shows for the first time that the conformally flat radial steady soliton appears as a finite time singularity model for the Yamabe flow. In the Mean curvature flow and the Ricci flow, examples of type II singularities and their asymptotic behavior were shown in both compact and non-compact settings under rotational symmetry (c.f. [AV], [IW] for Mean curvature flow and [AIK2], [AIK1], [W] for Ricci flow). Let us remark that unlike in the cases mentioned above our result also includes non-radial initial data.
To achieve our result we first construct sharp barriers (super and sub-solutions) for given fixed γ and A. The barriers are chosen sufficiently close to each other so that they give a model solution whose blow up limit at the tip is a radial steady soliton with the curvature blow up rate (1.4). In other words this already proves the result of Theorem 1.2 if the initial metric is in between sharp barriers. When the initial metric satisfies the condition of Theorem 1.2, the solution can be located between two model solutions which are differ by a translation in cylindrical direction. Then we do further analysis to show the solution also has the same asymptotic behavior and curvature blow up rate.
The barriers will be constructed to be radially symmetric though we don't assume it for initial metrics. Note that for a radially symmetric solution g := u 4 n+2 (r, t) (dr 2 + r 2 dg S n−1 ) of the Yamabe flow (1.2), it is often convenient to work in cylindrical coordinates where the metric is expressed as g = w(s, t) (ds 2 +dg S n−1 ), with s = ln r. The conformal factor in cylindrical coordinates is given by and satisfies the equation The outline of our paper is as follows: In the Section 2 we will begin by giving the formal matched asymptotics of the type II singularity at time t = T . Based on this analysis, we will introduce in Section 3 the two different regions outer and inner and the scalings in each region. Also some notation. We will refer to the notation of this section throughout the paper. Section 4 deals with the barrier construction in the outer region (c.f. Propositons 4.4 and 4.7) and Section 5 deals with the barrier construction in the inner region (c.f. Proposition 5.1). Combining the results from Sections 4 and 5, in Section 6 we will glue the barriers in the inner and outer region to construct suitable super and sub-solutions. In Section 7 we will show one of our main results, Theorem 7.1, which shows the type II convergence of any given conformally flat Yamabe flow to the steady soliton, assuming that its initial data is trapped between our super and sub-solutions. Finally, our last section 8 will be devoted to the proof of Theorem 1.2. In this section, along with the barriers constructed in previous sections, we will make use of the differential Harnack inequality in [Ch] and the classification result of Yamabe solitons (c.f. [DS2,CMM,CSZ]).

Formal matched asymptotics
In this section we will present the formal construction of our solutions which is based on matched asymptotic analysis. We hope that this will give our reader an intuition for our construction.
For any given parameters γ > 0 and A > 0 we will construct below a family of formal rotationally symmetric solutions where the curvature blows up in the type II rate Note that our main results Theorem 6.4 and 7.1 are not restricted on rotationally symmetric solutions, however the barriers which will be constructed in next sections are obtained from perturbations of this formal solution and those are radial. Motivated by condition (2.1) and in order to capture at the end a stationary solution, we perform the change of variables on our solution w(s, t) of (1.6) settinĝ (2.2) A direct calculation shows thatŵ(η, τ ) satisfies the evolution are negligible as τ → ∞, the above equation is reduced to the following ODE in η variable γηŵ η +ŵ − (n − 1)(n − 2) = 0.
(2.6) Solving this equation on η > 0 gives the solutionŵ 0 (η) given bŷ for a parameter κ ∈ R. We will assume from now on, without loss of generality, that κ > 0, although κ ≤ 0 also gives a solution. Moreover, there is a family of solutions to (2.6) on η < 0, but this case is exactly symmetric to the η > 0 case which we will handle below. Indeed, the solutions given by (2.7) describe non-compact surfaces moving in positive η (hence positive s) direction, while the corresponding solutions defined on η < 0 describe a symmetric surface just moving on the opposite direction. This ansatz, namely settinĝ approximates a solution of the equation (2.3). In fact, pluggingŵ(η, τ ) given by (2.8) into (2.3), we see that the error term becomes (2.9) Writing B[ŵ] = (n − 1)(ŵe γτ ) −2 ŵŵ ηη + n − 6 4ŵ 2 η , we see that it becomes arbitrarily small in the space-time region as τ → +∞ which we call the outer region (see Figure 1 below). This is the region where the diffusion doesn't play an important role and the remaining advection and reaction terms are dominant.
