The Existence of Type II Singularities for the Ricci Flow on S n+1
aa r X i v : . [ m a t h . DG ] J un The Existence of Type II Singularitiesfor the Ricci Flow on S n +1 Hui-Ling Gu and Xi-Ping Zhu
Department of MathematicsSun Yat-Sen UniversityGuangzhou, P. R. China
Abstract
In this paper we prove the existence of Type II singularities for theRicci flow on S n +1 for all n ≥
1. Introduction
In this paper, we consider the Ricci flow ∂g ij ∂t = − R ij ,g ij (0) = ˆ g ij , (1 . M, ˆ g ). This is a nonlinear(degenerate) parabolic system on metrics. In the seminal paper [19], Hamiltonproved the Ricci flow admits a unique solution on a maximal time interval [0 , T )so that either T = + ∞ or T < + ∞ and | Rm | is unbounded as t → T . We1all such a solution g ( t ) a maximal solution of the Ricci flow. If T < + ∞ andthe curvature becomes unbounded as t tends to T , we say the maximal solutiondevelops singularities as t tends to T and T is a singular time. It is well-knownthat the Ricci flow generally develops singularity.If a solution ( M, g ( t )) to the Ricci flow develops singularities at a maximal time T < + ∞ , according to Hamilton [24], we say it develops a Type I singularity ifsup t ∈ [0 ,T ) ( T − t ) K max ( t ) < + ∞ , and say it develops a Type II singularity ifsup t ∈ [0 ,T ) ( T − t ) K max ( t ) = + ∞ , where K max ( t ) = max {| Rm ( x, t ) | | x ∈ M } .Clearly, a round sphere, or more generally a finite product of several space-formswith positive curvature, shrinks to form Type I singularities. In [21, 13], Hamiltonand Chow proved the Ricci flow on two-sphere S (with an arbitrary metric) alwaysdevelops a Type I singularity and shrinks to a round point. In [19, 20], Hamiltonproved the Ricci flow on a compact three-manifold with positive Ricci curvature,or a compact four-manifold with positive curvature operator, develops a Type Isingularity and shrinks to a round point; recently, B¨ o hm-Wilking [7] had shownthat the Ricci flow on a general compact n -dimensional Riemannian manifold withpositive curvature operator also develops a Type I singularity and shrinks to around point.Intuitively, a compact manifold with the shape like a dumbbell will develop aType I singularity in the neck part. In views of the work [25] of Hamilton on four-manifolds with positive isotropic curvature (see also [12]), a Type I singularitywith neckpinch is expected. Indeed, such examples of Type I singularities withneckpinch for the mean curvature flow were known more than fifteen years ago(see for example [17] and [3]). It is very surprising that the existence of Type Isingularities with neckpinch for the Ricci flow was only known very recently. Thefirst rigorous examples of Type I singularity with neckpinch for the Ricci flow wereconstructed by Miles Simon [31] on noncompact warped products R × f S n . In [16],Feldman-Ilmanen-Knopf constructed another family of rigorous examples of TypeI with neckpinch on the holomorpic line bundle L − k over CP n − . Both of thesefamilies of examples live on noncompact manifolds. For the Type I singularity with2eckpinch on compact manifolds, the first rigorous examples were given by SigurdAngenent and Dan Knopf in [5] by constructing suitable rotationally symmetricmetrics on S n +1 , where the definition of a rotationally symmetric metric is thefollowing: Definition 1.1
A metric g on I × S n , where I is an interval, is called rotationallysymmetric if it has the following form: g = ϕ ( x ) dx + ψ ( x ) g can , x ∈ I, where g can is the standard metric of the round sphere S n with constant (sectional)curvature 1.For the Type II singularity for the Ricci flow, a rigorous example on R wasrecently given by Daskalopoulos and Hamilton in [15]. However, no rigorous ex-amples of Type II singularity for the Ricci flow on compact manifolds have yetappeared. We remark that some beautiful intuitions of the forming of Type II sin-gularity were described and explained by Chow-Knopf in [14] and Topping in [32].(For the mean curvature flow, the existence of Type II singularities was alreadyjustified by Altschuler-Angenent-Giga [1] and Angenent-Vel´ a zquez [4].)The purpose of this paper is to demonstrate the existence of Type II singularityon compact manifolds, in particular for rotationally-symmetric initial metrics on S n +1 . Our main result is the following: Theorem 1.2
For each n ≥ , there exist rotationally-symmetric metrics on S n +1 such that the Ricci flow starting at the metrics develop Type II singularitiesat some times T < + ∞ . This paper contains four sections. In Section 2, we recall some useful estimatesof Angenent-Knopf [5] on rotationally symmetric solutions to the Ricci flow. Ingeneral, to understand the structure of singularities, one usually needs to get aclassification for gradient shrinking solitons. The recent work [28] of Perelmangives a complete classification to positively curved gradient shrinking Ricci solitonin dimension three. In Section 3 we will extend Perelman’s classification to higherdimensions in the class of rotationally symmetric metrics. Finally in Section 4,based on the generalized classification, we will prove the main result Theorem 1.2.Our work in this paper benefits from a conversation with Professor R. S. Hamil-3on, who suggested the second author to consider the class of rotationally sym-metric metrics. The second author is partially supported by NSFC 10428102 andNKBRPC 2006CB805905.
