Uniqueness of compact ancient solutions to the higher dimensional Ricci flow
Simon Brendle, Panagiota Daskalopoulos, Keaton Naff, Natasa Sesum
aa r X i v : . [ m a t h . DG ] F e b UNIQUENESS OF COMPACT ANCIENT SOLUTIONS TO THEHIGHER DIMENSIONAL RICCI FLOW
SIMON BRENDLE, PANAGIOTA DASKALOPOULOS, KEATON NAFF AND NATASASESUM
Abstract.
In this paper, we study the classification of κ -noncollapsed ancientsolutions to n-dimensional Ricci flow on S n , extending the result in [13] tohigher dimensions. We prove that such a solution is either isometric to afamily of shrinking round spheres, or the Type II ancient solution constructedby Perelman. Contents
1. Introduction 12. Preliminary Results on Structure of Compact Ancient κ -Solutions 43. Rotational Symmetry of Compact Ancient κ -Solutions in HigherDimensions 74. A priori estimates for compact ancient κ -solutions with rotationalsymmetry 125. The tip region weights µ + ( ρ, τ ) and µ − ( ρ, τ ) 256. Overview of the proof of Theorem 1.3 277. Energy estimates in the tip region and proof of Proposition 6.5 358. Energy estimates in the cylindrical region and proof of Proposition 6.7 409. Analysis of the overlap region and proof of Proposition 6.8 4510. Analysis of the neutral mode and proof of Proposition 6.9 47Appendix A. The Bryant soliton 49References 491. Introduction
In this paper, we consider a solution to the Ricci flow ∂∂t g ( t ) = − g ( t ) ona compact manifold which exists for all times t ∈ ( −∞ , T ). We call such a so-lution an ancient solution. The main focus of this paper is the classification ofancient solutions to Ricci flow in dimensions n ≥
4, under natural isotropic curva-ture conditions that will be discussed below. Ancient solutions play an importantrole in singularity formation in geometric flows since these solutions occur as limitsof sequences of rescalings in regions of high curvature. For example, Perelman’s
The first author was supported by the National Science Foundation under grant DMS-1806190and by the Simons Foundation. The second author was supported by the National Science Foun-dation under grant DMS-1266172. The fourth author was supported by the National ScienceFoundation under grants DMS-1056387 and DMS-1811833. work on the Ricci flow [29] shows that high curvature regions in a three dimen-sional Ricci flow are modeled on ancient solutions with nonnegative curvature thatare κ -noncollapsed. In the same paper, Perelman also showed that even in higherdimensions, ancient solutions that occur as blow-up limits around points of highcurvature are κ -noncollapsed. We will focus on these κ -noncollapsed ancients so-lutions. Let us begin by briefly reviewing what is known in dimensions two andthree.In dimension two, ancient solutions to the Ricci flow have been completely clas-sified through a combination of work by Chu, the second author, Hamilton, andthe fourth author in three papers [18, 19, 21]. In particular, there is actually aclassification of both collapsed and κ -noncollapsed ancient solutions. Altogether,there are precisely three (non-flat, non-quotient) ancient solutions: the family ofshrinking round spheres, the King solution, and steady cigar soliton. The Kingsolution, independently discovered by King [26] and Rosenau [31], resembles twosteady cigar solitons which have been cut and glued together to form a compactsolution. Of course, the sphere is κ -noncollapsed, while the cigar and, hence, theKing solution are both collapsed.In dimension three, there are expected to be many more examples of collapsed ancient solutions. For κ - noncollapsed ancient solutions however, Perelman’s con-jecture [29], and its analogue in the compact setting, indicated a simple classifi-cation should exist. These conjectures stood for a number of years until severalrecent breakthroughs made it possible to resolve them in full. In dimension three,noncollapsed ancient solutions have now been completely classified through a com-bination of results by Angenent, and the first, second, and fourth authors in fourpapers [1, 6, 9, 13] (as well as a pinching result in [14]). Altogether, there areprecisely four (non-flat, non-quotient) κ -noncollapsed ancient solutions: the familyof shrinking round spheres, the family of shrinking round cylinders, Perelman’s an-cient oval solution on S , and the steady Bryant soliton. Perelman’s ancient oval isthe higher dimensional analogue of the King solution: it resembles a gluing of twoBryant solitons at very negative times.We now turn our attention to dimensions n ≥
4. We are interested in classi-fying ancient solutions which model singularity formation. The question is: whatclass of singularity models can we understand using the techniques developed indimension three? The classification of ancient κ -solutions in dimension three relieson a number of ingredients. As we have mentioned, Perelman’s κ -noncollapsing iscrucial and holds in all dimensions. There are two ingredients special to dimensionthree though, which do not apply in higher dimensions. The first ingredient is theHamilton-Ivey curvature pinching estimate, which ensures that all blow-up limitshave nonnegative curvature. Once one has nonnegative curvature, Hamilton’s Har-nack inequality holds. The Harnack inequality then implies bounded curvature atbounded distance and, finally, an argument of Perelman upgrades this to boundedcurvature. Therefore, in dimension three, ancient κ -solutions automatically havenonnegative and bounded curvature. No general Hamilton-Ivey-type estimate holdsin higher dimensions without an initial curvature positivity assumption. The secondingredient is that the cross section of a noncompact singularity model in dimen-sion three must be compact. Perelman used this property to establish an important“tube and cap” structure theorem for ancient κ -solutions in dimension three. When NIQUENESS OF COMPACT ANCIENT SOLUTIONS 3 n ≥ n ≥
12. In thesedimensions, PIC initial data does not ensure singularity models have nonnegativecurvature, but rather they must satisfy a weaker curvature condition known asPIC2. See Section 2 to recall the precise definitions of PIC and PIC2. Importantly,however, this latter curvature condition is still strong enough to ensure Hamilton’sHarnack inequality holds [10]. Hamilton and the first author’s work justifies thefollowing definition.
Definition 1.1.
Suppose n ≥
4. An ancient κ -solution is an n -dimensional, an-cient, complete, nonflat solution of the Ricci flow that is uniformly PIC and weaklyPIC2; has bounded curvature; and is κ -noncollapsed on all scales.To summarize, if the initial data of a Ricci flow is PIC and n = 4 or n ≥ κ -solutions in the sense above. We expecta similar result to be true for 5 ≤ n ≤ κ -solutions in thesense of Definition 1.1, extending the classification in dimension three. Havingidentified the correct curvature assumptions, the program is roughly the same.The first important step was accomplished in [6], where the first author showeduniqueness of the Bryant soliton in the class of steady solitons with asymptoticcylindricality. Subsequently, the first author and the second author used [6] andarguments in [9] to prove uniqueness of the Bryant soliton among noncompactancient κ -solutions in higher dimensions in [15]. It remains to extend the result of[13] for compact ancient solution to higher dimensions, which we complete here. Asin [13], the proof is accomplished in two steps. In the first step, we use argumentsfrom [15] to prove: Theorem 1.2.
Let ( S n , g ( t )) be an ancient κ -solution on S n . Then ( S n , g ( t )) isrotationally symmetric. Next, we give a complete classification of all ancient κ -solutions on S n withrotational symmetry: Theorem 1.3.
Let ( S n , g ( t )) and ( S n , g ( t )) be two ancient κ -solutions on S n which are rotationally symmetric. Assume that neither ( S n , g ( t )) nor ( S n , g ( t )) is a family of shrinking round spheres. Then ( S n , g ( t )) and ( S n , g ( t )) coincide upto a reparametrization in space, a translation in time, and a parabolic rescaling. Combining Theorem 1.2 and Theorem 1.3, we can draw the following conclusion:
Theorem 1.4.
Let ( S n , g ( t )) be an ancient κ -solution on S n which is not a familyof shrinking round spheres. Then ( S n , g ( t )) coincides with Perelman’s solution upto diffeomorphisms, translations in time, and parabolic rescalings. S.BRENDLE, P. DASKALOPOULOS, K. NAFF, N. SESUM
Let us mention some related work in the mean curvature flow setting. In [20], theauthors classified compact, convex ancient solutions to the curve shortening flow.In [11, 12], the authors proved that the bowl soliton is the only ancient solutionwhich is noncompact, noncollapsed, strictly convex, and uniformly two-convex. In[2], the authors showed that every ancient solution which is compact, noncollapsed,strictly convex, and uniformly two-convex is either the family of shrinking spheresor the ancient oval constructed by White (cf. [32]) and Haslhofer-Hershkovits (cf.[25]). Finally, compact ancient solutions which are collapsed were studied in [5].The outline of the paper is as follows: In Section 2, we recall some qualitativeproperties of ancient κ -solutions on S n . In particular, an ancient κ -solution on S n is either a family of shrinking round spheres, or it has the structure of two capsjoined by a tube (in which the solution is nearly cylindrical). In Section 3, we givethe proof of Theorem 1.2.In Section 4, we derive a-priori estimates for rotationally symmetric solutions.In Section 5, we introduce two weight functions µ + ( ρ, τ ) and µ − ( ρ, τ ) (one for eachcap). These will be used in Section 7 to prove weighted estimates for the linearizedequation in each tip region.In Section 6, we give an overview of the proof of Theorem 1.3. The proof relies ina crucial way on estimates for the linearized equation in the tip region (Proposition6.5) and in the cylindrical region (Proposition 6.7). These estimates are proved inSection 7 and Section 8.2. Preliminary Results on Structure of Compact Ancient κ -Solutions In this section, we record basic facts about the structure of ancient κ -solutionsas in Section 2 of [13]. We begin by recalling the definitions of uniformly PIC andweakly PIC2 Riemannian manifolds. Definition 2.1.
Suppose n ≥ M, g ) is a Riemannian manifold of dimension n ≥ • We say that (
M, g ) is uniformly PIC if there exists a real number α > R ( ϕ, ¯ ϕ ) ≥ α | Rm | | ϕ | for all complex two-vectors ofthe form ϕ = ( e + ie ) ∧ ( e + ie ), where { e , e , e , e } is an orthonormalfour-frame. • We say that (
M, g ) is weakly PIC2 if R ( ϕ, ¯ ϕ ) ≥ ϕ = ( e + iµe ) ∧ ( e + iλe ), where { e , e , e , e } is anorthonormal four-frame and λ, µ ∈ [0 , M, g ) is strictly PIC2.Equivalently, • ( M, g ) is uniformly PIC if there exists α > p ∈ M and every orthonormal four-frame e , e , e , e ∈ T p M , the curvature tensor R ijkl satisfies R + R + R + R − R ≥ α | Rm | . • ( M, g ) is weakly PIC2 if for every p ∈ M , every orthonormal four-frame e , e , e , e ∈ T p M , and every λ, µ ∈ [0 , R + λ R + µ R + λ µ R − λµR ≥ . NIQUENESS OF COMPACT ANCIENT SOLUTIONS 5
Four-dimensional ancient κ -solutions in the sense of Definition 1.1 automati-cally satisfy the restricted isotropic curvature pinching condition used to studyfour-dimensional ancient solutions in [17]. For a proof and to recall the meaningof the pinching condition, see Proposition A.2 in [15]. Note that the restrictedisotropic curvature pinching condition implies four-dimensional ancient κ -solutionshave nonnegative curvature operator. The restricted isotropic curvature conditionis the assumption under which the authors in [17] developed a theory for ancient κ -solutions in dimension four, following the work of Hamilton and Perelman. Im-portantly, for each of the structure results established for ancient κ -solutions indimensions n ≥ κ -solutions in dimension n = 4 under the restricted isotropic curvature pinching condition, which can befound in [17]. In particular, we note that compactness of ancient κ -solutions in thesense above is established in [8] for n ≥ n = 4.From now on, we assume ( M, g ( t )) is an ancient κ -solution which is compactand simply connected. We also assume ( M, g ( t )) is not a family of shrinking roundspheres. Note that because (
M, g ( t )) is compact, the strong maximum principle(Proposition 6.6 in [8]) implies ( M, g ( t )) is strictly PIC2. By the work of the firstauthor and Schoen, this implies M is diffeomorphic to S n . Proposition 2.2.