The inner-region is given by as τ → +∞ (2.10) (see Figure 1). In this region we perform another scaling, settinḡ w(ξ, τ ) = e γτŵ (η, τ ), η = A + e −γτ ξ (2.11) for some choice of A > 0, which combined with (2.2) gives w(ξ, τ ) = e γτŵ (A + e −γτ ξ, τ ) = e (1+γ)τ w(Ae γτ + ξ, T − e −τ ). (2.12) The evolution equation forw(ξ, τ )is given by (2.13) Thus assuming that in this region the first term having e −γτ becomes negligible as τ → ∞, the equation is reduced to (2.14) This can be seen as the equation satisfied by a traveling wave solution of (1.6) with speed γA. For each A > 0, this equation admits a τ independent entire solutionw(ξ) which is unique up to translation in ξ. For such profilew, the functionw(s − γAt) becomes a traveling wave solution of equation (1.6). From the geometric point of view,w(s) corresponds to a radially symmetric non-compact metric on R n via (1.5) and solutions with different A's are the only radially symmetric steady gradient solitons on locally conformally flat manifolds (c.f. [CSZ,DS2]). For such a solution w, the highest curvature point is at the origin (i.e. s = −∞) and one may formally compute that | Rm | max (t) = 2γA n(n − 1) , hence leading to (2.1). Remark 2.1. Note that a solutionw(ξ, τ ) of (2.14) which also depends on τ , can be written asw (ξ, τ ) =w 0 (ξ + C(τ )) (2.15) for a function C(τ ), wherew 0 is one τ −independent solution of (2.14). By plugging this into (2.13) again, we get an error term We may choose C(τ ) so that C is small and thus the error term above vanishes appropriately as τ → ∞. Later, we will use this C(τ ) to glue barriers from the two different regions, inner and outer.
Next, we will carry out a matching asymptotic analysis between the inner and outer regions and obtain a relation between κ > 0 in (2.7) and A > 0 in (2.13). It is known that a solutionW (ξ) of (2.14) satisfies the asymptotic behavior Therefore, recalling that η = A+e −γτ ξ, our solutionw(ξ, τ ) which is approximatelȳ W (ξ) satisfies On the other hand, from the outer region ansatz (2.7), by the first order Taylor approximation near η = κ γ we havê Thus, we can see that these two asymptotics are matched exactly if To see this in another way, we can argue that if A < κ γ the two asymptotics are inconsistent as η → (κ γ ) + and if κ γ < A the linearization of (2.7) near η = A is inconsistent with the asymptotic behavior from the inner region.

Notation and different scalings
In this section we will summarize the coordinates and different scalings of our solutions, as introduced in the previous section. We will refer to the notation introduced below throughout the paper.
3.1. Coordinate systems. Since our metric g(t) is conformally equivalent to the standard euclidean metric on R n denoted briefly by δ ij , we represent where the scalar function u(x, t) evolves by equation (1.2) under Yamabe flow.
In case when the metric is radial, it is often convenient to work in the cylindrical coordinates, that is g(t) = u 1−m (r, t)(dr 2 + r 2 g S n−1 ) = w(s, t)(ds 2 + g S n−1 ), r = |x| and w(s, t) = r 2 u 1−m (r, x), s = ln r = ln |x|. Under this coordinate change, w(s, t) evolves by equation (1.6).
The functionw(ξ, τ ) evolves by the equation (2.13). • The above scaling change from w(s, t) tow(ξ, τ ) corresponds to the following scaling change in euclidean coordinates from u 1−m (x, t) toū 1−m (y, l): where l is a new time variable l = (T − t) −γ /γ and This scaling is used only in Section 7.ū(y, l) evolves by equation (7.1).

3.3.
Relations. Let us summarize relations between different variables and functions appearing in the scalings introduced above.