2. Angenent-Knopf ’s Estimates
Consider a rotationally-symmetric metric g = ϕ ( x ) dx + ψ ( x ) g can (2 . − , × S n , in which g can is the metric of constant sectional curvature 1on S n . The coordinate x is ungeometric, a more geometric quantity is the distance s to the equator given by s ( x ) = Z x ϕ ( x ) dx. Then ∂∂s = 1 ϕ ( x ) ∂∂x and ds = ϕ ( x ) dx. With this notation the metric is g = ds + ψ g can . (2 . g to be a smooth Riemannian metric on S n +1 , it is sufficientand necessary to impose the boundary conditions: ψ ( ±
1) = 0 , lim x →± ψ s ( x ) = ∓ x →± d k ψ ( x ) ds k = 0for all k = 1 , , · · · . The Riemannian curvature tensor of (2.2) is determined bythe sectional curvatures K = − ψ ss ψ of the n { x }× S n , and the sectional curvatures K = 1 − ψ s ψ
4f the n ( n − x the Ricci tensor of the metric g given by (2.1) is Ric = n {− ψ xx ψ + ϕ x ψ x ϕψ } ( dx ) + {− ψψ xx ϕ − ( n − ψ x ϕ + ψϕ x ψ x ϕ + n − } g can . In the geometric coordinate this simplifies to
Ric = ( nK ) ds + ψ [ K + ( n − K ] g can . The scalar curvature is given by R = 2 nK + n ( n − K . The above computations can be found in [5] or the textbook [29].Suppose we have a time dependent family of metrics g ( · , t ) having the form(2.1). Then the family g ( · , t ) satisfies the Ricci flow if and only if ϕ and ψ evolveby: ϕ t = n ψ ss ψ ϕ, (2 . ψ t = ψ ss − ( n −
1) 1 − ψ s ψ . (2 . x ψ ( x, t ) are called “ necks ” and the (interior) local maximal pointsare called “ bumps ”. As long as the solution exists at a time t , the radius of thesmallest neck is given by r min ( t ) = min { ψ ( x, t ) | ψ x ( x, t ) = 0 } . Of course, if the solution has no necks at the time t , we let r min not be defined.Denote by x + ( t ) , x − ( t ) the right-most bump (i.e. the largest local maximal pointon ( − , +1)) and left-most bump (i.e. the least local maximal point on ( − , +1))respectively. The region right of x + ( t ) and left of x − ( t ) are called the “ right polarcap ” and “ left polar cap ” respectively. In [5], Angenent and Knopf obtainedseveral useful estimates for the Ricci flow via the equations (2.3) and (2.4). Werecall some of their estimates as follows. Proposition 2.1 (Angenent-Knopf [5])
Let g ( t ) be a solution to the Ricci flowof the form (2.2) such that | ψ s | ≤ and the scalar curvature R > and ψ s hasfinitely many zeroes initially. Then
51) (Proposition 5.1 of [5])
As long as the solution exists, | ψ s | ≤ . (2) (Lemma 7.1 of [5]) There exists C = C ( n, g (0)) such that as long as thesolution exists, | Rm | ≤ Cψ . (3) (Lemma 5.6 and Lemma 7.2 of [5]) If the left polar cap is strictly concave(i.e., ψ ss < )at initial, then as long as the solution exists, left polar cap existsand remains strictly concave, and D = lim t ր T ψ ( x − ( t ) , t ) exists. Furthermore, nosingularity occurs on the left polar cap if D > . (4) (Lemma 9.1 of [5]) There exists C = C ( n, g ) such that KL [ logL + 2 − logL min (0)] ≤ C, where K = − K = ψ ss ψ and L = K = − ψ s ψ .
3. Classification of Shrinking Solitons
To understand the structure of singularities, one usually needs to get a classifi-cation for gradient shrinking solitons. In [28], Perelman obtained a complete classi-fication for nonnegatively curved gradient shrinking soliton in dimension three. Anopen question is how to generalize Perelman’s classification to higher dimensions.In the next proposition, we obtain such a classification for the class of rotationallysymmetric solitons.
Proposition 3.1
Let ( M, g ij ( t )) , −∞ < t < , be a nonflat gradient shrinkingsoliton to the Ricci flow on a complete ( n + 1) -dimensional manifold and assumethe metric g ij ( t ) is rotationally symmetric. Suppose ( M, g ij ( t )) has bounded andnonnegative sectional curvature and is κ -noncollapsed on all scales for some κ > .Then ( M, g ij ( t )) is one of the followings: (i) the round sphere S n +1 ; (ii) the round infinite cylinder ( −∞ , + ∞ ) × S n . roof. Note that for a rotationally symmetric metric, the nonnegativity of sec-tional curvatures is equivalent to the nonnegativity of curvature operator. Indeed,we can choose a coordinate system ( x , x , · · · , x n ) (where x is the radial directionand x i , i = 1 , · · · , n, are the spherical directions) on M such that all components ofthe Riemannian curvature tensor vanish in the coordinate system except the sec-tional curvatures R i i = ψ K and R ijij = ψ K ( i = j ), and then the equivalencefollows directly from Proposition 1.1 and 1.2 of [29].Firstly, we consider the case that the gradient shrinking soliton is compact andhas strictly positive sectional curvature everywhere. By the Theorem 1 in [7] wesee that the compact gradient shrinking soliton is getting round