The asymptotic shrinking soliton associated with ( M, g ( t )) is acylinder.Proof. The only gradient shrinking Ricci solitons which are uniformly PIC andweakly PIC2 are round the sphere S n , the round cylinder S n − × R , or a quotientof one of these two by a discrete group of isometries (see Theorem A.1 in [15]). Ifthe asymptotic soliton has constant curvature, then ( M, g ( t )) would have constantcurvature by the pinching result in [14]. This would contradict our assumption that( M, g ( t )) is not a family of shrinking round spheres. The asymptotic soliton cannotbe a compact quotient of the cylinder for a number of reasons. Perhaps the clearestis that these compact quotients of S n − × R do not move self-similarly under theRicci flow. Alternatively, if the asymptotic soliton is compact, then by smoothCheeger-Gromov convergence M must be diffeomorphic to a compact quotient ofthe cylinder, but M is diffeomorphic to S n . Finally, if the asymptotic soliton is anoncompact quotient of the cylinder, then by Theorem A.1 in [7] the fundamentalgroup of some nontrivial quotient of S n − would inject into the fundamental groupof M , which is trivial by assumption. It follows the asymptotic soliton must beisometric to a round cylinder. (cid:3) Proposition 2.3.
Let ( x k , t k ) be an arbitrary sequence of points in space-timesatisfying lim k →∞ t k = −∞ . Consider the family of rescaled metrics g k ( t ) := R ( x k , t k ) g (cid:0) t k + R ( x k , t k ) − t (cid:1) . After passing to a subsequence, the sequence ofpointed flows ( M, g k ( t ) , x k ) converges in the Cheeger-Gromov sense to either a fam-ily of shrinking cylinders or the Bryant soliton.Proof. We have compactness of ancient κ -solutions for n ≥ (cid:3) We now fix a large number
L < ∞ and a small number ε > λ ( x, t ) thesmallest eigenvalue of the Ricci tensor at ( x, t ), we fix a small number θ > S.BRENDLE, P. DASKALOPOULOS, K. NAFF, N. SESUM property that if ( x, t ) is a spacetime point satisfying λ ( x, t ) ≤ θR ( x, t ), then thepoint ( x, t ) lies at the center of an evolving ε -neck. The existence of θ is based ona standard contradiction argument which uses compactness of ancient κ -solutionsin higher dimensions. See Lemma A.2 in [15] for a proof in the noncompact case.The proof in the compact case is nearly identical. Proposition 2.4.
Consider a sequence of times t k → −∞ . If k is sufficiently large,then we can find two disjoint compact domains Ω ,k and Ω ,k with the followingproperties: • Ω ,k and Ω ,k are each diffeomorphic to B n . • For each x ∈ M \ (Ω ,k ∪ Ω ,k ) , we have λ ( x, t k ) < θR ( x, t k ) . In particular,the point ( x, t k ) lies at the center of an evolving ε -neck. • For each x ∈ Ω ,k ∪ Ω ,k , we have λ ( x, t k ) > θR ( x, t k ) . • ∂ Ω ,k and ∂ Ω ,k are leaves of Hamilton’s CMC foliation of ( M, g ( t k )) . • For each k , there exists a leaf Σ k of Hamilton’s CMC foliation with theproperty that Ω ,k and Ω ,k lie in different connected components of M \ Σ k ,and sup x ∈ Σ k λ ( x,t k ) R ( x,t k ) → . • The domains (Ω ,k , g ( t k )) and (Ω ,k , g ( t k )) each converge, after rescaling,to a corresponding subset of the Bryant soliton.Proof. The proof is the same as the proof of Proposition 2.3 in [13]. (cid:3)
Corollary 2.5. If k is sufficiently large, then the set { x ∈ M : λ ( x, t k ) > n R ( x, t k ) , ∇ R ( x, t k ) = 0 } consists of exactly two points. One of these pointslies in Ω ,k and the other lies in Ω ,k . In particular, these points are contained indifferent connected components of M \ Σ k .Proof. The proof is the same as the proof of Corollary 2.4 in [13]. We note thatin higher dimensions, at the tip of the n -dimensional Bryant soliton we have Ric = n R g , ∇ R = 0, and ∇ R <
0. Moreover, the tip is the unique point on the Bryantsoliton where ∇ R = 0 and λ > n R . The claim now follows from the previousproposition. (cid:3) Definition 2.6.
We say that p is a tip of ( M, g ( t )) if λ ( p, t ) > n R ( p, t ) and ∇ R ( p, t ) = 0.By the previous corollary, ( M, g ( t )) has exactly two tips − t is sufficiently large,and the two tips are contained in different components of { x ∈ M : λ ( x, t ) > θR ( x, t ) } . Let us denote the tips of ( M, g ( t )) by p ,t and p ,t . The followingproposition is an immediate consequence of Proposition 2.3. Proposition 2.7.
Consider a sequence of times t k → −∞ . Let p ,t k and p ,t k denote the tips in ( M, g ( t k )) . If we rescale the flow around ( p ,t k , t k ) or ( p ,t k , t k ) as in Proposition 2.3, then the rescaled flows subsequentially converge to the Bryantsoliton in the Cheeger-Gromov sense. Proposition 2.8.
Consider a sequence of times t k → −∞ . Let p ,t k and p ,t k denote the tips in ( M, g ( t k )) . Then, we have R ( p ,t k , t k ) d g ( t k ) ( p ,t k , p ,t k ) → ∞ and R ( p ,t k , t k ) d g ( t k ) ( p ,t k , p ,t k ) → ∞ .Proof. We have Perelman’s long-range curvature estimate for n = 4 by Proposition3.6 in [17] and for n ≥ (cid:3) NIQUENESS OF COMPACT ANCIENT SOLUTIONS 7
Corollary 2.9.
For a given positive real number A , if − t is sufficiently large (de-pending on A ), then the balls B g ( t ) ( p ,t , AR ( p ,t , t ) − ) and B g ( t ) ( p ,t , AR ( p ,t , t ) − ) are disjoint. Proposition 2.10.
For a sequence of spacetime points ( x k , t k ) with t k → −∞ denote by p ,t k and p ,t k the tips of ( M, g ( t k )) . If R ( p ,t k , t k ) d g ( t k ) ( p ,t k , x k ) → ∞ and R ( p ,t k , t k ) d g ( t k ) ( p ,t k , x k ) → ∞ , then λ ( x k ,t k ) R ( x k ,t k ) → .Proof. The proof is the same as the proof of Proposition 2.9 in [13] (cid:3)
Corollary 2.11.
Given ε > , we can find a time T ∈ ( −∞ , and a large con-stant A with the following property. If t ≤ T and x B g ( t ) ( p ,t , AR ( p ,t , t ) − ) ∪ B g ( t ) ( p ,t , AR ( p ,t , t ) − ) , then ( x, t ) lies at the center of an evolving ε -neck. Rotational Symmetry of Compact Ancient κ -Solutions in HigherDimensions In this section, we give a proof of rotational symmetry, extending Section 3in [13] to higher dimensions. The arguments are essentially the same, except wewill use results from [15], which is the higher dimensional analogue of part twoof [9], where the first author first established rotational symmetry of noncompactancient κ -solutions in dimension three. Throughout this section, we assume n ≥ M, g ( t )) is an n -dimensional ancient κ -solution which is compact and simplyconnected. We also assume that ( M, g ( t )) is not a family of shrinking round spheres.The proof of rotational symmetry is by contradiction. Therefore: We will assumethroughout this section that ( M, g ( t )) is not rotationally symmetric. As in the previous section, let us fix a large number
L < ∞ and small number ε > θ > λ ( x, t ) ≤ θR ( x, t ), the spacetime point ( x, t ) lies atthe center of an evolving ε -neck.We begin with a definition of ε -symmetry of the caps based on the definitionused in the noncompact case in [15]. Definition 3.1 (Symmetry of Caps) . We will say the flow is ε -symmetric at time¯ t if there exists a compact domain D ⊂ M and a family of time-independent vectorfields U = { U ( a ) : 1 ≤ a ≤ (cid:0) n (cid:1) } which are defined on an open subset containing D such that the following statements hold: • The domain D is a disjoint union of two domains D and D , each of whichis diffeomorphic to B n . • λ ( x, ¯ t ) < θR (¯ x, ¯ t ) for all points x ∈ M \ D . • λ ( x, ¯ t ) > θR ( x, ¯ t ) for all points x ∈ D . • ∂D and ∂D are leaves of Hamilton’s CMC foliation of ( M, g (¯ t )). • For each x ∈ M \ D , the point ( x, ¯ t ) is ε -symmetric in the sense of Definition4.2 in [15]. • The Lie derivative L U ( a ) ( g ( t ))) satisfies the estimatesup D × [¯ t − ρ , ¯ t ] 2 X l =0 ( n ) X a =1 ρ l (cid:12)(cid:12) D l ( L U ( a ) ( g ( t ))) | ≤ ε , where ρ − := sup x ∈ D R ( x, ¯ t ). S.BRENDLE, P. DASKALOPOULOS, K. NAFF, N. SESUM • The Lie derivative L U ( a ) ( g ( t ))) satisfies the estimatesup D × [¯ t − ρ , ¯ t ] 2 X l =0 ( n ) X a =1 ρ l (cid:12)(cid:12) D l ( L U ( a ) ( g ( t ))) | ≤ ε , where ρ − := sup x ∈ D R ( x, ¯ t ). • If Σ ⊂ D is a leaf of Hamilton’s CMC foliation of ( M, g (¯ t )) that hasdistance at most 50 r neck ( ∂D ) from ∂D , thensup Σ ( n ) X a =1 ρ − |h U ( a ) , ν i| ≤ ε , where ν is the unit normal vector to Σ in ( M, g (¯ t )) and r neck ( ∂D ) is definedby the identity area g (¯ t ) ( ∂D ) = area g Sn − ( S n − ) r neck ( ∂D ) n − . • If Σ ⊂ D is a leaf of Hamilton’s CMC foliation of ( M, g (¯ t )) that hasdistance at most 50 r neck ( ∂D ) from ∂D , thensup Σ ( n ) X a =1 ρ − |h U ( a ) , ν i| ≤ ε , where ν is the unit normal vector to Σ in ( M, g (¯ t )) and r neck ( ∂D ) is definedby the identity area g (¯ t ) ( ∂D ) = area g Sn − ( S n − ) r neck ( ∂D ) n − . • If Σ ⊂ D is a leaf of Hamilton’s CMC foliation of ( M, g (¯ t )) that hasdistance at most 50 r neck ( ∂D ) from ∂D , then( n ) X a,b =1 (cid:12)(cid:12)(cid:12)(cid:12) δ ab − area g (¯ t ) (Σ) − n +1 n − Z Σ h U ( a ) , U ( b ) i g (¯ t ) dµ g (¯ t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε . • If Σ ⊂ D is a leaf of Hamilton’s CMC foliation of ( M, g (¯ t )) that hasdistance at most 50 r neck ( ∂D ) from ∂D , then( n ) X a,b =1 (cid:12)(cid:12)(cid:12)(cid:12) δ ab − area g (¯ t ) (Σ) − n +1 n − Z Σ h U ( a ) , U ( b ) i g (¯ t ) dµ g (¯ t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε . By Proposition 2.7 and Corollary 2.11, the solution is increasingly symmetricback in time.