• For every A > 0, we defineŵ 0 (η) to be the outer region ansatẑ which is a solution of (2.6). • For the same A > 0 as above, we denote byw 0 (ξ) the inner region approximation which the solution of equation This solution represents a steady gradient soliton of the flow and it is unique up to translation in ξ. • FinallyŪ denotes the representation ofw 0 on R n by the following change of coordinate |x| 2Ū 1−m (x) =w 0 (ln |x|). (3.10)

Barrier construction in the outer region
Let us fix parameters γ > 0 and A > 0 as they appear in the curvature blow up rate of our singularity (2.1). In this section we will construct appropriate super and sub solutions in the outer region (e γτŵ (η, τ )) −1 = o(1), as τ → +∞, which will be given by Recall that for a rotationally symmetric solution u(r, t), r = |x| of the conformally flat Yamabe flow (1.2), we perform the cylindrical change of coordinates (1.5) leading to a solution w(s, t) of (1.6). As already seen in section 2, to capture the behavior in the outer region we perform a further change of variables (2.2) leading to a solutionŵ(η, τ ) of (2.3). Assuming that near our singularity (2.5) holds, we find that the zero order behavior ofŵ near the singularity in the outer region is given by a solution of the ODE (2.6). The general solution of (2.6) is given by (2.7) for a parameter κ > 0. Thus, setting A := κ 1/γ > 0 we define the zero order approximation ofŵ in the outer region to bê In this section, we are going to construct sub and super solutions of equation for a parameter θ ∈ R. To this end, we will chooseŵ 1 (η) andŵ 2 (η) to be solutions of a first order linear ODE with different inhomogeneous terms. By setting we will choose w 1 , w 2 to be solutions of the equations Pluggingŵ(η, τ ) given by (4.2) into the equation gives that where B[·] is given by (2.4) Thus, from the proposed choice ofŵ 1 ,ŵ 2 to satisfy (4.4), one expects thatŵ is a subsolution of equation (2.3) if θ < n−6 4 and a supersolution if θ > n−6 4 , for all parameters γ > 0. The rest of this section is devoted to the justification of this idea which requires a rather delicate calculation. The case of parameters γ ≥ 1/2 is shown in Proposition 4.4 below. As we will see in the proof Proposition 4.7 below, in the case of parameters γ < 1/2 one needs to add correction term toŵ.
Recalling the definition ofŵ 0 in (4.1) and f 1 , f 2 in (4.3) we have hence the explicit solutions w i , i = 1, 2 of (4.4) are given bŷ (4.7) We will now fix the functionsŵ 1 andŵ 2 by fixing their values at a given point η 0 > A. While doing so, we will chooseŵ 2 to be a positive function on η > A.
. For the functionŵ 1 we may choose any value asŵ 1 (η 0 ), since we do not need to choose it to be positive. Note that by choosingŵ 2 to be positive, the family of functionŝ w 1 + θŵ 2 is monotone in θ ∈ R. To simplify the notation we will simply set from now onĥ :=ŵ 1 + θŵ 2 . (4.8) We will investigate the behavior of the familyĥ(η) := ŵ 1 + θŵ 2 (η), θ ∈ R, as η → A + and η → +∞. We will first see that the behavior near η = A + is governed byŵ 2 .
Lemma 4.1. For any linear combinationĥ :=ŵ 1 + θŵ 2 of the solutionsŵ 1 ,ŵ 2 chosen above we have, we havê Proof. By (4.7) we havê Now, by Taylor's theorem, (4.6) and derivatives of these equations imply the following behavior as η → A + . In particular we see from the above thatŵ 2 dominates as η → A + and by L'Hôpital's rule on Similarly, taking derivatives in (4.9) and using the asymptotics for (f 1 + θf 2 ) and (f 1 + θf 2 ) , we obtain Remark 4.1. Lemma 4.1 shows thatŵ(η, τ ) defined by (4.2), which is the ansatz for super and sub solutions, for θ > n−6 4 and θ < n−6 4 respectively, predicts the correct lower order asymptotic behavior in the outer region which is then matched with the inner region where a steady soliton comes in. Indeed, in our previous work [CD] we showed that a steady soliton of (1.2) satisfies the asymptotics On the other hand, Lemma 4.1 yieldŝ Hence, if we match the inner-outer variables by setting η = A + ξe −γτ , we obtain Also, this suggests that in the inner regionw should bew 0 , the translating soliton which satisfies asymptotics (4.10) with κ = 0.
Lemma 4.2. For any linear combinationĥ :=ŵ 1 + θŵ 2 of the solutionsŵ 1 ,ŵ 2 chosen above we have, we havê Proof. The proof is similar as in Lemma 4.1 if we check the asymptotics and corresponding asymptotics for f 1 and f 2 as η → ∞.
In the next lemma we give more precise asymptotics which will be used later when we have to add higher order terms to barrier in the case 0 < γ ≤ 1 2 . Notice that η − 1 γ −2 is a solution of the homogenous equation of (4.4) and we have constants C in the lemma below since we haven't chosen a specificŵ 1 .
Proof. Can be shown in the same manner as in Lemma 4.2.
We will now show thatŵ(η, τ ) given by (4.2) is a sub or super -solution of equation (2.3) in the appropriate regions. We will first deal with the case of parameters γ > 1/2. The case γ ≤ 1/2 is more delicate and will be considered later.