and tends to aspace form (with positive constant curvature) as the time tends the maximal time t = 0. Since the shape of a gradient shrinking Ricci soliton is unchanging up toreparameterizations and homothetical scalings, the gradient shrinking soliton hasto be the round ( n + 1)-sphere S n +1 .Next, we consider the case that the sectional curvature of the nonflat gradientshrinking soliton vanishes somewhere. Note that a rotationally symmetric metricis defined on I × S n for some interval I . By Hamilton’s strong maximum principlein [20], we know that the soliton splits off a line and then the soliton is the roundcylinder R × S n . (We remark that R k × S n +1 − k is not rotationally symmetric if k > n + 1)-dimensional noncompact κ -noncollapsedgradient shrinking soliton g ij ( t ), −∞ < t <
0, satisfies ∇ i ∇ j f + R ij + 12 t g ij = 0 , on − ∞ < t < , (3 . f and g ( t ) = ds + ψ ( s, t ) g can and with boundedand positive sectional curvature at each time t ∈ ( −∞ , t = −
1. Arbitrarily fix a point x in M . By thesame arguments as in the proof of Lemma 1.2 of Perelman [28] (or see the proofof Lemma 6.4.1 of [10] for the details), one has the followings:(1) at large distance from the fixed point x the function f has no critical point,and its gradient makes small angle with the gradient of the distance function from x ; 72) at large distance from x , the scalar curvature R is strictly increasing alongthe gradient curves of f , and lim sup d ( − ( x,x ) → + ∞ R ( x, − ≤ n f satisfies V ol { f = a } < V ol ( S n ( q n − . a .In the three-dimension case, Perelman (in Lemma 1.2 of [28]) argued by usingGauss-Bonnet formula to the level set { f = a } to derive a contradiction. But nowwe are considering the general dimensional case, in particular, the (generalized)Gauss-Bonnet formulas are not available. So we need a new argument in thefollowing.By using Gauss equation and (3.1), the intrinsic sectional curvature ˜ R ijij of thelevel set { f = a } can be compute as˜ R ijij = R ijij + ( h ii h jj − h ij )= R ijij + |∇ f | ( f ii f jj − f ij ) ≤ R ijij + |∇ f | ( f ii + f jj ) = R ijij + |∇ f | (1 − R ii − R jj ) . (3 . X = ∇ f |∇ f | the unit normal vector to the level set { f = a } . Then set X = δ ∂∂x + δ α ∂∂x α , and e i = u i ∂∂x + u αi ∂∂x α , i = 1 , , · · · , n. where the summation convention of summing over repeated indices is used and { x , x , · · · , x n } is the local coordinate on the ( n + 1)-dimensional rotationallysymmetric manifold with g = ds + ψ g can with x = s ∈ R and P = ( x , · · · , x n ) ∈ n and g αβ = δ αβ at ( s, P ). In these coordinates all components of the Riemanntensor and Ricci tensor vanish except R α α = K and R αβαβ = K ( α = β ) and R = nK and R αα = K + ( n − K , ( α = 1 , , · · · , n ) where K = − ψ ss ψ and K = − ψ s ψ . And then the scalar curvature R = 2 nK + n ( n − K . So we have R ijij = R ( u αi ∂∂x α , u βj ∂∂x β , u γi ∂∂x γ , u ηj ∂∂x η )= P αβγη u αi u βj u γi u ηj R αβγη = P αβ ( u αi u βj ) R αβαβ − P αβ u αi u βj u βi u αj R αβαβ = P nβ =1 [( u i ) ( u βj ) + ( u βi ) ( u j ) ] K + P nα,β =1 ( u αi u βj ) K − P nβ =1 u i u j u βi u βj K − P nα,β =1 u αi u βj u βi u αj K = ( P nβ =1 [( u i ) ( u βj ) + ( u βi ) ( u j ) ] + 2( u i ) ( u j ) ) K + P nα,β =1 [( u αi u βj ) − u αi u βj u βi u αj ] K = [( u i ) + ( u j ) ] K + P nα,β =1 [( u αi u βj ) − u αi u βj u βi u αj ] K = [( u i ) + ( u j ) ] K + [(1 − ( u i ) )(1 − ( u j ) ) − ( u i ) ( u j ) ] K = [( u i ) + ( u j ) ] K + [1 − ( u i ) − ( u j ) ] K . (3 . n X β =1 u βi u βj = − u i u j and n X β =0 ( u βj ) = 1since { e , · · · , e n } is an orthonormal basis of the level set { f = a } . Then by (3.4)9e have R ii = R iXiX + P nj =1 ,j = i R ijij = [( u i ) + ( δ ) ] K + [1 − ( u i ) − ( δ ) ] K + P nj =1 ,j = i [( u i ) + ( u j ) ] K + P nj =1 ,j = i [1 − ( u i ) − ( u j ) ] K = [ n ( u i ) + 1 − ( u i ) ] K + [ n − n ( u i ) − u i ) ] K = [1 + ( n − ε ] K + [( n − − ε ) K ] (3 . ε = ( u i ) ≪
1, if a is large enough.Obviously by (3.5) we get R ii < K + ( n − K = Rn < , and then 1 − R ii − R jj > . Again by (3.5) we know R ii > K + ( n − − ε ) K (3 . R ijij < εK + (1 − ε ) K . (3 . R ijij ≤ R ijij + |∇ f | (1 − R ii − R jj ) < εK + (1 − ε ) K + |∇ f | [1 − K + ( n − − ε ) K )] = 2 εK + n − { − K + ( n − − ε ) K + 2( n − − ε ) K − [1 − K + ( n − − ε ) K )] } + |∇ f | [1 − K + ( n − − ε ) K )] = n − { − − ε ( n − K − [1 − K + ( n − − ε ) K )]+ n − |∇ f | [1 − K + ( n − − ε ) K )] } < n − (3 . a , since 2(1 − ε ( n − K > − K +( n − − ε ) K ) > |∇ f | is large as a large. Then by (3.8) and the volume comparison theoremwe know V ol { f = a } > V ol ( S n ( q n − a and then it is a contradiction with (3.2).Therefore we have proved the proposition.