Proposition 3.2.
Let ε > be given. If − t is sufficiently large (depending on ε ),then the flow is ε -symmetric at time t . Remark 3.3.
For − ¯ t sufficiently large, the flow is ε -symmetric and the tips of( M, g (¯ t )) are contained in different connected components of the domain D . Inparticular, after relabeling D and D if necessary, we may assume p , ¯ t ∈ D and p , ¯ t ∈ D . In view of Corollary 2.11 and Definition 3.1, we have diam g (¯ t ) ( D ) ≤ CR ( p , ¯ t , ¯ t ) − and diam g (¯ t ) ( D ) ≤ CR ( p , ¯ t , ¯ t ) − for some constant C , which de-pends only on our choice of θ . By the long-range curvature estimate, this implies C R ( p , ¯ t , ¯ t ) ≤ R ( x, ¯ t ) ≤ CR ( p , ¯ t , ¯ t ) for all x ∈ D and C R ( p , ¯ t , ¯ t ) ≤ R ( x, ¯ t ) ≤ CR ( p , ¯ t , ¯ t ) for all x ∈ D .Recall by Lemma 9.5 in [9] (cf. Lemma 5.5 in [15]): NIQUENESS OF COMPACT ANCIENT SOLUTIONS 9
Lemma 3.4.
Suppose that the flow is ε -symmetric at time ¯ t . If ˜ t is sufficientlyclose to ¯ t , then the flow is ε -symmetric at time ¯ t . Now to proceed with the proof by contradiction, consider an arbitrary sequenceof positive real numbers ε k →
0. For k large, define t k := inf { t ∈ ( −∞ ,
0] : The flow is not ε k -symmetric at time t } . We must have lim sup k →∞ t k = −∞ since otherwise the flow would be symmetricfor − t sufficiently large, in contradiction with our assumption.For sufficiently negative times, we denote by p ,t and p ,t the tips of ( M, g ( t )).Let and r − ,k := R ( p ,t k t k ). Since t k → −∞ , Proposition 2.7 implies that if werescale the flow about either p ,t k by the factor r − ,k := R ( p ,t k , t k ) or p ,t k bythe factor r − ,k := R ( p ,t k , t k ), then the sequence subsequentially converges to theBryant soliton in the pointed Cheeger-Gromov sense. This gives us the analogue ofLemma 3.5 in [13]. Lemma 3.5.
There exists a sequence of real numbers δ k → such that the followingstatements hold when k is sufficiently large: • For each t ∈ [ t k − δ − k r ,k , t k ] , we have d g ( t ) ( p ,t k , p t ) ≤ δ k r ,k and − δ k ≤ r ,k R ( p ,t , t ) ≤ δ k . • For each t ∈ [ t k − δ − k r ,k , t k ] , we have d g ( t ) ( p ,t k , p t ) ≤ δ k r ,k and − δ k ≤ r ,k R ( p ,t , t ) ≤ δ k . • The scalar curvature satisfies r ,k R ( x, t ) ≤ and K ( r − ,k d g ( t ) ( p ,t k , x ) + 1) − ≤ r ,k R ( x, t ) ≤ K ( r − ,k d g ( t ) ( p ,t k , x ) + 1) − for all points ( x, t ) ∈ B g ( t k ) ( p ,t k , δ − k r ,k ) × [ t k − δ − k r ,k , t k ] . • The scalar curvature satisfies r ,k R ( x, t ) ≤ and K ( r − ,k d g ( t ) ( p ,t k , x ) + 1) − ≤ r ,k R ( x, t ) ≤ K ( r − ,k d g ( t ) ( p ,t k , x ) + 1) − for all points ( x, t ) ∈ B g ( t k ) ( p ,t k , δ − k r ,k ) × [ t k − δ − k r ,k , t k ] . • There exists a nonnegative function f := f ,k : B g ( t k ) ( p ,t k , δ − k r ,k ) × [ t k − δ − k r ,k , t k ] → R such that | Ric − D f | ≤ δ k r − ,k and | ∆ f + |∇ f | − r − ,k | ≤ δ k r − ,k and | ∂∂t f + |∇ f | | ≤ δ k r − ,k . Moreover, function f satisfies K ( r − ,k d g ( t ) ( p ,t k , x ) + 1) ≤ f ( x, t ) + 1 ≤ K ( r − ,k d g ( t ) ( p ,t k , x ) + 1) for all points ( x, t ) ∈ B g ( t k ) ( p ,t k , δ − k r ,k ) × [ t k − δ − k r ,k , t k ] . • There exists a nonnegative function f := f ,k : B g ( t k ) ( p ,t k , δ − k r ,k ) × [ t k − δ − k r ,k , t k ] → R such that | Ric − D f | ≤ δ k r − ,k and | ∆ f + |∇ f | − r − ,k | ≤ δ k r − ,k and | ∂∂t f + |∇ f | | ≤ δ k r − ,k . Moreover, function f satisfies K ( r − ,k d g ( t ) ( p ,t k , x ) + 1) ≤ f ( x, t ) + 1 ≤ K ( r − ,k d g ( t ) ( p ,t k , x ) + 1) for all points ( x, t ) ∈ B g ( t k ) ( p ,t k , δ − k r ,k ) × [ t k − δ − k r ,k , t k ] .Here, K := K ( n ) ≥ is a universal constant. The following three results are restatements of Lemma 3.6, Lemma 3.7, andLemma 3.8 in [13]. See also Lemma 5.14 in [15].
Lemma 3.6.
By a suitable choice of δ k , we can arrange the following holds. If t ∈ [ t k − δ − k r ,k , t k ] and d g ( t ) ( p ,t k , x ) ≤ δ − k r ,k , then the time derivative of thedistance function satisfies the estimate ≤ − ddt d g ( t ) ( p ,t k , x ) ≤ n r − ,k . Similarly,if t ∈ [ t k − δ − k r ,k , t k ] and d g ( t ) ( p ,t k , x ) ≤ δ − k r ,k , then the time derivative of thedistance function satisfies the estimate ≤ − ddt d g ( t ) ( p ,t k , x ) ≤ n r − ,k . Lemma 3.7.
By a suitable choice of δ k , we can arrange so that the following holds:the two balls B g ( t ) ( p ,t , δ − k R ( p ,t , t ) − ) and B g ( t ) ( p ,t , δ − k R ( p ,t , t ) − ) are disjointfor t ∈ ( −∞ , t k ] . Lemma 3.8. If t ∈ ( −∞ , t k ) , then the flow is ε k -symmetric at time t . In particular,if ( x, t ) ∈ M × ( −∞ , t k ) is a spacetime point satisfying λ ( x, t ) < θR ( x, t ) , thenthe point ( x, t ) is ε k -symmetric in the sense of Definition 4.2 in [15] . By Corollary 2.11, we can find a time T ∈ ( −∞ ,
0] and a large constant Λ withthe following properties: • L q n K Λ ≤ − . • If (¯ x, ¯ t ) ∈ M × ( −∞ , T ] satisfies d g (¯ t ) ( p , ¯ t , ¯ x ) ≥ Λ2 R ( p , ¯ t , ¯ t ) − and d g (¯ t ) ( p , ¯ t , ¯ x ) ≥ Λ2 R ( p , ¯ t , ¯ t ) − , then λ ( x, t ) < θR ( x, t ) for all points ( x, t ) ∈ B g (¯ t ) (¯ x, Lr neck (¯ x, ¯ t )) × [¯ t − Lr neck (¯ x, ¯ t ) , ¯ t ], where r neck (¯ x, ¯ t ) − = n − n − R (¯ x, ¯ t ).The next two results are extensions of Lemma 3.9 and Lemma 3.10 in [13] tohigher dimensions. The proofs are exactly the same. Lemma 3.9. If (¯ x, ¯ t ) ∈ M × ( −∞ , t k ] satisfies d g (¯ t ) ( p , ¯ t , ¯ x ) ≥ Λ2 R ( p , ¯ t , ¯ t ) − and d g (¯ t ) ( p , ¯ t , ¯ x ) ≥ Λ2 R ( p , ¯ t , ¯ t ) − , then (¯ x, ¯ t ) is ε k -symmetric. Lemma 3.10. If (¯ x, ¯ t ) ∈ M × [ t k − δ − k r ,k , t k ] satisfies Λ r ,k ≤ d g (¯ t ) ( p ,t k , ¯ x ) ≤ δ − k r ,k , then d g (¯ t ) ( p , ¯ t , ¯ x ) ≥ Λ2 R ( p , ¯ t , ¯ t ) − and d g (¯ t ) ( p , ¯ t , ¯ x ) ≥ Λ2 R ( p , ¯ t , ¯ t ) − . Sim-ilarly, if (¯ x, ¯ t ) ∈ M × [ t k − δ − k r ,k , t k ] satisfies Λ r ,k ≤ d g (¯ t ) ( p ,t k , ¯ x ) ≤ δ − k r ,k ,then d g (¯ t ) ( p , ¯ t , ¯ x ) ≥ Λ2 R ( p , ¯ t , ¯ t ) − and d g (¯ t ) ( p , ¯ t , ¯ x ) ≥ Λ2 R ( p , ¯ t , ¯ t ) − . The proof of the following proposition is the same as the proof of Proposition3.11 in [13], except for minor difference with how we define the scale of a neck inhigher dimensions. Recall that if (¯ x, ¯ t ) lies at the center of an evolving neck, thenwe define r neck (¯ x, ¯ t ) − := n − n − R (¯ x, ¯ t ). For the convenience of the reader, weverify this minor difference here. Proposition 3.11. If (¯ x, ¯ t ) ∈ M × [ t k − − j δ − k r ,k , t k ] satisfies j Λ r ,k ≤ d g (¯ t ) ( p ,t k , ¯ x ) ≤ (400 n KL ) − j δ − k r ,k , then (¯ x, ¯ t ) is − j − ε k -symmetric. Simi-larly, if (¯ x, ¯ t ) ∈ M × [ t k − − j δ − k r ,k , t k ] satisfies j Λ r ,k ≤ d g (¯ t ) ( p ,t k , ¯ x ) ≤ (400 n KL ) − j δ − k r ,k , then (¯ x, ¯ t ) is − j − ε k -symmetric.Proof. The proof is by induction on j . The assertion for j = 0 follows from theprevious two lemmas. NIQUENESS OF COMPACT ANCIENT SOLUTIONS 11
Assume j ≥ j −
1. Suppose (¯ x, ¯ t ) ∈ M × [ t k − − j δ − k r ,k , t k ] such that 2 j Λ r ,k ≤ d g (¯ t ) ( p ,t k , ¯ x ) ≤ (400 n KL ) − j δ − k r ,k . Notethat since λ (¯ x, ¯ t ) < θR (¯ x, ¯ t ), (¯ x, ¯ t ) lies at the center of an ε -neck. By Lemma3.5, r neck (¯ x, ¯ t ) = ( n − n − R (¯ x, ¯ t ) − ≤ Kn r ,k d g (¯ t ) ( p ,t k , ¯ x ) . Therefore, ¯ t − L r neck (¯ x, ¯ t ) ≥ ¯ t − KLn r ,k d g (¯ t ) ( p ,t k , ¯ x ) ≥ ¯ t − KLn (400 n KL ) − j δ − k r ,k ≥ ¯ t − − j δ − k r ,k ≥ t k − − j +1 δ − k r ,k . On the other hand, r neck (¯ x, ¯ t ) ≤ Kn r ,k d g (¯ t ) ( p ,t k , ¯ x ) ≤ Kn Λ d g (¯ t ) ( p ,t k , ¯ x ) .Since L q n K Λ ≤ − , we obtain r neck (¯ x, ¯ t ) ≤ − L − d g (¯ t ) ( p ,t k , ¯ x ) . Consequently, if x ∈ B g (¯ t ) (¯ x, L r neck (¯ x, ¯ t )), then d g (¯ t ) ( p ,t k , x ) ≥ d g (¯ t ) ( p ,t k , ¯ x ) − Lr neck (¯ x, ¯ t ) ≥ (1 − − ) d g (¯ t ) ( p ,t k , ¯ x ) ≥ (1 − − )2 j Λ r ,k ≥ j − Λ r ,k . Now on the other hand, r ,k ≤ R (¯ x, ¯ t ) − ≤ r neck (¯ x, ¯ t ). Putting this together with r neck (¯ x, ¯ t ) ≤ Kn r ,k d g (¯ t ) ( p ,t k , ¯ x ), for all x ∈ B g (¯ t ) (¯ x, L r neck (¯ x, ¯ t )) we obtain d g (¯ t ) ( p ,t k , x ) + 2 nL r neck (¯ x, ¯ t ) r − ,k ≤ d g (¯ t ) ( p ,t k , ¯ x ) + L r neck (¯ x, ¯ t ) + 2 nL r neck (¯ x, ¯ t ) r − ,k ≤ d g (¯ t ) ( p ,t k , ¯ x ) + (2 n + 1) L r neck (¯ x, ¯ t ) r − ,k ≤ n KL d g (¯ t ) ( p ,t k , ¯ x ) ≤ (400 n KL ) − j +1 δ − k r ,k . Since by Lemma 3.6 d g (¯ t ) ( p ,t k , x ) ≤ d g ( t ) ( p ,t k , x ) ≤ d g (¯ t ) ( p ,t k , x ) + 2 nL r neck (¯ x, ¯ t ) r − ,k we conclude 2 j − Λ r ,k ≤ d g (¯ t ) ( p ,t k , x ) ≤ (400 n KL ) − j +1 δ − k r ,k for all ( x, t ) ∈ B g (¯ t ) (¯ x, L r neck (¯ x, ¯ t )) × [¯ t − L r neck (¯ x, ¯ t ) , ¯ t ]. It follows by the inductionhypothesis and the Neck Improvement Theorem that the point (¯ x, ¯ t ) is 2 − j − ε k -symmetric. (cid:3) The remaining arguments in the proof of rotational symmetry in Section 5 of[15], go through without change to give us the following final proposition.