Proposition 4.4. For any γ > 1/2 and given θ = n−6 4 , there exist τ 0 ∈ R and ξ 0 > 0 depending on n, A, γ and θ such that the function Proof of Proposition 4.4. As before, let us denote bŷ We need to show that holds in the region Proposition 4.4 follows from the the two claims below Claim 4.5. For any γ > 0 and given θ = n−6 4 , there exist ξ 0 > 0 and δ > 0 Proof of Claim 4.5. By Lemma 4.1, we may find κ = κ(n, A, γ) > 0 such that holds on the region A < η < A + 1. Moreover, by Taylor's theorem we may choose the constant κ so that Hence by restricting to 0 < δ < 1 small and ξ 0 > 0 large, we may assume that Using the above we find that in the considered region we have Similarly, we estimate from below Thus, for a fixed > 0 to be determined later, we may find small δ > 0 and large Recalling the formula for B[ŵ] in (4.5) and using the triangle inequality successively we obtain Finally, by choosing := 1 2 θ − n − 6 4 |θ| + n − 6 4 + 2 −1 we conclude that (4.5) holds, finishing the proof of the claim.
We will next proceed in our next claim which holds for γ > 1/2.
Claim 4.6. For γ > 1/2 and given θ = n−6 Proof of Claim 4.6. For a given δ 0 > 0, Lemma 4.2 and the asymptotic behavior ofŵ 0 ,ŵ 0 andŵ 0 can be used to find a constant κ > 0 such that Thus, we may start with some large τ 0 such that w,ŵ η ,ŵ ηη have same the sign asŵ 0 ,ŵ 0 ,ŵ 0 respectively on η > A + δ and τ > τ 0 . In particular, they are nonzero. Using the formula for We will show that the first two terms become arbitrarily small in comparison with the last term, for τ 0 1 (and hence τ 1). Indeed, we have Thus, the asymptotics in Lemma 4.3 imply that by choosing τ 0 1 we have (4.13) for all τ > τ 0 . In the case γ > 1 2 , by ln η η 2 , e −2γτ terms in the previous estimate and asymptotics (4), we can make τ 0 large and conclude (4.14) on the considered region. This proves that for τ ≥ τ 0 1, (4.11) holds, finishing the proof of the claim.
(4.15) They satisfy the following relation and to simplify the notation we also set v k,−1 (η) = 0 and v k,−2 (η) = 0. We will show the following.
Proof of Claim 4.8. Let us assume θ < n−6 4 because the other case follows similarly. Supposeŵ is of the form of (4.8). We will choose the coefficients c k,l later.
As we fixedŵ 1 andŵ 2 in the proof of Proposition 4.4, from now on let us fix c k,l so that Claim 4.8 holds, by choosing c k,0 = 0. Next, we give the analogue of Claim 4.5 in this case.

Barrier construction in the inner region
We will now construct the appropriate barrier in the inner region which is the region where e γτŵ (η, τ ) = O(1), as τ → +∞.
In this region we definew(ξ, τ ) as in (2.11), that is we setw(ξ, τ ) = e γτŵ (η, τ ), ξ = (η − A) e γτ . We have seen in section 2 thatw(ξ, τ ) satisfies the equation given by (2.13). Let us assume that in this region the first term in (2.13) having e −γτ becomes negligible as τ → ∞. Then, we expect that the solutionw 0 (ξ) of equation (w 0 ) 2 ξ w 2 − (n − 1)(n − 2) + γA (w 0 ) ξ = 0 is the leading order term forw(ξ, τ ) in this region. We are going to find super and sub solutionsw + andw − , respectively in the following form Here, > 0 is a small constant and C 1 (τ ) and C 2 (τ ) are smooth functions of τ . Both will be chosen later and will depend on ξ 0 which appears in the construction of our barriers in the outer region. As we will see below, the construction is rather straightforward. If we plug these into I[·], we get ξ + e −γτ (C 2 (τ )w + ξ − (1 + γ)w − ). We will next show the following.

Construction of super and sub-solutions
In this section we will combine the results from Sections 4 and 5 to construct a family of super-solutions w + and sub-solutions w − of equation (1.6) which is equivalent to the conformally flat Yamabe flow (1.2) under rotational symmetry and after the cylindrical change of variables (1.5). This will give a family of rotationally symmetric super and sub solutions of equation (1.2) which we will then be used in the next section to analyze the type II blow up behavior of any solution u(x, t) of (1.2) which satisfies the assumptions of Theorem 6.4.