4. Type II Singularity Happens
Suppose we have a family of rotationally symmetric solutions { ( S n +1 , g α ( t )) | α ∈ [0 , } of the Ricci flow with g α (0) = ds + ψ α g can , α ∈ [0 , g can is the standardmetric of constant sectional curvature 1 on S n . We specify the initial metrics asfollows.When α = 1, let the initial metric g (0) be a symmetric dumbbell with twoequally-sized hemispherical regions joined by a thin neck. By the work in [5], wecan assume the two hemispheres are suitably large and the neck is suitably thinso that this initial metric g (0) leads to a neckpinch singularity of the Ricci flowat some time T < + ∞ . (see Figure 1.)Figure 1: A neckpinch formingWhen α = 0, let the initial metric g (0) be a lopsided and degenerate dumbbellwhere g (0) = ds + ψ g can with ψ (0) has only one bump and it is nonincreasingon the right polar cap and strictly concave on the left polar cap. (see Figure 2.)11igure 2: A degenerate dumbellClearly, we may choose the g (0) , g (0) to have positive scalar curvatures. Let { g α (0) | α ∈ [0 , } (see Figure 3.) be a smooth family of dumbbells (includingdegenerate dumbbells) connecting the g (0) to the g (0) and satisfying the follow-ings:(i) for each α ∈ [0 , ψ α (0) has exactly two bumps or one bump,(ii) for each α ∈ [0 , ψ α ) s (0) has only finitely many zeros, and satisfies | ( ψ α ) s | (0) ≤ , (iii) for each α ∈ [0 , ψ α (0) is strictly concave on the left polar cap,(iv) each initial metric g α (0), α ∈ [0 , g (0) ❅■ g α (0) ✻ g (0) (cid:0)(cid:0)(cid:0)✒ Figure 3: The smooth family of dumbellsSince the scalar curvature is positive, each solution g α ( t ), α ∈ [0 , T α < + ∞ and develops a singularity. The main purpose of this12ection is to show that there exists α ∈ [0 ,
1) such that the solution g α ( t ), withthe metric g α (0) as initial datum, develops a Type II singularity. We remark thata Type II singularity might occur in such family of metrics had been conjecturedand the intuition had already described in [14] and [32].Let us first consider the case that the solutions with degenerate dumbbells asinitial data. Lemma 4.1
Suppose g α ( t ) is a rotationally symmetric solution of the Ricci flowon S n +1 with g α (0) = ϕ ( x, dx + ψ α ( x, g can , x ∈ [ − , . If at initial, thescalar curvature R ( α ) > , ψ α ( x, has only one bump, it is nonincreasing on theright polar cap and is strictly concave on the left polar cap, and | ( ψ α ) s | ( x, ≤ on [ − , , then either the solution g α ( t ) develops a Type II singularity or it shrinksto a round point. Proof.
By the assumption of R ( α ) > ∂R∂t = ∆ R + 2 | Ric | we know that the maximal time T < + ∞ .Now we consider the geometric quantity s defined by s ( x, t ) = Z x ϕ ( x, t ) dx. Then the metric can be written as g = ds + ψ α ( s, t ) g can . In the following if we write a relation of the type f = f ( s ) , it is to be understoodas shorthand for f = f ( s ( x, t )) for evolving metrics. Since ψ α ( ± , t ) = 0, weknow that for any time 0 ≤ t < T , the bump exists. By the standard Sturmiancomparison [2], we know that ψ α ( x, t ) also has a unique bump for each t ∈ [0 , T ).Let x ∗ ( t ) denote the unique bump. By Proposition 2.1(3), we can define D = lim t ր T ψ α ( x ∗ ( t ) , t ) . We divide it into two cases: 13 ase 1:
D > ψ α is strictly con-cave on the left polar cap, we know that no singularity occurs on the left polarcap. Thus the singularity must occur on the right polar cap. Take the maximalpoints ( ˜ P m , t m ), i.e., choose the points ( ˜ P m , t m ) such that | Rm ( ˜ P m , t m ) | = sup t ≤ t m ,Q ∈ S n +1 | Rm ( Q, t ) | → + ∞ , as m → + ∞ .Since for any time t ∈ [0 , T ) we have ( ψ α ) s ( P, t ) = −
1, where P is the pole ofthe right polar cap (i.e. the point with x = 1), we can choose the nearest point P ′ m to P such that ( ψ α ) s ( P ′ m , t m ) = − . If d t m ( P ′ m , P ) > d t m ( ˜ P m , P ), then we set P m = ˜ P m , otherwise set P m = P ′ m . Clearly in the region between P m and P , wehave | ( ψ α ) s | ≥ .We first claim that the curvature at ( P m , t m ) is comparable to the curvature atthe maximal point ( ˜ P m , t m ). Indeed, if P m = ˜ P m , then there is nothing to show.If P m = ˜ P m , then by the estimate in Proposition 2.1(2) and by the condition that ψ α is nonincreasing on the right polar cap and by the choice of the point P m , weknow that | Rm ( ˜ P m , t m ) | ≤ Cψ α ( ˜ P m , t m ) ≤ Cψ α ( P m , t m ) . On the other hand K ( P m , t m ) = 1 − ( ψ α ) s ( P m , t m ) ψ α ( P m , t m ) = 34 ψ α ( P m , t m ) . So | Rm ( P m , t m ) | ≥ K ( P m , t m ) = 34 ψ α ( P m , t m ) ≥ C | Rm ( ˜ P m , t m ) | . Obviously since ( ˜ P m , t m ) is the maximal point, we have | Rm ( P m , t m ) | ≤ | Rm ( ˜ P m , t m ) | . So the curvatures at ( P m , t m ) and ( ˜ P m , t m ) are comparable, where we used thedefinition of | Rm | to be the largest absolute value of the eigenvalues of the curvatureoperator Rm . 14pplying the maximum principle to the evolution equation of the scalar cur-vature R : ∂R∂t = ∆ R + 2 | Ric | and using the pinching estimate in Proposition 2.1(4) we get dR max dt ≤ CR max . Then R max ( t ) ≥ CT − t for some constant C .We now argue by contradiction to show that the solution develops a Type IIsingularity in this case.Suppose not, then the singularity is of Type I. That is, there exists some con-stant C > C − T − t m ≤ R ( α ) ( P m , t m ) ≤ CT − t m . (4 . g ( m ) ij ( · , t ) = R ( α ) ( P m , t m )( g α ) ij ( · , t m + tR ( α ) ( P m , t m ) ) , for t ∈ [ − t m R ( α ) ( P m , t m ) , P m to the pole P measured in the rescaled metric g ( m ) ij ( · ,