Proposition 3.12. If k is sufficiently large, then the flow is ε k -symmetric at time t k . The proposition above contradicts the definition of t k in view of Lemma 3.4.This completes the proof of rotational symmetry. A priori estimates for compact ancient κ -solutions withrotational symmetry We begin by recalling some basic facts about the Bryant soliton in higher di-mensions. For the convenience of the reader we include some further discussion inAppendix A.
Proposition 4.1 (R. Bryant [16]) . Consider the n -dimensional Bryant soliton,normalized so that the scalar curvature at the tip is equal to . Then the metric canbe written in the form Φ( r ) − dr ⊗ dr + r g S n − , where Φ( r ) = 1 − r n ( n − + O ( r ) as r → and Φ( r ) = ( n − r − + (5 − n )( n − r − + O ( r − ) as r → ∞ . Proof.
See [16], Theorem 1 on p. 17.
Proposition 4.2.
Let η > be given. If s is sufficiently small (depending on η ),then (cid:12)(cid:12) Φ((1 + s ) r ) − − Φ( r ) − (cid:12)(cid:12) ≤ η (cid:0) Φ( r ) − − (cid:1) for all r ≥ . Proof.
We define χ ( r ) = r − (Φ( r ) − − χ ( r ) is a positivesmooth function which satisfies χ ( r ) = n ( n − + O ( r ) as r → χ ( r ) = n − + O ( r − ) as r → ∞ . Hence, if s is sufficiently small (depending on η ), then | χ ((1 + s ) r ) − χ ( r ) | ≤ η χ ( r )for all r >
0. Therefore, if s is sufficiently small (depending on η ), then | (1 + s ) χ ((1 + s ) r ) − χ ( r ) | ≤ (1 + s ) | χ ((1 + s ) r ) − χ ( r ) | + | (1 + s ) − | χ ( r ) ≤ η χ ( r )for all r >
0. This implies (cid:12)(cid:12)
Φ((1 + s ) r ) − − Φ( r ) − (cid:12)(cid:12) ≤ η (cid:0) Φ( r ) − − (cid:1) for all r > Lemma 4.3.
Consider the Bryant soliton, normalized so that the scalar curvatureat the tip is equal to . Then, r Φ r + 2Φ = 2( n − n − r + o ( r − ) , for r ≫ . As a result we have r Φ r + 2Φ − n − n −
2) Φ = o ( r − ) . Proof.
According to Proposition 4.1, we haveΦ( r ) = ( n − r − + (5 − n )( n − r − + o ( r − )implying that r Φ r ( r ) = − n − r − − − n )( n − r − + o ( r − )for r ≫
1. Hence r Φ r + 2Φ = 2( n − n − r + o ( r − ) . NIQUENESS OF COMPACT ANCIENT SOLUTIONS 13
This proves the first formula. The second one follows from the first and the factthat Φ( r ) = ( n − r − + o ( r − ) . Corollary 4.4.
Consider the n -dimensional Bryant soliton, normalized so thatthe scalar curvature at the tip is equal to . Let us write the metric in the form dz ⊗ dz + B ( z ) g S n − . Let α = 0 if n = 4 and α > ( n − / ( n − if n ≥ . Then,there exists a large constant L and such that d dz (cid:16) B ( z ) (cid:17) − α (cid:16) ddz B ( z ) (cid:17) < holdsif B ( z ) ≥ L . Proof.
Since r Φ ′ ( r ) + 2Φ( r ) = 2( n − n − r − + O ( r − ) as r → ∞ , we con-clude that r Φ ′ ( r ) + 2Φ( r ) < r sufficiently large only if n = 4. We next observethat (cid:0) ddz B ( z ) (cid:1) = Φ( B ( z )). Differentiating this identity with respect to z gives2 d dz B ( z ) = Φ ′ ( B ( z )). Thus, we conclude that ( B ( z ) ) zz = 2 B ( z ) B zz + 2 B z = B ( z ) Φ ′ ( B ( z )) + 2Φ( B ( z )) < B ( z ) is sufficiently large and n = 4. When n ≥ α Φ > n − n − r − . SinceΦ = ( n − r − + O ( r − ) we obtain a negative sign provided α > ( n − / ( n − Corollary 4.5.
Consider the Bryant soliton, normalized so that the scalar curva-ture at the tip is equal to . Let us write the metric in the form dz ⊗ dz + B ( z ) g S n − .Then B ( z ) ddz B ( z ) → n − as z → ∞ . Proof.
Note that r Φ( r ) → n − r → ∞ . Using the identity (cid:0) ddz B ( z ) (cid:1) =Φ( B ( z )), we obtain B ( z ) ddz B ( z ) = B ( z ) Φ( B ( z )) → n − z → ∞ .We now assume that ( S n , g ( t )) is an ancient κ -solution which is not a family ofshrinking round spheres. Let q ∈ S n be a reference point chosen as in [1]. The sameproof as the one in [1] implies that if t j → −∞ and if we dilate the flow aroundthe point ( q, t j ) by the factor ( − t j ) − , then the rescaled manifolds converge to acylinder of radius p n − F ( z, t ) denote the radius of a sphere of symmetryin ( S n , g ( t )) which has signed distance z from the point q . The function F ( z, t )satisfies the PDE F t ( z, t ) − F zz ( z, t ) = − n − F ( z, t ) (1 − F z ( z, t ) ) − ( n − F z ( z, t ) Z z F zz ( z ′ , t ) F ( z ′ , t ) dz ′ . Furthermore, if G ( ξ, τ ) = e τ F ( e − τ ξ, − e − τ ) − p n − G τ = G ξξ − ξ G ξ + G + ( n − G ξ p n −
2) + G − G p n −
2) + G ) − ( n − G ξ Z ξ G ξξ ( ξ ′ , τ ) p n −
2) + G ( ξ ′ , τ ) dξ ′ , or, equivalently, G τ = G ξξ − ξ G ξ + G − G ξ p n −
2) + G − G p n −
2) + G )+ ( n − G ξ G ξ (0 , τ ) p n −
2) + G (0 , τ ) − Z ξ G ξ ( ξ ′ , τ )( p n −
2) + G ( ξ ′ , τ )) dξ ′ ! . We can write this equation as G τ = L G − G ξ p n − − G p n −
2) + E ( ξ, τ ) , where E ( ξ, τ ) is the error term and L G = G ξξ − ξ G ξ + G . As in [1], let P + , P and P − be orthogonal projections associated with the direct sum H = H + ⊕ H ⊕ H − ,where H + , H and H − are the positive, zero and negative eigenspaces with respectto operator L , respectively. Exactly the same reasoning and arguments as in [1]yield that for τ sufficiently small the positive mode, i.e. the projection onto H dominates and that Z | ξ | <δ ( τ ) − e − ξ E ( ξ, τ ) ( ξ − dξ = O ( A ( τ ) ) , where A ( τ ) is the norm of the orthogonal projection of ˆ G ( ξ, τ ) = G ( ξ, τ ) χ ( δ ( τ ) ξ )onto H . Note that χ is a cut off function with the support in a parabolic regionand lim τ →−∞ δ ( τ ) = 0, where both, χ and δ ( τ ) are defined in the same way as in[1]. Having the equation for G and the integral estimate above, the same argumentsas in [1] imply the following asymptotics: Theorem 4.6.
Let ( S n , g ( t )) be a rotationally symmetric ancient κ -solution whichis not isometric to a family of shrinking spheres. Then we can find a reference point q ∈ S n such that the following holds. Let F ( z, t ) denote the radius of the sphereof symmetry in ( S n , g ( t )) which has signed distance z from the reference point q .Then the profile F ( z, t ) has the following asymptotic expansions: (i) Fix a large number L . Then, as t → −∞ , we have F ( z, t ) = ( n − h ( − t ) − z + 2 t − t ) i + o (cid:16) ( − t )log( − t ) (cid:17) for | z | ≤ L √− t (ii) Fix a small number θ > . Then as t → −∞ , we have F ( z, t ) = ( n − (cid:2) ( − t ) − z + 2 t − t ) (cid:3) + o ( − t ) for | z | ≤ p (1 − θ ) p ( − t ) log( − t ) . (iii) The reference point q has distance (2 + o (1)) p ( − t ) log( − t ) from each tip.The scalar curvature at each tip is given by (1 + o (1)) log( − t )( − t ) . Finally, ifwe rescale the solution around one of the tips, then the rescaled solutionsconverge to the Bryant soliton as t → −∞ . We next let H ( z, t ) := F ( z, t ) + ( n − t , K ( z, t ) := F z ( z, t ) and consider thequantity Q ( z, t ) := H zz − α ( n ) K. NIQUENESS OF COMPACT ANCIENT SOLUTIONS 15 where α ( n ) = 0 if n = 4 and α ( n ) = 1 > ( n − / ( n −
2) if n ≥
5. This definitionof α ( n ) ensures both Lemma 4.7 and Lemma 4.10 will hold. Lemma 4.7.
Let L be the constant in Corollary 4.4. There exists a time T < with the following property. If t ≤ T , F ( z, t ) = L
20 ( − t )log( − t ) , then Q ( z, t ) < . Proof.