We are now ready to proceed to the proof of Proposition 6.1.
We will finish this section with the following result which is an immediate consequence of the comparison principle and Proposition 6.1.

Asymptotic shape of the singularity in the inner region and geometric properties
Throughout this section we will fix ξ 1 > 0 and > 0 so that w + (ξ, τ ) and w − (ξ, τ ) given by (6.4) are barriers in view of Proposition 6.1 and Theorem 6.4. To simplify the notation we will denote them by w + (ξ, τ ) and w − (ξ, τ ) respectively. They are super and sub-solutions of equation (2.3) on R × [τ 1 , ∞), respectively.
We will first prove that if our initial conformally flat metric of the Yamabe flow (1.2) u 0 (·) is bounded from above and below by w + (·, − ln T ) and w − (·, − ln T ), for some − ln T ≥ τ 1 (c.f. (6.13)), then the rescaled solution converges to a steady gradient solitonw 0 (ξ), which is the unique entire solution of the equation (3.8) with asymptotic behavior (3.9) as ξ → ∞.
Since we are not assuming that our solution u(x, t) of (1.2) is radially symmetric, it is more convenient to work in euclidean coordinates on R n , rather than cylindrical coordinates. We have seen that in order to see the steady statew 0 in the inner region one needs to perform the coordinate change (2.11) on radially symmetric solutions in cylindrical coordinates. Under the transformation (1.5) which brings us back to the plane, this change of variables corresponds the coordinate change (3.1)-(3.2) which transforms a solution u(x, t) of (1.2) to a solutionū(y, l) of equation We denoteŪ (y) the steady solitonw 0 in euclidean coordinates, namelȳ This is the unique radial solution of n − 1 m ∆u m + γA(y · ∇u) + 2γA 1 − m u = 0 (7.2) with asymptotic behavior We will next prove the following result.
Theorem 7.1. Under the assumptions of Theorem 6.4, the rescaled solutionū(y, l) converges, as l → +∞, smoothly on compact sets of R n to the radial steady soliton U 1−m (y).
Proof. Let l 0 := γ −1 T −γ be the initial rescaled time, corresponding to t = 0. By Theorem 6.4, for l > l 0 > 0 we have These two bounds give upper and lower bounds away from zero forū(·, l) on every compact set in R n which are uniform in time l ≥ l 0 1. Hence, by standard higher order regularity estimates for uniformly parabolic equations and a compactness argument, we conclude that for any sequence l i → ∞, the solutionsū i (y, l) := u(y, l i + l) converge, passing to a subsequence, to a limitū ∞ (y, l). The convergence is smooth on compact subsets of R n ×R. Therefore, in view of (7.1) and the uniform local upper bound of our sequence, the limitū ∞ is a smooth eternal solution of To finish the proof we need to show that u ∞ (y, l) =Ū (y) which would also imply that our limit is unique, thus concluding thatū(·, l) →Ū , as l → ∞. To this end, we first observe that by our barrier construction (6.4), we have w ± (ln |y|, τ ) → (n − 1)(n − 2) γA ln |y| + (n − 1) Aγ θ ± 1 ln |y| as τ → +∞, uniformly on e ξ1 ≤ |y| ≤ K, for any fixed K > e ξ1 . In particular, this implies that our limitū 1−m ∞ has these bounds and thus as |y| → ∞ uniformly in l ∈ R. For λ > 0, if we denoteŪ λ (y) := λ 2 1−mŪ (λy), this is again a radial solution of (7.2) with This is just a time translation of the radial steady solitonŪ and they are isometric. Since on the soliton the scalar curvature R > 0 everywhere, the solution pointwise decreases as time increases and henceŪ λ1 >Ū λ2 for λ 1 > λ 2 . Thus we may define Our proof will finish if we show that λ + = λ − = 1. Let us prove that λ + = 1. Since inf B(e ξ 1 ,0)Ūλ → ∞ as λ → ∞ (see the observation in Corollary 3.3 in [CD]), the construction of w + in the inner region and (7.7) imply that we can find large λ > 1 such thatŪ λ (·) >ū ∞ (·, l) for all l. By (7.6) and (7.7),Ū λ (·) ū ∞ (·, l) for λ < 1. Therefore, λ + is a well defined number with λ + ≥ 1. Assume that λ + > 1. For each U λ+−2 −n , there is a point (x n , l n ) withŪ λ+−2 −n (x n ) <Ū ∞ (x n , l n ). Moreover, the sequence of points {x n } such that λ + − 2 −n > 1 is bounded due to (7.6) and (7.7). By standard regularity estimates on the equation (7.5), we can find a subsequence of (x n , l n ) such thatū * nj (x, l) :=ū ∞ (x, l nj + l) →ū * ∞ (x, l) smoothly on compact sets and x nj → x * .