0) is bounded. Indeed, by the estimatein Proposition 2.1(2), we know | Rm | ≤ Cψ α for some constant C . Then we have ψ α ( P m , t m ) ≤ C | Rm ( P m , t m ) | and using | ( ψ α ) s | ≥ in the region between P m and P , we have d t m ( P m , P ) ≤ ψ α ( P m , t m ) ≤ C q | Rm ( P m , t m ) | (4 . d t is the distance measured with the metric g α ( t ). Therefore by the pinchingestimate in Proposition 2.1(4) we know that the distance from P m to the pole P measured in the rescaled metric g ( m ) ij ( · ,
0) is bounded.The rescaled g ( m ) ij ( t ) is a solution of the Ricci flow defined for t ∈ [ − t m R ( α ) ( P m , t m ) , < R ( m ) ( · , t ) ≤ R ( m ) ( P m ,
0) = 1 and has bounded curvature. After tak-ing a subsequence of g ( m ) ij , we can assume that the marked manifold ( S n +1 , g ( m ) ij ( t ) , P )converges to a marked manifold ( R n +1 , g ij ( t ) , P ) , −∞ < t ≤
0, which is a solutionof the Ricci flow on R n +1 with nonnegative curvature operator ( by the pinchingestimate in Proposition 2.1(4) ), has bounded curvature with R ( P ∗ ,
0) = 1 at somepoint P ∗ , and is κ -noncollapsed for all scales. So the limit is a nonflat ancient κ -solution on R n +1 .The reduced distance, due to Perelman [27], is defined by l ( α ) ( q, τ ) = √ τ inf { R τ √ s ( R ( α ) ( γ ( s ) , t m − s ) + | ˙ γ ( s ) | g α ) ij ( t m − s ) ) ds | γ : [0 , τ ] → S n +1 with γ (0) = P, γ ( τ ) = q } . where τ = t m − t, for t < t m . Then by the Type I assumption, we have l ( α ) ( P, τ ) ≤ √ τ R τ √ s CT − t m + s ds ≤ C √ τ R τ √ s ds = C. (4 . t k → −∞ such that the scaling of g ij ( · , t )around P with the factor | t k | − and with the times t k shifting to the new time zeroconverge to a nonflat gradient shrinking soliton in C ∞ loc topology. Indeed, in theProposition 11.2 of [27], Perelman takes a limit around some points q ( τ ) wherethe reduced distance at q ( τ ) are uniformly bounded above by ( n + 1) /
2. Instead,in our situation, we want to take a backward limit around the fixed point P . Byinspecting the proof of Proposition 11.2 of [27] (see also the proof of Theorem6.2.1 of [10] for the details), one only needs to have a uniform upper bound for thereduced distance at the fixed point P . This is just our estimate (4.3) by the TypeI assumption. Then the same argument as Perelman in section 11.2 in [27] appliesto the present situation. 16y combining with the above Proposition 3.1 and noting that the gradientshrinking soliton ¯ g ij is noncompact, we conclude that the backward limit is S n × R . But since the limit is taking around the pole and the metric is rotationallysymmetric, it can not be S n × R , so we get a contradiction! Hence we have provedthat the singularity is of Type II. Case 2: D = 0.In this case, if the singularity is of Type II, then there is nothing to prove.Thus we may assume that the singularity is of Type I. By the same argument asin the case 1, we can first take a rescaling limit around the pole P at the maximaltime T to get an ancient κ -solution and then take a backward limit around thepole P again to get a nonflat gradient shrinking soliton. If the shrinking soliton iscompact, then by Proposition 3.1 we know that it is the round S n +1 . This impliesthat the original solution shrinks to a round point as the time tends to the maximaltime T . While if the shrinking soliton is noncompact, then by Proposition 3.1 weknow that it is S n × R ; so the same reason in the proof of the case 1 gives acontradiction! Therefore we have proved Lemma 4.1. Lemma 4.2
The set A of α ∈ [0 , such that the initial metric g α (0) leads to aneckpinch singularity of the Ricci flow at some time T α < + ∞ is open in [0 , . Proof.
Obviously it is not empty for 1 ∈ A .Suppose α ∈ A , then we claim that ψ α (0) has two bumps. Otherwise it hasonly one bump. Since it satisfies the assumptions of Lemma 4.1 by the aboveconditions (i), (ii), (iii) and (iv), then we know that either it shrinks to a roundpoint, or forms a Type II singularity. Consequently, the initial metric does notlead to a neckpinch for such α . This contradicts with α ∈ A . Similarly, ψ α ( t )has two bumps as long as the solution exists. Take a small perturbation g ( k ) α (0) of g α (0) (in C topology). Then g ( k ) α (0) still has two bumps. We need to show that g ( k ) α (0) leads to a neckpinch singularity at the maximal time T ( k ) α < + ∞ .Since g ( k ) α (0) are very close to g α (0) in C topology, the scalar curvatures ofthe metrics g ( k ) α (0) have a uniform positive lower bound. Thus it follows fromthe evolution equation of the scalar curvature that the maximal times T ( k ) α areuniformly bounded. After passing to a subsequence, we can then assume that17 ( k ) α → ˜ T as k → ∞ . Claim: ˜ T ≥ T .Indeed, suppose not, then there exists ε >