By our assumptions on α ( n ), the proof of this is identical to the proofof Lemma 4.5 in [13] once we use Corollary 4.4. Lemma 4.8.
The function H ( z, t ) satisfies the equation H t ( z, t ) − H zz ( z, t ) = ( n − H z ( z, t ) F ( z, t ) − ( n − H z Z z F zz ( z ′ , t ) F ( z ′ , t ) dz ′ . Proof.
We have H t = F F t , H z = F F z , H zz = F F zz + F z . Hence, H t − H zz = − F z − ( n − − F z ) + ( n − − ( n − F F z Z z F zz ( z ′ , t ) F ( z ′ , t ) dz ′ which gives H t − H zz = ( n − H z F − ( n − H z Z z F zz ( z ′ , t ) F ( z ′ , t ) dz ′ . (cid:3) Lemma 4.9.
The function H zz ( z, t ) satisfies the evolution equation H zzt ( z, t ) − H zzzz ( z, t )= (cid:18) ( n − F z ( z, t ) F ( z, t ) − ( n − Z z F zz ( z ′ , t ) F ( z ′ , t ) dz ′ (cid:19) H zzz ( z, t ) − n − F z ( z, t ) F zz ( z, t ) F ( z, t ) − F zz ( z, t ) . The function K ( z, t ) satisfies the evolution equation K t ( z, t ) − K zz ( z, t )= (cid:18) ( n − F z ( z, t ) F ( z, t ) − ( n − Z z F zz ( z ′ , t ) F ( z ′ , t ) dz ′ (cid:19) K z ( z, t )+ 8 F z ( z, t ) F zz ( z, t ) F ( z, t ) − F z ( z, t ) F zz ( z, t ) + 4( n −
2) 1 − F z ( z, t ) F ( z, t ) F z ( z, t ) . Proof.
We differentiate twice the equation of H to find H zzt − H zzzz = ( n − (cid:0) H z F (cid:1) zz − ( n − H zzz Z z F zz ( z ′ , t ) F ( z ′ , t ) dz ′ − n − H zz F zz F − ( n − H z (cid:0) F zz F (cid:1) z . Next we use H z = F F z , H zz = F F zz + F z and H zzz = F F zzz + 3 F z F zz to compute H z (cid:0) F zz F (cid:1) z = F F z F F zzz − F zz F z F = H zzz F z F − F z F zz F and also use (cid:0) H z F (cid:1) zz = ( F z ) zz = 2( F z F zz ) z = 2 F z F zzz + 2 F zz = 2 F z F H zzz − F z F zz F + 2 F zz . Combining the above yields H zzt − H zzzz = H zzz (cid:16) − ( n − Z z F zz ( z ′ , t ) F ( z ′ , t ) dz ′ + ( n − F z F (cid:17) − n − F z F zz F + 2( n − F zz − n − F zz − n − F z F zz F + 4( n − F z F zz F = H zzz (cid:16) − ( n − Z z F zz ( z ′ , t ) F ( z ′ , t ) dz ′ + ( n − F z F (cid:17) − n − F z F zz F − F zz . Next, a simple computation shows that F z satisfies the equation F zt = F zzz + (cid:16) ( n − F z F − ( n − Z z F zz ( z ′ , t ) F ( z ′ , t ) dz ′ (cid:17) F zz + ( n − − F z ) F z F . Set K = F z . Using the previous equation and K t = 4 F z F zt , K z = 4 F z F zz , K zz = 4 F z F zzz + 12 F z F zz we find that K satisfies K t − K zz = − F z F zz + 4 F z (cid:16) ( n − F z F − ( n − Z z F zz ( z ′ , t ) F ( z ′ , t ) dz ′ (cid:17) F zz + 4( n − − F z ) F z F = (cid:16) ( n − F z F − ( n − Z z F zz ( z ′ , t ) F ( z ′ , t ) dz ′ (cid:17) K z + 8 F z F zz F − F z F zz + 4( n − − F z ) F z F This completes the proof of the lemma.Combining evolution equations for H zz and K we obtain that Q = H zz − αK satisfies Q t − Q zz = (cid:16) ( n − F z F − ( n − Z z F zz ( z ′ , t ) F ( z ′ , t ) dz ′ (cid:17) Q z − n − F z F zz F − F zz − α F z F zz F + 12 α F z F zz − α ( n − − F z ) F z F = (cid:16) ( n − F z F − ( n − Z z F zz ( z ′ , t ) F ( z ′ , t ) dz ′ (cid:17) Q z − (4( n −
4) + 8 αF z ) F z F zz F − (4 − αF z ) F zz − α ( n −
2) 1 − F z F F z . We will use the last equation, Lemma 4.7 and the maximum principle to provethe following auxiliary lemma, which will be used to establish Proposition 4.11below.
Lemma 4.10.
Let L be chosen as in Corollary 4.4. Then, there exists T ≪− (depending on L and dimension n ) such that if for some t ≤ T , we have NIQUENESS OF COMPACT ANCIENT SOLUTIONS 17 Q ( z , t ) := max F ( z ,t ) ≥ L − t − t Q ( z, t ) > , then sup F ( z,t ) ≥ L − t )log( − t ) Q ( z, t ) ≥ Q ( z , t ) for all t ≤ t . Proof.
When n = 4, α ( n ) = 0, and Q ( z, t ) = H zz ( z, t ). The maximum principleapplied to the evolution equation of H yields the claimed inequality, as in the proofof Proposition 4.7 in [13]. Hence, we will assume that n ≥
5. Let q ( t ) = sup F ( z,t ) ≥ L − t )log( − t ) Q ( z, t ) . Let T be determined by Lemma 4.7 and assume t ≤ T . Assume Q ( z , t ) := q ( t ) >
0. We must show q ( t ) ≥ q ( t ) for all t ≤ t .Suppose not. Let t = sup { t ≤ t : q ( t ) < q ( t ) } . Then q ( t ) = q ( t ) and thereexists a sequence of times t ,j ր t such that q ( t ,j ) < q ( t ). Choose z such that F ( z , t ) ≥ L ( − t )log( − t ) and q ( t ) = Q ( z , t ) = Q ( z , t ) >
0. Since t ≤ T , Lemma4.7 implies Q ≤ F ( z, t ) = L
20 ( − t )log( − t ) . Hence z is an interior spacialmaximum. Applying the maximum principle to the evolution equation for Q showsthat at such interior maximum we have Q t ( z , t ) ≤ − (cid:0) n −
4) + 8 αF z (cid:1) F z F zz F − (4 − αF z ) F zz − α ( n − − F z ) F z F , where the right hand side is evaluated at ( z , t ). Using that | F z | ( z , t ) ≪ − T is sufficientlylarge), we have 4 − αF z >
0, and hence we conclude the inequality Q t ( z , t ) ≤ − (cid:0) n −
4) + 8 αF z (cid:1) F z F zz F − α ( n − − F z ) F z F . Next express F zz = Q − F z + αF z F which gives F z F zz F = F z Q − F z + αF z F so that Q t ( z , t ) ≤ − (cid:0) n −
4) + 8 αF z (cid:1) F z F Q + 4 (cid:0) ( n − − α ( n − (cid:1) F z F + O ( F z F ) . Using again that | F z | ( z , t ) ≪
1, we conclude that if Q ( z , t ) >
0, then Q t ( z , t ) <
0. But this implies Q ( t ,j , z ) > Q ( z , t ) for t ,j sufficiently close to t and hence q ( t ,j ) > q ( t ) = q ( t ), a contradiction. Therefore, we must have q ( t ) ≥ q ( t ) forall t ≤ t , completing the proof.As consequence we can show the following analogue of Proposition 4.7 in [13]. Proposition 4.11.
Let L be chosen as in Corollary 4.4, T be chosen as in Lemma4.7. If t ≤ T and F ( z, t ) ≥ L
20 ( − t )log( − t ) , then Q ( z, t ) ≤ . Proof.
Suppose this is false. Then we can find a point ( z , t ) such that t ≤ T , F ( z , t ) ≥ L
20 ( − t )log( − t ) , and Q ( z , t ) >
0. In view of Lemma 4.7 and Lemma 4.10,we have sup F ( z,t ) ≥ L
20 ( − t )log( − t ) Q ( z, t ) ≥ Q ( z , t ) > t ≤ t . Let us consider a sequence t j → −∞ . For j large, we can finda point z j such that F ( z j , t j ) ≥ L
20 ( − t j )log( − t j ) and Q ( z j , t j ) ≥ Q ( z , t ) >
0. Usingthe inequality F zz ≤
0, we obtain F z ( z j , t j ) ≥ Q ( z j , t j ) ≥ Q ( z , t ) > j large. Hence, if we rescale around the points ( z j , t j ) and pass to the limit, thenthe limit cannot be a cylinder. Consequently, the limit of these rescalings must bethe Bryant soliton. Hence, after passing to the limit, we obtain a point z ∞ on theBryant soliton such that B ( z ∞ ) ≥ L and d dz (cid:16) B ( z ) (cid:17) − α ( n ) (cid:16) ddz B ( z ) (cid:17) ≥ , at z ∞ . This contradicts Corollary 4.4.We next recall a crucial estimate from [1].
Proposition 4.12 (cf. [1]) . Fix a small number θ > and a small number η > .Then (cid:12)(cid:12)(cid:12) F ( z, t ) + ( n − t + ( n − z + 2 t − t ) (cid:12)(cid:12)(cid:12) ≤ η z − t log( − t ) if F ( z, t ) ≥ θ √− t and − t is sufficiently large (depending on η and θ ). Proof.
The proof is analogous to the proof of Proposition 4.8 in [13] and isbased on the asymptotics of the ancient solution in radially symmetric setting. In[1] we showed precise asymptotics in the case n = 3, but the proof carries overwithout any changes to higher dimensions due to new results in [15] and [28]. Proposition 4.13.