Let us remark the following.
Remark 7.1 (Scalar curvature blow up rate near the origin). u(x, t) of (1.2) represents conformally flat solution of the Yamabe flow g(t) = u 1−m (x, t) δ ij on R n . The metric of rescaled solutionū(y, l) can be written as where φ t is a one parameter family of diffeomorphisms φ t (x) = e A(T −t) −γ x. Therefore, Theorem 7.1 can be rephrased as convergence of the pointed manifold in Cheeger-Gromov sense. This, in particular, implies that concluding that In particular the above implies the following blow up rate of the Riemmannian and Scalar curvature at the origin We will next show that the global supremum of the curvature occurs asymptotically at the origin as t → T −.
The result of this proposition follows from the following curvature estimate lemma in the outer region. This lemma is useful in the sense that it also gives the curvature blow up rate in other regions. For example, it shows that the curvature blows up in a type I manner near the infinite cylindrical region.
Lemma 7.3. There exist C > 0, r 0 1 and 0 < t 0 < T such that the following holds on |y| ≥ r 0 and t 0 < t < T .
Let us first show that Lemma 7.3 implies the proposition and then finish this section by proving the lemma.
Proof of Proposition 7.2. Assuming that the Lemma 7.3 holds, then sinceŵ 0 is increasing function, for each fixed r 1 ≥ r 0 we have and, by the Taylor expansion ofŵ 0 (·) at A, taking the limit t → T −, yields Choose r 1 ≥ r 0 sufficiently large so that Now on the remaining region |x| ≤ r 1 , due to the smooth convergence ofū 1−m toŪ 1−m on compact sets and the fact that the steady solitonŪ 1−m attains its maximum curvature at the origin we have and also the second statement of the proposition holds, concluding the proof.
Proof of Lemma 7.3. For r 0 and t 0 to be chosen later, we pick and fix a point (y 1 , t 1 ) with |y 1 | ≥ r 0 and t 0 < t 1 < T and consider the following scaling of the solution Since u satisfies (1.2), it follows that v(z, σ) evolves by Claim 7.4. There are r 0 > 1 and t 0 > 0 so that if |y 1 | ≥ r 0 and t 1 ≥ t 0 , then there exist positive constants c and C which are independent of (y 1 , t 1 ) such that where A is the annulus A = {z ∈ R n | 1/2 ≤ |z| ≤ 3/2 }.
Since v 1−m (·, 0)δ ij and u 1−m (·, t 1 )δ ij are isometric, we conclude that The constant C is independent from (y 1 , t 1 ). This finishes the proof of the lemma.
Proof of the Claim 7.4. Although the computations below might look intimidating, the idea is simple. Since u is trapped between the two barriers in the outer region where |x| ≥ e A (T −t) γ e ξ1 , namelŷ and sinceŵ + andŵ − are close toŵ 0 , we expect that different values of v are similar in the whole annulus. Indeed, suppose r 0 > 1 and t 0 ∈ (0, T ) are first chosen to satisfy ln r 0 2 > ξ 1 and − ln(T − t 0 ) > − ln T + (n − 1)(n − 2).
Since t ≤ t 1 and |y 1 | ≥ r 0 , we have |y 1 ||z|e , thus we could bound above v 1−m (z, σ) using our barrierŵ + of the outer region in the second line and use (7.16) in the last line. Similarly we get a lower bound .
If we could find constants 0 < c < 1 < C such that holds on (v, σ) ∈ A × [−1, 0], then sinceŵ 0 (A + x) is an increasing function on x ≥ 0, concave andŵ 0 (A + 0) = 0, we would deduce that which combined with the above would finish the proof of our claim. Now let us now find the bounds (7.17). First, it is clear that 1 + ln 1 2 ln r 0 ≤ ln |y 1 | + ln |z| ln |y 1 | ≤ 1 + ln 3 2 ln r 0 (7.18) and notice that 1 + ln 1 2 ln r 0 = ln r0 2 ln r 0 is positive since ln r 0 2 > ξ 1 > 0. Thus, it suffices to prove by some constant C 1 < ∞. If we introduce θ := (T − t 1 ) γ ln |y 1 | > 0, then by the definition of σ in (7.12) By the mean value theorem, there exists 0 < θ 0 < θ such that Here, we used the facts thatŵ 0 (A + x) is a nonnegative concave function on the set x ≥ 0 witĥ Combining the last two inequalities implies that (7.19) holds with This finished the proof of the claim and also the proof of the lemma.