0, such that˜ T − ε < T ( k ) α < ˜ T + ε < T − ε < T for all sufficiently large k . Consider the time interval [0 , ˜ T − ε ]. By the assumptionthat g ( k ) α (0) is sufficiently close to g (0) and g ( t ) is smooth on [0 , ˜ T − ε ], we first showthat the curvature of g ( k ) α ( t ) is uniformly bounded on [0 , ˜ T − ε ] for all sufficientlylarge k .For each 0 ≤ t ≤ ˜ T − ε , set M ( t ) = sup {| Rm ( k ) ( x, t ) || k ≥ , x ∈ S n +1 } and t = sup { t ≥ | M ( t ) < + ∞} , where Rm ( k ) denotes the curvature of g ( k ) α ( t ). We want to show that t = ˜ T − ε .By Shi’s local derivative estimate in [30], we know that t >
0. Suppose t < ˜ T − ε , then for any small ε ′ >
0, consider the time interval [0 , t − ε ′ ]. Bythe above definition of M ( t ), we know that the curvature of g ( k ) α ( t ) is uniformlybounded by M ( t − ε ′ ) on [0 , t − ε ′ ]. Take a limit of g ( k ) α ( t ) and by the uniquenessof the solution to the Ricci flow [19] or [24], we get the limit must be the originalsolution g ( t ) on [0 , t − ε ′ ]. So we have the curvature of g ( k ) α ( t ) is uniformly boundedby some constant C which does not depend on ε ′ . Then by Shi’s local derivativeestimate in [30] again, we know that the curvature of g ( k ) α ( t ) is uniformly boundedon [0 , t − ε ′ + C ]. By choosing ε ′ small enough, we get t − ε ′ + C > t and thenit is a contradiction! So we have proved that t = ˜ T − ε , that is the curvature of g ( k ) α ( t ) is uniformly bounded on [0 , ˜ T − ε ] for all sufficiently large k . Similarly asabove, we can take a limit of g ( k ) α ( t ) and by the uniqueness of the Ricci flow [19] or[24], we get the limit must be g ( t ) on [0 , ˜ T − ε ]. Using g ( t ) is smooth on [0 , ˜ T ], weget the curvature of g ( k ) α ( t ) is uniformly bounded by some constant C ′ which doesnot depend on ε . Again by Shi’s local derivative estimate in [30], we know thatthe curvature of g ( k ) α ( t ) is uniformly bounded on [0 , ˜ T − ε + C ′ ] for all sufficientlylarge k . Choose ε > T ( k ) α → ˜ T as k → ∞ , we get˜ T − ε + C ′ > T ( k ) α for all sufficiently large k , which contradicts with the definitionof the maximal time. So ˜ T ≥ T . 18ext we show that each g ( k ) α (0) leads to a neckpinch singularity at the maximaltime T ( k ) α < + ∞ .For all sufficiently small ε >
0, since g α (0) leads to a neckpinch, we have( r α ) min ( T − ε ) ψ α ( x ± ( T − ε ) , T − ε ) ≪ . (4 . g ( k ) α (0) is sufficiently close to g α (0) and T ( k ) α → ˜ T ≥ T , weknow that as k large enough, T ( k ) α > T − ε and g ( k ) α ( T − ε ) is sufficiently close to g α ( T − ε ). So ( r ( k ) α ) min ( T − ε ) ψ ( k ) α ( x ± ( T − ε ) , T − ε ) ≪ . (4 . g (0) = ds + ψ (0) g can on S n +1 which has two bumps x ± (0) and r min (0) ψ ( x ± (0) , < C − for some universal constant C >
0, (for example we cantake C = 100), then it leads to a neckpinch singularity. By (4.5), we know that g ( k ) α (0) leads to a neckpinch singularity. Therefore we proved that A is open in[0 , Proposition 4.3
Suppose g ij ( t ) , t ∈ [0 , T ) , is a rotationally symmetric solutionof the Ricci flow on S n +1 with g (0) = ds + ψ (0) g can . If at initial the scalarcurvature R > , then for any given ε > , there exists K = K ( ε, g (0)) > max { ε − , Q ( T ) } > , where Q ( T ) denotes the upper bound of the curvaturefor the times t ≤ T , such that for any point ( x , t ) with t ≥ T and Q = R ( x , t ) ≥ K , the solution in { ( y, t ) | d t ( y, x ) < ε − Q − , t − ε − Q − ≤ t ≤ t } is, after scaling by the factor Q , ε -close to the corresponding subset of some ori-entable ancient κ -solution, where κ is a positive constant depending only on T andthe initial metric g (0) . Consequently, in the region, we have the following gradientestimates |∇ ( R − ) | ≤ η and | ∂∂t ( R − ) | ≤ η for some constant η = η ( κ ) > . roof. This is just a higher dimensional version of Perelman’s singularity result(Theorem 12.1 of [27]) for the rotationally symmetric class. In Theorem 12.1 of[27], Perelman obtained this singularity structure result for any three-dimensionalsolution. For the details, one can consult [26] (from page 83 to 88) or [10] (frompage 399 to 405). By inspecting Perelman’s argument, when one tries to general-ize Perelman’s singularity structure result to higher dimensions, one only needs tohave a higher-dimensional version of the (three-dimensional) Hamilton-Ivey cur-vature pinching estimate and shows the canonical neighborhoods of an ancient κ -solution consisting ε -necks or ε -caps. For our case, since the metric is rotation-ally symmetric, the estimate due to Angenent-Knopf in Proposition 2.1(4) givesthe desired curvature pinching estimate. While for a rotationally symmetric an-cient κ -solution, it is clear that any canonical neighborhood is either an ε -neckor an ε -cap. So by repeating Perelman’s argument, we obtain the proof of theproposition. Proof of Theorem 1.2.