Let us fix a small number θ > and a small number η > .Then (cid:12)(cid:12)(cid:12) F ( z, t ) F z ( z, t ) + ( n − z − t ) (cid:12)(cid:12)(cid:12) ≤ η | z | + √− t log( − t ) if F ( z, t ) ≥ θ √− t and − t is sufficiently large (depending on η and θ ). Proof. If n = 4, by Proposition 4.11 we have that F is still concave on theset F ( z, t ) ≥ L
20 ( − t )log( − t ) for appropriately chosen L ≫ − t sufficientlylarge, and the proof is the same to the proof of Proposition 4.9 in [13].Now assume n ≥
5. The beginning of the proof is similar to the proof of Propo-sition 4.9 in [13]. The difference comes from the fact that
F F z is not monotonedecreasing, but instead one needs to use Proposition 4.11.Let θ ∈ (0 , ) and η ∈ (0 , ) be given. We can find a small positive number µ ∈ (0 , ηn − ) and time T with the property that F ((1 + µ ) z, t ) ≥ θ √− t whenever F ( z, t ) ≥ θ √− t and t ≤ T . Moreover, by Proposition 4.12, we can find a time NIQUENESS OF COMPACT ANCIENT SOLUTIONS 19 T ≤ T such that (cid:12)(cid:12)(cid:12) F ( z, t ) + ( n − t + ( n − z + 2 t − t ) (cid:12)(cid:12)(cid:12) ≤ η µ z
16 log( − t )whenever z ≥ √− t , F ( z, t ) ≥ θ √− t , and t ≤ T .Suppose now that ( z , t ) is a point in spacetime satisfying z ≥ √− t , F ( z , t ) ≥ θ √− t , and t ≤ T . We assume − T is sufficiently large so that F z ( z, t ) < z ≥ √− t . By the above, we have F ( z, t ) ≥ θ √− t for all z ∈ [(1 − µ ) z , (1 + µ ) z ]. Consequently, (cid:12)(cid:12)(cid:12) F ( z, t ) + ( n − t + ( n − z + 2 t − t ) (cid:12)(cid:12)(cid:12) ≤ η µ z − t )for all z ∈ [(1 − µ ) z , (1 + µ ) z ]. This impliesinf z ∈ [(1 − µ ) z ,z ] (cid:16) F ( z, t ) F z ( z, t ) + ( n − z − t ) (cid:17) ≤ η z − t )and sup z ∈ [ z , (1+ µ ) z ] (cid:16) F ( z, t ) F z ( z, t ) + ( n − z − t ) (cid:17) ≥ − η z − t ) . Define a function S ( z, t ) := 1 F ( z, t ) F z ( z, t ) − αF ( z, t ) = 1 − αF z ( z, t ) F ( z, t ) F z ( z, t ) . Since F ( z, t ) ≥ θ √− t for all z ∈ [(1 − µ ) z , (1 + µ ) z ], we have F z ( z, t ) ≪ S ( z, t ) > z ∈ [(1 − µ ) z , (1 + µ ) z ]. Moreover, we may assume − t is sufficiently large, depending on θ and η , so that F ( z, t ) | F z ( z, t ) | ≤ S ( z, t ) − ≤ (1 + η ) F ( z, t ) | F z ( z, t ) | . Since F z <
0, this means(1 + η ) F ( z, t ) F z ( z, t ) ≤ − S ( z, t ) − ≤ F ( z, t ) F z ( z, t ) . We compute S z = − F F z − F zz F F z + 2 αF z F = − F F z (cid:0) F z + F F zz − αF z (cid:1) = 2 QF ( − F z ) . Since F z < Q ≤
0, the function S is monotone decreasingin the variable z for z ∈ [(1 − µ ) z , (1 + µ ) z ]. This means − S − is monotonedecreasing as well. So although the function F F z is not monotone decreasing inhigher dimensions, it is very close to the monotone function − S − .Now it follows from our estimates above thatinf z ∈ [(1 − µ ) z ,z ] (cid:16) − S ( z, t ) − + ( n − z − t ) (cid:17) ≤ η z − t )and sup z ∈ [ z , (1+ µ ) z ] (cid:16) −
11 + η S ( z, t ) − + ( n − z − t ) (cid:17) ≥ − η z − t ) . Since − S − is monotone decreasing in the relevant region, this implies − S ( z , t ) − + ( n − − µ ) z − t ) ≤ η z − t )and −
11 + η S ( z , t ) − + ( n − − µ ) z − t ) ≥ − η z − t ) . Hence (1 + η ) F ( z , t ) F z ( z , t ) + ( n − z − t ) ≤ (( n − µ + η ) z − t )and 11 + η F ( z , t ) F z ( z , t ) + ( n − z − t ) ≥ − (( n − µ + η ) z − t ) . Since µ ∈ (0 , ηn − ) and evidently η < < η , it follows that (cid:12)(cid:12)(cid:12) F ( z , t ) F z ( z , t ) + ( n − z − t ) (cid:12)(cid:12)(cid:12) ≤ η z log( − t ) . To summarize, we have verified the assertion for z ≥ √− t . An analogous argumentshow the assertion holds for z ≤ − √− t . Finally, suppose | z | ≤ √− t . In dimensionthree, this case follows from Proposition 5.10 in [1]. An analogous result holds inhigher dimensions; namely ( − τ ) G ( ξ, τ ) → − √ n − ( ξ −
2) in C ∞ loc . The assertionin the region | z | ≤ √− t (which is equivalent to | ξ | ≤
4) follows directly from thisresult. This completes the proof of Proposition 4.13.
Corollary 4.14.
Let us fix a small number θ > . Then | F z ( z, t ) | ≤ C ( θ ) p log( − t ) if F ( z, t ) ≥ θ √− t and − t is sufficiently large (depending on θ ). Proof.
Given Theorem 4.6, the same reasoning as in [1] imply that | z | ≤ (2 + o (1)) p ( − t ) log( − t ). Hence, the assertion follows from Proposition 4.13. Proposition 4.15.
Let us fix a small number θ > . Then F ( z, t ) | F zz ( z, t ) | + F ( z, t ) | F zzz ( z, t ) | ≤ C ( θ ) p log( − t ) if F ( z, t ) ≥ θ √− t and − t is sufficiently large (depending on θ ). Proof.
The proof is analogous to the proof of Proposition 4.11 in [13].
Proposition 4.16.
Let us fix a small number θ > . Then | n − F F t | ≤ C ( θ ) p log( − t ) whenever F ≥ θ √− t , and − t is sufficiently large (depending on θ ). NIQUENESS OF COMPACT ANCIENT SOLUTIONS 21
Proof.
Using the evolution equation for F , we obtain( n −
2) + F ( z, t ) F t ( z, t )= F ( z, t ) F zz ( z, t ) − F z ( z, t ) + ( n − F ( z, t ) F z ( z, t ) (cid:20) F (0 , t ) − F z (0 , t ) − Z z F z ( z ′ , t ) F ( z ′ , t ) dz ′ (cid:21) . The same arguments in the proof of Proposition 4.12 in [13] yield the proof in higherdimensional case as well.
Proposition 4.17.
Let ε > be given. Then there exists a large number L (de-pending on ε ) and a time T such that the following holds. If F ≥ L q ( − t )log( − t ) and t ≤ T at some point in space-time, then that point lies at the center of an evolving ε -neck. Proof.
This follows from the fact, that follows in an analogous way as in [1],that the scalar curvature at each tip is comparable log( − t )( − t ) . Corollary 4.18.
Let η > be given. Then there exists a large number L (dependingon η ) and a time T such that | F z | + F | F zz | + F | F zzz | ≤ η whenever F ≥ L q ( − t )log( − t ) and t ≤ T . Proof.
This follows directly from Proposition 4.17.
Proposition 4.19.
Let η > be given. Then there exist a large number L ∈ ( η − , ∞ ) and a small number θ ∈ (0 , η ) (depending on η ), and a time T with theproperty that (cid:12)(cid:12)(cid:12)(cid:12) ( n − − s log( − t )( − t ) F | F z | (cid:12)(cid:12)(cid:12)(cid:12) ≤ η whenever L q ( − t )log( − t ) ≤ F ≤ θ √− t and t ≤ T . Proof.
By Corollary 4.5, we can find a large number L ∈ ( η − , ∞ ) such that (cid:12)(cid:12) ( n − − B ( z ) ddz B ( z ) (cid:12)(cid:12) ≤ η for z ≥ L . Recall that the solution looks like the Bryantsoliton near each tip, and the scalar curvature at each tip equals (1 + o (1)) log( − t )( − t ) .Consequently, (cid:12)(cid:12)(cid:12)(cid:12) ( n − − s log( − t )( − t ) F | F z | (cid:12)(cid:12)(cid:12)(cid:12) ≤ η , if F = L q ( − t )log( − t ) and − t is sufficiently large. Now recall the function S ( z, t )defined in Proposition 4.13: S ( z, t ) := 1 F ( z, t ) F z ( z, t ) − αF ( z, t ) . Let µ = min { η n − , } . The function z S ( z, t ) − is monotone decreasing.Moreover, whenever F ≥ L q ( − t )log( − t ) , L is large, and − t is sufficiently large, | F z | is small and hence F | F z | ≤ S − ≤ (1 + µ ) F | F z | . Consequently,( n − − s log( − t )( − t ) S − ≥ ( n − − (1 + µ ) s log( − t )( − t ) F | F z |≥ − (1 + µ ) (cid:12)(cid:12)(cid:12)(cid:12) ( n − − s log( − t )( − t ) F | F z | (cid:12)(cid:12)(cid:12)(cid:12) − η n − − s log( − t )( − t ) S − ≤ ( n − − s log( − t )( − t ) F | F z |≤ (1 + µ ) (cid:12)(cid:12)(cid:12)(cid:12) ( n − − s log( − t )( − t ) F | F z | (cid:12)(cid:12)(cid:12)(cid:12) + η F ≥ L q ( − t )log( − t ) and − t is sufficiently large. In other words, (cid:12)(cid:12)(cid:12)(cid:12) ( n − − s log( − t )( − t ) S − (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) ( n − − s log( − t )( − t ) F | F z | (cid:12)(cid:12)(cid:12)(cid:12) + η F ≥ L q ( − t )log( − t ) and − t is sufficiently large. Similarly, we can show (cid:12)(cid:12)(cid:12)(cid:12) ( n − − s log( − t )( − t ) F | F z | (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) ( n − − s log( − t )( − t ) S − (cid:12)(cid:12)(cid:12)(cid:12) + η . In particular, from the first estimate in the proof, we obtain (cid:12)(cid:12)(cid:12)(cid:12) ( n − − s log( − t )( − t ) S − (cid:12)(cid:12)(cid:12)(cid:12) ≤ η F = L q ( − t )log( − t ) and − t is sufficiently large. On the other hand, for each θ ∈ (0 , ), Proposition 4.12 implies z = (4 + o (1)) (cid:16) − (100 θ ) n − (cid:17) ( − t ) log( − t )if F = 100 θ √− t . Using Proposition 4.13, we obtain F | F z | = (( n −
2) + o (1)) | z | − t )if F = 100 θ √− t . Consequently,( n − − s log( − t )( − t ) F | F z | = ( n − (cid:16) − r − (100 θ ) n − o (1) (cid:17) if F = 100 θ √− t . Therefore, if we choose θ sufficiently small (depending on η ),then we obtain (cid:12)(cid:12)(cid:12)(cid:12) ( n − − s log( − t )( − t ) F | F z | (cid:12)(cid:12)(cid:12)(cid:12) ≤ η , NIQUENESS OF COMPACT ANCIENT SOLUTIONS 23 if F = 100 θ √− t and − t is sufficiently large. This implies (cid:12)(cid:12)(cid:12)(cid:12) ( n − − s log( − t )( − t ) S − (cid:12)(cid:12)(cid:12)(cid:12) ≤ η , if F = 100 θ √− t and − t is sufficiently large. Because S − is monotone, we con-clude that (cid:12)(cid:12)(cid:12)(cid:12) ( n − − s log( − t )( − t ) S − (cid:12)(cid:12)(cid:12)(cid:12) ≤ η , whenever L q ( − t )log( − t ) ≤ F ≤ θ √− t and − t is sufficiently large. Finally, thisimplies (cid:12)(cid:12)(cid:12)(cid:12) ( n − − s log( − t )( − t ) F | F z | (cid:12)(cid:12)(cid:12)(cid:12) ≤ η whenever L q ( − t )log( − t ) ≤ F ≤ θ √− t and − t is sufficiently large. This completesthe proof of Proposition 4.19. Proposition 4.20.
Let us fix a small number θ > . If − τ is sufficiently large(depending on θ ), then C ( θ ) ( − τ ) − ≤ V + ( ρ, τ ) ≤ C ( θ ) ( − τ ) − and (cid:12)(cid:12) ∂∂ρ V + ( ρ, τ ) (cid:12)(cid:12) ≤ C ( θ ) for every ρ ∈ [ θ , θ ] . Proof.