8. Proof of Theorem 1.2 In this final section, we will give the proof of Theorem 1.2 as stated in the introduction. We will need the following rigidity result for eternal solutions of conformally flat Yamabe flow.
Proposition 8.1. Let g(t) = u 1−m (x, t) δ ij be a smooth eternal solution to the conformally flat Yamabe flow (1.2) on R n ×(−∞, ∞), with positive Ricci and uniformly bounded sectional curvature. We further assume that u is bounded from below by a radial steady gradient soliton centered at the origin with maximum scalar curvature 2γA > 0, that is Then, u(x, t) must be a radial gradient steady soliton, that is Proof. By the Harnack inequality for the Yamabe flow (c.f. Theorem 3.7 in [Ch]), for any 1-form X i Note that since our solution exists from t = −∞ we could drop R/t term from the original Harnack expression in [Ch]. This inequality, in particular implies that (n − 1)∆R + R 2 = ∂ t R > 0.
Proof of Claim 8.2. The proof is simple and uses that ∂ t R > 0 and On the other hand, this contradicts to which holds for t ≥ t 0 . Similarly as before, if R g (0, 0) < 2γA, then as t → −∞ which again contradicts to (8.1).
According to the classification of conformally flat radial solitons (c.f. Propositions 1.4 and 1.5 in [DS2]) the one parameter family of solutions are all possible conformally flat radial steady gradient solitons whose maximum scalar curvature is 2γA at the origin. It also is known that these solutions attain a strict curvature maximum at the origin. Meanwhile, due to the Claim 8.2, g(x, t) attains its maximum scalar curvature at an interior space-time point (0, 0). Furthermore, since g(x, t) has positive Ricci curvature and bounded sectional curvature, Corollary 5.1 in [DS2] implies that g(x, t) must be a steady gradient soliton. Also, under the nonnegative Ricci condition, such steady gradient solitons are (globally) conformally flat and radially symmetric (c.f. Theorem 3.2 and Corollary 3.3 in [CMM] or Corollary 1.6 and Remark 1.2 in [CSZ]). Since g(x, t) has its maximum scalar curvature at the origin, it must be symmetric with respect to this point. In view of Liouville's rigidity theorem on conformal mappings on R n with n ≥ 3, u(x, t) must be a radially symmetric function which represents a steady gradient soliton. Hence u(x, t) =Ū ξ0 (x, t) by the classification theorem in [DS2].
We are now in position to finally give the proof of our main result, Theorem 1.2.
Proof of Claim 8.3. For our fixed T , it is not hard to check that We can also check that if ln |x| = ξ + AT −γ , |x| 2 u 1−m 0 (x) also satisfies these asymptotics. Indeed, Condition ii) in Theorem 1.2 implies that Using these asymptotic behaviors, we can first find ξ a,0 > ξ b,0 such that (8.5) holds asymptotically (outside of compact interval in ξ). Next, we may use condition i) of Theorem 1.2 to find possibly smaller ξ b ≤ ξ b,0 so that the second inequality of (8.5) holds everywhere. Finally, by a similar argument which uses the fact |x| 2 u 0 (x) is uniformly bounded away from zero on |x| ≥ r 0 for all r 0 > 0, we may find larger ξ a ≥ ξ a,0 so that the first inequality of (8.5) holds everywhere. This argument is very similar to the proof of Claim 4.4 in [CD].
Letū(y, l) be the rescaled solution obtained from u(x, t) under (3.1)-(3.2). By Theorem 6.4, 7.1 and the claim, we have local uniform upper and lower bounds on u, namelyū a ≤ū ≤ū b wherē In order to show the blow up rate (1.4), we first need the following claim which asserts that Lemma 7.3 holds for our given solution.
Note that the time dilation parameter ξ 0 might be different for different limits along sequences l i → ∞, but metrics with different ξ 0 represent the same soliton and thus this proves Cheeger-Gromov convergence of the metricū 1−m (y, l)δ ij to the same limit soliton as l → ∞. Also this convergence and ( U ξa (y, l) ≤ u ∞ (y, l) ≤Ū ξ b (y, l).
(8.9) g ∞ (l) = u 1−m ∞ (y, l)δ ij is an eternal solution of the flow which has nonnegative Ricci curvature.