Suppose g α ( t ) is the family of the solutions to the Ricci flow satisfies the aboveconditions (i), (ii), (iii) and (iv). We want to show that there exists α ∈ [0 ,
1) suchthat for the solution g α ( t ) of the Ricci flow on S n +1 with initial data g α (0) = ds + ψ α g can , exists up to a maximal time T α < + ∞ and develops a Type IIsingularity.Since g (0) = ds + ψ g can and by our assumption that g (0) leads to a neckpinchsingularity. Then by Lemma 4.2, we know that A is not empty and open in [0 , g ( t ) with the initial data g (0)either develops a Type II singularity or shrinks to a round point, so 0 ∈ A . Let( α,
1] be a connected component of A . We want to show that the α is the numberwe want.If g α (0) develops a Type II singularity, then there is nothing to show. So in thefollowing we assume it does not develop a Type II singularity. Claim 1: ψ α (0) exactly has two bumps. Indeed, if ψ α (0) has only one bump, then ψ α is nonincreasing on the right polar20ap and by our construction we know that ψ α is strictly concave on the left polarcap. So by Lemma 4.1, we know either the singularity is of Type II or it shrinksto a round point at the maximal time T α < + ∞ . By our assumption that thesingularity is not of Type II. So it is shrinking to a round point, and then thereexists a time ˜ t < T α close to T α , such that the curvature is positive for t ≥ ˜ t .Whenever β ∈ ( α, ⊂ A is sufficiently close to α , the metric g β (0) is sufficientlyclose to the metric g α (0) ( in the C topology). Then by Lemma 4.2 we canchoose β ∈ ( α, ⊂ A sufficiently close to α so that the maximal time T β of g β ( t )satisfies T β > ˜ t + ( T α − ˜ t ) /
2; moreover, by the continuous dependence of the initialmetric, the curvature operator of g β ( t ) is also positive at the time t = ˜ t . Hence byTheorem 1 in [7] we know that g β (0) will shrink to a round point at the maximaltime T β < + ∞ , which contradicts with β ∈ ( α, ⊂ A . Therefore we have provedthe Claim 1. Claim 2: ψ α ( t ) exactly has two bumps as long as the solution exists. Indeed, since g α ( t ) is a rotationally symmetric solution of the Ricci flow on S n +1 , we know that at the poles of the right and left polar caps ψ α ( t ) = 0 for anytime 0 ≤ t < T α , so there always exists one bump. By the standard Sturmiancomparison in [2] we know that the number of the bumps is nonincreasing in time.Suppose at some time t ∈ (0 , T α ) such that the right-most bump disappeared, then ψ α ( t ) has only one bump. Thus by Lemma 4.1 and our assumption, it shrinksto a round point. Particularly, there exists a time t < ˜ t < T α such that thecurvature is positive for all times t ≥ ˜ t . By the same argument as above, we canchoose β ∈ ( α, ⊂ A sufficiently close to α so that the maximal time T β of g β ( t )is greater than ˜ t and the curvature of g β ( t ) at the time t = ˜ t is also positive. Byapplying Theorem 1 in [7] again we know that g β (0) will shrink to a round pointat the maximal time T β < + ∞ , which also contradicts with β ∈ ( α, ⊂ A . Thuswe have proved the Claim 2.So in the following we always assume that ψ α ( t ) has two bumps for all times t ∈ [0 , T α ).Since at the maximal time T α , the solution g β ( t ) does not develop a neckpinch.In views of Angenent-Knopf’s result [5], the smaller polar cap must collapse. So,without loss of generality, we may assume that singularity occurs on the rightpolar cap. Similarly as in Lemma 4.1, we first take the maximal points ( ˜ P m , t m ) on S n +1 (i.e., | Rm ( ˜ P m , t m ) | = sup t ≤ t m ,Q ∈ S n +1 | Rm ( Q, t ) | ). We then take the nearestpoint P ′ m to the pole P on the right polar cap, such that ( ψ α ) s ( P ′ m , t m ) = − .21f d t m ( P ′ m , P ) > d t m ( ˜ P m , P ), then we set P m = ˜ P m ; otherwise we set P m = P ′ m .Clearly in the region between P m and P , we have | ( ψ α ) s | ≥ .Define g ( m ) ij ( · , t ) = R ( α ) ( P m , t m )( g α ) ij ( · , t m + tR ( α ) ( P m , t m ) )for t ∈ [ − t m R ( α ) ( P m , t m ) , Claim 3:
A subsequence of g ( m ) ij ( · , t ) around the point P will converge to anonflat complete ancient κ -solution on a smooth manifold M , where κ is somepositive constant depending only on the initial metric g α (0) . Indeed, if the maximal point ( ˜ P m , t m ) is on the right polar cap, then by theestimate in Proposition 2.1(2) and by the condition that ψ α is nonincreasing onthe right polar cap and by the choice of the point P m , we know that | Rm ( ˜ P m , t m ) | ≤ Cψ α ( ˜ P m , t m ) ≤ Cψ α ( P m , t m ) . On the other hand K ( P m , t m ) = 1 − ( ψ α ) s ( P m , t m ) ψ α ( P m , t m ) = 34 ψ α ( P m , t m ) . So | Rm ( P m , t m ) | ≥ K ( P m , t m ) = 34 ψ α ( P m , t m ) ≥ C | Rm ( ˜ P m , t m ) | . Obviously since ( ˜ P m , t m ) is the maximal point, we have | Rm ( P m , t m ) | ≤ | Rm ( ˜ P m , t m ) | . So the curvatures at ( P m , t m ) and ( ˜ P m , t m ) are comparable, where we used thedefinition of | Rm | to be the largest absolute value of the eigenvalues of the curvatureoperator Rm . So by repeating (part of) the argument as in case 1 in Lemma 4.1,we know that a subsequence of g ( m ) ij ( · , t ) around the point P will converge to anonflat complete ancient κ -solution on a smooth manifold M for some positiveconstant κ depending only on the initial metric g α (0).We remain to consider the case that the maximal point ( ˜ P m , t m ) does not lieon the right polar cap, then it must lie in the region between the two bumps.22n this case, we first prove the following assertion: For any