Similarly to Proposition 4.16 in [13].In the remainder of this section, we define functions U + ( r, t ) and U − ( r, t ) so that U + ( r, t ) = (cid:16) ∂∂z F ( z, t ) (cid:17) for r = F ( z, t ) and z ≥ √− t and U − ( r, t ) = (cid:16) ∂∂z F ( z, t ) (cid:17) for r = F ( z, t ) and z ≤ − √− t . Let us consider the rescaled functions V + ( ρ, τ ) := q U + ( e − τ ρ, − e − τ ) ,V − ( ρ, τ ) := q U − ( e − τ ρ, − e − τ ) . For each ρ ∈ (0 , ξ + ( ρ, τ ) the unique positive solution of the equa-tion F ( e − τ ξ, − e − τ ) = e − τ ρ ; moreover, we denote by ξ − ( ρ, τ ) the unique negativesolution of the equation F ( e − τ ξ, − e − τ ) = e − τ ρ . Proposition 4.21.
Fix a small number η > . Then we can find a small number θ ∈ (0 , η ) (depending on η ) such that, for − τ sufficiently large, we have | V + ( ρ, τ ) − − Φ(( − τ ) ρ ) − | ≤ η ( V + ( ρ, τ ) − − in the region { ρ ≤ θ } . Here, Φ denotes the profile of the Bryant soliton. Proof.
Similarly to Proposition 4.17 in [13].
Proposition 4.22.
Fix a small number η > . Then we can find a large number L (depending on η ) such that, for − τ sufficiently large, we have V + ( ρ, τ ) ≤ η, (cid:12)(cid:12)(cid:12) ∂∂ρ V + ( ρ, τ ) (cid:12)(cid:12)(cid:12) ≤ η ρ − V + ( ρ, τ ) − and (cid:12)(cid:12)(cid:12) ∂ ∂ρ V + ( ρ, τ ) (cid:12)(cid:12)(cid:12) ≤ η ρ − V + ( ρ, τ ) − in the region { L ( − τ ) − ≤ ρ ≤ } . Proof.
Similarly to Proposition 4.18 in [13].
Corollary 4.23.
Fix a small number η > . Then, for − τ sufficiently large, wehave (cid:12)(cid:12)(cid:12) ∂∂τ V + ( ρ, τ ) (cid:12)(cid:12)(cid:12) ≤ η ρ − ( V + ( ρ, τ ) − − in the region { ρ ≤ } . Proof.
Similarly to Proposition 4.19 in [13].
Proposition 4.24.
Fix a small number η > . Then we can find a small number θ ∈ (0 , η ) (depending on η ) such that, for − τ sufficiently large, we have (cid:12)(cid:12)(cid:12) ∂∂ρ (cid:16) ξ + ( ρ, τ ) (cid:17) + ( n − ρ − ( V + ( ρ, τ ) − − (cid:12)(cid:12)(cid:12) ≤ η ρ − ( V + ( ρ, τ ) − − in the region { θ ≤ ρ ≤ θ } . Proof.
Similarly to Proposition 4.20 in [13].
Proposition 4.25.
Fix a small number θ > . Then, for − τ large, we have (cid:12)(cid:12)(cid:12) ∂∂ρ (cid:16) ξ + ( ρ, τ ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ C ( θ ) ( − τ ) and (cid:12)(cid:12)(cid:12) ∂ ∂ρ (cid:16) ξ + ( ρ, τ ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ C ( θ ) ( − τ ) in the region { θ ≤ ρ ≤ θ } . Proof.
Similarly to Proposition 4.21 in [13].
Proposition 4.26.
Fix a small number θ > . Then, for − τ large, we have (cid:12)(cid:12)(cid:12) ∂∂τ (cid:16) ξ + ( ρ, τ ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ o (1) ( − τ ) in the region { θ ≤ ρ ≤ θ } . Proof.
Similarly to Proposition 4.22 in [13].
NIQUENESS OF COMPACT ANCIENT SOLUTIONS 25 The tip region weights µ + ( ρ, τ ) and µ − ( ρ, τ )In this section, we define weights µ + ( ρ, τ ) and µ − ( ρ, τ ) which will be neededin the analysis of the linearized equation in the tip region. Let θ > ζ : R → [0 ,
1] be a smooth, monotone increasing cutofffunction satisfying ζ ( ρ ) = 0 for ρ ≤ θ and ζ ( ρ ) = 1 for ρ ≥ θ . We define the weight µ + ( ρ, τ ) by µ + ( ρ, τ ) = − ζ ( ρ ) ξ + ( ρ, τ ) − Z θρ ζ ′ (˜ ρ ) ξ + (˜ ρ, τ ) d ˜ ρ − ( n − Z θρ (1 − ζ (˜ ρ )) ˜ ρ − (cid:0) Φ(( − τ ) ˜ ρ ) − − (cid:1) d ˜ ρ, where Φ denotes the profile of the Bryant soliton. We can define a weight µ − ( ρ, τ )in analogous fashion. Of course, the cutoff function ζ and the weights µ + ( ρ, τ ) and µ − ( ρ, τ ) depend on the choice of the parameter θ , but we suppress that dependencein our notation. Lemma 5.1.
The weight µ + ( ρ, τ ) satisfies µ + ( ρ, τ ) = − ξ + ( ρ,τ ) for ρ ≥ θ . More-over, µ + ( ρ, τ ) ≤ for all ρ ≤ θ . Proof.
This follows immediately from the definition of µ + ( ρ, τ ). Lemma 5.2.
Fix a small number η > . Then we can find a small number θ ∈ (0 , η ) (depending on η ) such that, for − τ sufficiently large, we have (cid:12)(cid:12)(cid:12) ∂µ + ∂ρ ( ρ, τ ) − ( n − ρ − ( V + ( ρ, τ ) − − (cid:12)(cid:12)(cid:12) ≤ η ρ − ( V + ( ρ, τ ) − − in the tip region { ρ ≤ θ } . Proof.
The assertion follows from Proposition 4.21 and Proposition 4.24 (seeLemma 5.2 in [13] for details).
Lemma 5.3.
If we choose θ > sufficiently small, then the following holds. If − τ is sufficiently large (depending on θ ), then ∂ µ + ∂ρ ( ρ, τ ) ≤ (cid:16) ∂µ + ∂ρ ( ρ, τ ) (cid:17) + K ∗ ρ − in the tip region { ρ ≤ θ } . Here, K ∗ is a constant which depends on the dimension b but is independent of θ . Proof.
We compute ∂ µ + ∂ρ ( ρ, τ ) = − ζ ( ρ ) ∂ ∂ρ (cid:16) ξ + ( ρ, τ ) (cid:17) − ζ ′ ( ρ ) ∂∂ρ (cid:16) ξ + ( ρ, τ ) (cid:17) − ( n − − ζ ( ρ ) + ρ ζ ′ ( ρ )] ρ − (cid:0) Φ(( − τ ) ρ ) − − (cid:1) − ( n − − ζ ( ρ )) ( − τ ) ρ − Φ(( − τ ) ρ ) − Φ ′ (( − τ ) ρ ) . Recall that 0 ≤ ζ ≤ ζ ′ ≥
0. Moreover, we have Φ( r ) − − ≥ K r and | Φ( r ) − Φ ′ ( r ) | ≤ Kr for all r ∈ [0 , ∞ ), where K is a universal constant dependingonly on dimension n . This implies ∂ µ + ∂ρ ( ρ, τ ) ≤ − ζ ( ρ ) ∂ ∂ρ (cid:16) ξ + ( ρ, τ ) (cid:17) − ζ ′ ( ρ ) ∂∂ρ (cid:16) ξ + ( ρ, τ ) (cid:17) + K (1 − ζ ( ρ )) ( − τ ) , where K is a constant depending on the dimension n but is independent of θ . UsingProposition 4.25, we obtain ∂ µ + ∂ρ ( ρ, τ ) ≤ o (1) ( − τ ) in the region { θ ≤ ρ ≤ θ } , and ∂ µ + ∂ρ ( ρ, τ ) ≤ K ( − τ )in the region { ρ ≤ θ } . On the other hand, applying Lemma 5.2 with η = andusing that n − ≥ n ≥
4, we obtain ∂µ + ∂ρ ( ρ, τ ) ≥ ρ − ( V + ( ρ, τ ) − − ≥ ρ − (Φ(( − τ ) ρ ) − − ≥ K ( − τ ) ρ in the region { ρ ≤ θ } , where again K is a constant depending on dimension but isindependent of θ . Hence, if − τ is sufficiently large (depending on θ ), then we have ∂ µ + ∂ρ ( ρ, τ ) ≤ (cid:16) ∂µ + ∂ρ ( ρ, τ ) (cid:17) + 16 K ρ − in the region { ρ ≤ θ } . This completes the proof of Lemma 5.3. Lemma 5.4.
Let us fix a small number θ > . Then, for − τ large, we have (cid:12)(cid:12)(cid:12) ∂µ + ∂τ ( ρ, τ ) (cid:12)(cid:12)(cid:12) ≤ o (1) ( − τ ) in the tip region { ρ ≤ θ } . Proof.
It follows from Proposition 4.26, similarly to Proposition 5.3 in [13].We finish this section with the following weighted Poincar´e inequality.
Proposition 5.5.
If we choose θ > sufficiently small, then the following holds.If − τ is sufficiently large (depending on θ ), then Z θ (cid:16) ∂µ + ∂ρ (cid:17) f e − µ + dρ ≤ Z θ (cid:16) ∂f∂ρ (cid:17) e − µ + dρ + K ∗ Z θ ρ − f e − µ + dρ for every smooth function f which is supported in the region { ρ ≤ θ } . Here, K ∗ isthe constant in Lemma 5.3; in particular, K ∗ depends only on the dimension andis independent of θ . Note that the right hand side is infinite unless f (0) = 0 . NIQUENESS OF COMPACT ANCIENT SOLUTIONS 27
Proof.
We compute ∂∂ρ (cid:16) ∂µ + ∂ρ f e − µ + (cid:17) = ∂ µ + ∂ρ f e − µ + + 2 ∂µ + ∂ρ f ∂f∂ρ e − µ + − (cid:16) ∂µ + ∂ρ (cid:17) f e − µ + . Using Young’s inequality, we obtain ∂∂ρ (cid:16) ∂µ + ∂ρ f e − µ + (cid:17) ≤ ∂ µ + ∂ρ f e − µ + + 2 (cid:16) ∂f∂ρ (cid:17) e − µ + − (cid:16) ∂µ + ∂ρ (cid:17) f e − µ + . Hence, Lemma 5.3 gives ∂∂ρ (cid:16) ∂µ + ∂ρ f e − µ + (cid:17) ≤ (cid:16) ∂f∂ρ (cid:17) e − µ + − (cid:16) ∂µ + ∂ρ (cid:17) f e − µ + + K ∗ ρ − f e − µ + . From this, the assertion follows.6.
Overview of the proof of Theorem 1.3
In this section, we state the four main estimates needed for the proof of Theorem1.3, generalizing Section 6 in [13]. At the end of this section, we give the proof ofTheorem 1.3 assuming these key results. To that end, we consider two ancient κ -solutions, ( S n , g ( t )) and ( S n , g ( t )) such that neither solution is a family ofshrinking round spheres. By the main result of the previous section, we know bothsolutions are rotationally symmetric. We first choose reference points q , q ∈ S n such thatlim sup t →−∞ ( − t ) R g ( t ) ( q ) ≤ n −
1) and lim sup t →−∞ ( − t ) R g ( t ) ( q ) ≤ n − . The existence of these points is ensured by the Neck Stability Theorem of Kleinerand Lott. See Proposition 3.1 in [1] for a proof in dimension three, which also worksin higher dimensions.Since ( S n , g ( t )) is rotationally symmetric, we can define a profile function F ( z, t ) to be the radius of the sphere of symmetry that has signed distance z from the reference point q . Similarly we can define F ( z, t ) on ( S n , g ( t )) withrespect to q . These functions, F ( z, t ) and F ( z, t ), satisfy the PDE F t ( z, t ) = F zz ( z, t ) − ( n − F ( z, t ) − (1 − F z ( z, t ) ) − ( n − F z ( z, t ) Z z F zz ( z ′ , t ) F ( z ′ , t ) dz ′ . Our goal is to show that the profile functions F and F will agree after areparametrization in space, a translation in time, and a parabolic rescaling. We thuswill now define a new function F αβγ ( z, t ) obtained from F ( z, t ) through a spacialreparametrization, a time translation, and a parabolic rescaling. Here, ( α, β, γ ) isa triplet of real numbers satisfying the following admissibility condition previouslydefined in [13]: Definition 6.1.