To apply Proposition 8.1 to u ∞ (y, l), we need to show it has actually strictly positive Ricci curvature and uniformly bounded |Rm|. We first show uniform boundedness of curvature. By (8.7),ḡ ∞ (l) has bounded curvature on R n \ B r1 (0), for some large r 1 . We also have a uniform curvature bound ofḡ ∞ (l) on B r1 (0) by two bounds (8.8) and interior uniformly parabolic regularity estimate of the equation (7.5). Sinceḡ ∞ (l) and g ∞ are isometric, this gives uniform bound of |Rm|. Next, the proof for positive Ricci uses Theorem 8.1, the classification of locally conformally flat nonnegative Ricci Yamabe flow having a nontrivial null eigenvector. It solely an interesting result, so we prove it in a separate theorem. (R n ,ḡ ∞ (l)) can not be flat by bounds (8.9). Also an eternal solution can not be isometric to a cylinder solution which exists up to a finite time. Hence Ricci ofḡ ∞ (l) is positive definite everywhere by Theorem 8.1.
We will finish with proving of the following result which was used above in the proof of Theorem 1.2.
Theorem 8.1. For n ≥ 3, let (M, g(t)) for t ∈ (0, T ) be a complete locally conformally flat solution of the Yamabe flow which has nonnegative Ricci and uniformly bounded Riemann curvature. If the Ricci tensor has a null eigenvector at some point (p 0 , t 0 ), then (M, g(t)) is either locally isometric to flat Euclidean space or a cylinder solution (R × S n−1 , f (t)(dr 2 × g can )) where g can is the round metric on S n−1 and f (t) = (n − 1)(n − 2)(T − t) for some T > T .
Proof. The uniform boundedness of the Riemann curvature tensor will only be used to apply the (strong) maximum principle. For a locally conformally flat solution of the Yamabe flow, the evolution of Ricci tensor R ij is shown in Lemma 2.4 [Ch] as where B ij is a quadratic expression of R ij . It was shown in (2.11) and (2.12) of [Ch] that, with respect to an orthonormal basis which diagonalize Ricci tensor by Let λ 1 ≤ λ 2 ≤ . . . ≤ λ n be the eigenvalues of R ij in an increasing order. Note that for any 1 ≤ k ≤ n, we have m k := λ 1 + λ 2 + · · · + λ k = inf{Tr g (R ij (V, V )) | V ⊂ T p M is a subspace of dim k} is a concave function of R ij . Since the solution has nonnegative Ricci, m j = 0 implies λ i = 0 for all i ≤ j. From equation (8.10), it is easy to check that the ODE ∂ t R ij = B ij preserves m k ≥ 0 under the nonnegative Ricci condition. Therefore, we can apply the strong maximum principle (Lemma 8.1 in [Ha2]) on m k ≥ 0. The lemma and the continuity of m k imply that either m k ≡ 0 or m k > 0 everywhere at each time t = t . Furthermore, if m k > 0 at t = t , m k > 0 for all t > t . As a consequence, there is a well defined decreasing functionk(t) ∈ {0, . . . , n} such that m k (p, t) = 0 if k ≤k(t) and m k (p, t) > 0 if k >k(t). Since m k = 0 iff dim(Null(R ij )) ≥ k, we conclude that the rank of R ij is constant in space and it is equal to n −k(t), which is increasing with respect to time.
Under the assumption that there is a point (p 0 , t 0 ) where Ricci curvature has a null eigenvector, we will show that the rank of Ricci curvature is either 0 or n − 1 for all time. By the previous argument, the Ricci tensor can't have full rank for t ≤ t 0 . Also since it is increasing, there is an interval of time (t 1 , t 2 ) with t 2 ≤ t 0 such that dim(Null(R ij )) = k, for some fixed k ∈ {1, . . . , n − 1, n} on this time interval. If k = 0, then it is clear that the solution must be stationary for all time and the solution must be Ricci flat. Since on a locally conformally flat manifold the Riemann curvature tensor is determined by the Ricci tensor, this implies that the solution is locally euclidean. Next, in case where 1 ≤ k ≤ n − 1 we can exactly follow the argument of Lemma 8.2 [Ha2] on the time interval (t 1 , t 2 ) to conclude that the null space of the Ricci tensor is invariant under parallel translation and also it is invariant in time. Moreover, it lies in the null space of B ij . By this last property and (8.10), we see that k has to be 1 and other λ i s except λ 1 should be the same positive number (possibly different at each point). In this case, the manifold locally splits off along this parallel 1-dimensional null eigenvector distribution (see the lemma which follows after Theorem 8.3 [Ha2]) i.e. (M, g(t)) is locally splits