A < + ∞ , there exists a positive constant C ( A ) such that the curva-tures of g ( m ) ij ( · , t ) at the new time t = 0 satisfy the estimate | Rm ( m ) ( y, | ≤ C ( A ) whenever d g ( m ) ( · , ( y, P m ) ≤ A and m ≥ , where Rm ( m ) denotes the curvature ofthe metric g ( m ) ij . This assertion in the three-dimensional Ricci flow has been verified by Perelmanin his proof of the Theorem 12.1 in [27] (the first detailed exposition of this part ofPerelman’s argument appeared in the first version of Kleiner-Lott [26]), where theonly three-dimension features he used are the Hamliton-Ivey curvature pinchingestimate and the canonical neighborhood condition of an ancient κ -solution con-sisting the ε -necks and ε -caps. In our case, by noting that the metric is rotationallysymmetric, the canonical neighborhood condition can be easily obtained as pointedout before, and the pinching estimate has already given in Proposition 2.1(4). Soby some slight modifications, Perelman’s argument also works for our case. In thefollowing we only give the details for the modified parts. For the complete details,one can compare with [26] (from page 85 to 87) or [10] (from page 400 to 402).For each ρ ≥
0, set M ( ρ ) = sup { R ( m ) ( x, | m ≥ , x ∈ S n +1 with d ( x, P m ) ≤ ρ } and ρ = sup { ρ ≥ | M ( ρ ) < + ∞} . By the pinching estimate in Proposition 2.1(4), it suffices to show ρ = + ∞ . We need to adapt Perelman’s argument to show that ρ > ε >
0, by Proposition 4.3, we know that there exists K = K ( ε, g α (0)) > max { ε − , Q ( T α ) } >
0, where Q ( T α ) denotes the upperbound of the curvature for the times t ≤ T α , such that for any point ( x , t ) with t ≥ T α and Q = R ( α ) ( x , t ) ≥ K , the solution in { ( y, t ) | d t ( y, x ) < ε − Q − , t − ε − Q − ≤ t ≤ t } is, after scaling by the factor Q , ε -close to the correspondingsubset of some orientable ancient κ -solution for some positive constant κ dependingonly on the initial metric g α (0). Consequently we have the gradient estimate in23he region |∇ ( R − ) | ≤ η and | ∂∂t ( R − ) | ≤ η (4 . η = η ( κ ) > . If R ( α ) ( P m , t m ) ≥ K , then by the above gradient estimate (4.6), we know thatthere exists some constant c = c ( η ) > R ( α ) ( x, t m ) ≤ R ( α ) ( P m , t m )for any point x ∈ B t m ( P m , c ( R ( α ) ( P m , t m )) − ). Hence in this case we have ρ ≥ c > R ( α ) ( P m , t m ) < K , then we prove that ρ ≥ ˜ c for some constant ˜ c = ˜ c ( c, K, ¯ c ),where ¯ c is the positive lower bound of the scalar curvature R ( α ) on S n +1 at initialtime. In fact, consider the points x ∈ B t m ( P m , c ( R ( α ) ( P m , t m )) − ), if R ( α ) ( x, t m )
0. Set ˜ c = min { c , c ′ } , then in this case we have ρ ≥ ˜ c > ρ > ρ = + ∞ . That is, the curvatures of g ( m ) ij ( · , t ) at thenew times t = 0 stay uniformly bounded at bounded distances from P m for all m .Furthermore, by the estimate in Proposition 2.1(2) and using | ( ψ α ) s | ≥ in theregion between P m and P , we know that the distance from P m and P measuredin the rescaled metric g ( m ) ij ( · ,
0) is bounded. So we obtained that the curvature24f g ( m ) ij ( · , t ) at the new times t = 0 stay uniformly bounded at bounded distancesfrom P for all m . This completes the proof of the assertion.By the gradient estimate in Proposition 4.3 and Shi’s local derivative estimatein [30] and Hamilton’s compactness theorem in [23], we can take a C ∞ loc subsequentlimit to obtain ( M, g ∞ ( · , t ) , P ) which is complete, κ -noncollapsed on all scales andis defined on a space-time open subset of M × ( −∞ ,
0] containing the time slice M × { } . Clearly it follows from the pinching estimate in Proposition 2.1(4) thatthe limit ( M, g ∞ ( · , t ) , P ) has nonnegative curvature operator. Then exactly asPerelman’s argument in Theorem 12.1 of [27] (see also [26] and [10] for details),we can get that the curvature of the limit g ∞ ( · , t ) at t = 0 has bounded curvatureand also that the limit g ∞ ( · , t ) can be defined on ( −∞ , g ∞ ( · , t ) is an ancient κ -solution on M and Claim 3 holds.Since by our assumption that the singularity is not of Type II. Then thereexists some constant ˜ C > ≤ R ( P m , t ) ≤ ˜ CT α − t . Then by Claim 3 we know that a subsequence of g ( m ) ij ( · , t ) around P converges toa nonflat ancient κ -solution g ij on M . Then by the same proof as in the case 1 inLemma 4.1, we obtain that there exists a sequence of times t k → −∞ such that thescaling of g ij ( · , t ) around P with the factor | t k | − and with the times t k shifting tothe new time zero converge to a nonflat gradient shrinking soliton in C ∞ loc topology.If the nonflat gradient shrinking soliton is noncompact, Proposition 3.1 gives usthat it is R × S n . But since the limit is taking around the pole P and the metricis rotationally symmetric, it can not be R × S n . So this contradiction implies thatthe nonflat gradient shrinking soliton is compact. By Proposition 3.1 again, weknow that it is the round S n +1 . Consequently the curvature of the original solutionbecomes positive as the time t close to the maximal time T α . Then repeat the sameproof as in Claim 1, we can choose β ∈ ( α, ⊂ A sufficiently close to α suchthat g β (0) will also shrink to a round point at the maximal time T β < + ∞ , whichcontradicts with β ∈ ( α, ⊂ A . So the singularity must be of Type II. Thereforewe have proved our theorem 1.2. Remark 1.
During the proof of the main theorem, we actually proved the exis-tence of Type II singularities on noncompact manifolds. More precisely, we proved25hat for each n ≥
2, there exists complete and rotationally symmetric metrics on R n +1 with bounded curvatures such that the Ricci flow starting at the metricsdevelop Type II singularities at some times T < + ∞ . In particular, we can takethe initial metrics on R n +1 to be the complete and rotationally symmetric, withnonnegative sectional curvature and positive scalar curvature, and asymptotic tothe round cylinder of scalar curvature 1 at infinity. Remark 2.
In the unpublished preprint [9], Robert Bryant proved the existenceof the nontrivial steady Ricci solitons on R n by solving certain nonlinear ODEsystem. These steady Ricci solitons are complete, rotationally symmetric withpositive curvatures. By combining with the work of Hamilton [22], this papergives another proof for the existence of the nontrivial steady Ricci solitons on R n for all dimensions n ≥
3, which are also complete, rotationally symmetric and havepositive curvatures.
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