Given a real number ε ∈ (0 , α, β, γ ) is ε -admissible with respect to time t ∗ if | α | ≤ ε √− t ∗ , | β | ≤ ε ( − t ∗ )log( − t ∗ ) , | γ | ≤ ε log( − t ∗ ) . Consider a time t ∗ < − t ∗ is very large. Suppose ( α, β, γ ) is a triplet ofreal numbers satisfying the criteria of ε -admissibility with respect to the time t ∗ ,for some ε ∈ (0 , t ≤ t ∗ , we define a time-translated and parabolically-rescaled metric by g βγ ( t ) := e γ g ( e − γ ( t − β )) . Of course, ( S n , g βγ ( t )) is again a rotationally symmetric ancient κ -solution. Wedefine the time-translated and parabolically-rescaled profile function F βγ ( z, t ) onthe ancient κ -solution ( S n , g βγ ( t )) to be the radius of the sphere of symmetry withsigned distance z from the reference point q . Evidently, F βγ ( z, t ) = e γ F (cid:0) e − γ z, e − γ ( t − β ) (cid:1) . Even after a time translation and a parabolic rescaling, it is possible for theprofile functions to differ by a translation in space. To account for this, we definea new reference point q αβγ with the property that q αβγ has signed distance α fromthe original reference point q with respect to the metric g βγ ( t ∗ ). For t ≤ t ∗ , wedefine a function s αβγ ( t ) to be the signed distance between the sphere of symmetrythrough q αβγ and the point q , with respect to g βγ ( t ). The function s αβγ ( t ) is theunique solution of the ODE ddt s αβγ ( t ) = ( n − Z s αβγ ( t )0 F βγ ,zz ( z ′ , t ) F βγ ( z ′ , t ) dz ′ , s αβγ ( t ∗ ) = α, for t ≤ t ∗ . This ODE, of course, is just the usual evolution of distance along theRicci flow and the integrand is the radial component of the Ricci curvature. Now,for t ≤ t ∗ , we define F αβγ ( z, t ) to be the radius of the sphere of symmetry in( S n , g βγ ( t )) which has signed distance z from the point q αβγ . The three profilefunctions are related by the equation F αβγ ( z, t ) = F βγ ( z + s αβγ ( t ) , t ) = e γ F (cid:0) e − γ ( z + s αβγ ( t )) , e − γ ( t − β ) (cid:1) . In particular, at time t = t ∗ , we have F αβγ ( z, t ∗ ) = F βγ ( z + α, t ) = e γ F (cid:0) e − γ ( z + α ) , e − γ ( t − β ) (cid:1) . In the next lemma, we show that for a ε -admissible triplet ( α, β, γ ) at t ∗ , weexpect the new reference point q αβγ to remain suitably close to the original point q for all earlier times t ≤ t ∗ . Lemma 6.2. If − t ∗ is sufficiently large, then the following holds. Suppose thetriplet ( α, β, γ ) is ε -admissible with respect to time t ∗ , where ε ∈ (0 , . Let s αβγ ( t ) be the solution of the ODE ddt s αβγ ( t ) = ( n − Z s αβγ ( t )0 F βγ ,zz ( z ′ , t ) F βγ ( z ′ , t ) dz ′ with terminal condition s αβγ ( t ∗ ) = α . Then | s αβγ ( t ) | ≤ ε √− t for all t ≤ t ∗ . Proof.
The proof is essentially the same as the proof of Lemma 6.2 in [13].Recall that if we rescale the ancient κ -solution ( S n , g ( t )) around the referencepoint q by the factor ( − t ), then the solution converges to a round cylinder inpointed Cheeger-Gromov sense at t → −∞ . In particular, in the region | z | ≤ √− t , NIQUENESS OF COMPACT ANCIENT SOLUTIONS 29 the radial component of the Ricci curvature tends to zero. Consequently, if − t ∗ issufficiently large, then we will have0 ≤ − F ,zz ( z, t ) F ( z, t ) ≤ − n − t )whenever t ≤ e − γ t ∗ and | z | ≤ √− t . The first inequality follows from nonneg-ativity of the Ricci curvature. For the profile function F βγ ( z, t ) we replace t by e − γ ( t − β ) and z by e − γ z . This implies0 ≤ − F βγ ,zz ( z, t ) F βγ ( z, t ) ≤ − n − t − β )) . whenever t − β ≤ t ∗ and | z | ≤ p − t − β ). By admissibility, | β | ≤ ε ( − t ∗ )log( − t ∗ ) ≤ ε ( − t )log( − t ) ≤ ( − t ) whenever t ≤ t ∗ and − t ∗ is sufficiently large. This ensures that2 t ≤ t − β ≤ t whenever t ≤ t ∗ . Consequently,0 ≤ − F βγ ,zz ( z, t ) F βγ ( z, t ) ≤ − n − t ) . whenever t ≤ t ∗ and | z | ≤ √− t . Plugging this estimate into the ODE for s αβγ ( t ),we obtain (cid:12)(cid:12)(cid:12)(cid:12) ddt s αβγ ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ − t ) | s αβγ ( t ) | , whenever t ≤ t ∗ and | s αβγ ( t ) | ≤ √− t . This gives ddt (cid:16) | s αβγ ( t ) |√− t (cid:17) ≥ t ≤ t ∗ and | s αβγ ( t ) | ≤ √− t . At time t ∗ , we have | s αβγ ( t ∗ ) | = | α | ≤ ε √− t ∗ . The the differential inequality above implies | s αβγ ( t ) | ≤ ε √− t . This com-pletes the proof of Lemma 6.2.Using the admissibility conditions for ( α, β, γ ) and the previous lemma, we canestimate the profile function F αβγ : Proposition 6.3.
Fix a small number θ > and a small number η > . Then thereexists a small number ε > (depending on θ and η ) with the following property.If the triplet ( α, β, γ ) is ε -admissible with respect to time t ∗ and − t ∗ is sufficientlylarge, then (cid:12)(cid:12)(cid:12) F αβγ ( z, t ) + ( n − t + ( n − z + 2 t − t ) (cid:12)(cid:12)(cid:12) ≤ η z − t log( − t ) and (cid:12)(cid:12)(cid:12) F αβγ ( z, t ) F αβγ z ( z, t ) + ( n − z − t ) (cid:12)(cid:12)(cid:12) ≤ η | z | + √− t log( − t ) whenever F αβγ ( z, t ) ≥ θ √− t and t ≤ t ∗ . Proof.
The proof is essentially the same as the proof of the Proposition 6.3 in[13]. Using Proposition 4.12 and Proposition 4.13, we obtain (cid:12)(cid:12)(cid:12) F ( z, t ) + ( n − t + ( n − z + 2 t − t ) (cid:12)(cid:12)(cid:12) ≤ η z − t log( − t ) and (cid:12)(cid:12)(cid:12) F ( z, t ) F z ( z, t ) + ( n − z − t ) (cid:12)(cid:12)(cid:12) ≤ η | z | + √− t log( − t )whenever F ( z, t ) ≥ θ √− t and − t is sufficiently large. To estimate F βγ , we replace t by e − γ ( t − β ) and z by e − γ z . This gives (cid:12)(cid:12)(cid:12) F βγ ( z, t ) + ( n − t − β ) + ( n − z + 2( t − β )4 log( − ( t − β )) − γ (cid:12)(cid:12)(cid:12) ≤ η z − ( t − β )log( − ( t − β )) − γ and (cid:12)(cid:12)(cid:12) F βγ ( z, t ) F βγ z ( z, t ) + ( n − z − ( t − β )) − γ (cid:12)(cid:12)(cid:12) ≤ η | z | + p − ( t − β )log( − ( t − β )) − γ whenever F βγ ( z, t ) ≥ θ p − ( t − β ) and − e − γ ( t − β ) is sufficiently large. The ε -admissibility assumptions on ( α, β, γ ) at time t ∗ ensure that | β | ≤ ε ( − t )log( − t ) and γ ≤ ε log( − t ) for t ≤ t ∗ . If ε is sufficiently small (depending on θ and η ) and − t ∗ is sufficiently large (depending on θ and η ), then we obtain (cid:12)(cid:12)(cid:12) F βγ ( z, t ) + ( n − t + ( n − z + 2 t − t ) (cid:12)(cid:12)(cid:12) ≤ η z − t log( − t )and (cid:12)(cid:12)(cid:12) F βγ ( z, t ) F βγ z ( z, t ) + ( n − z − t ) (cid:12)(cid:12)(cid:12) ≤ η | z | + √− t log( − t )whenever F βγ ( z, t ) ≥ θ √− t and t ≤ t ∗ . By Lemma 6.2, | s αβγ ( t ) | ≤ ε √− t for t ≤ t ∗ . Replacing z by z + s αβγ ( t ) and using this estimate, we obtain (cid:12)(cid:12)(cid:12) F αβγ ( z, t ) + ( n − t + ( n − z + 2 t − t ) (cid:12)(cid:12)(cid:12) ≤ η z − t log( − t )and (cid:12)(cid:12)(cid:12) F αβγ ( z, t ) F αβγ z ( z, t ) + ( n − z − t ) (cid:12)(cid:12)(cid:12) ≤ η | z | + √− t log( − t )whenever F αβγ ( z, t ) ≥ θ √− t and t ≤ t ∗ . This completes the proof of Proposition6.3.As in [13], we need to use different functions to describe our solutions and es-tablish estimates in the tip regions. These functions labeled by U are analogousto the profile function Φ used to describe the Bryant soliton. As in [13], we definefunctions U ( r, t ) and U − ( r, t ) by U ( r, t ) = (cid:16) ∂∂z F ( z, t ) (cid:17) for r = F ( z, t ) and z ≥ √− t and U − ( r, t ) = (cid:16) ∂∂z F ( z, t ) (cid:17) for r = F ( z, t ) and z ≤ − √− t . Similarly, we define functions U ( r, t ) and U − ( r, t ) by U ( r, t ) = (cid:16) ∂∂z F ( z, t ) (cid:17) for r = F ( z, t ) and z ≥ √− t and U − ( r, t ) = (cid:16) ∂∂z F ( z, t ) (cid:17) for r = F ( z, t ) and z ≤ − √− t . Then, we define U βγ ( r, t ) := U ( e − γ r, e − γ ( t − β )) ,U βγ − ( r, t ) := U − ( e − γ r, e − γ ( t − β )) . Recalling that F αβγ ( z, t ) = F βγ ( z + s αβγ ( t ) , t ), observe that U βγ ( r, t ) = (cid:16) ∂∂z F αβγ ( z, t ) (cid:17) for r = F ( z, t ), z ≥ √− t , and t ≤ t ∗ , and U βγ − ( r, t ) = (cid:16) ∂∂z F αβγ ( z, t ) (cid:17) for r = F ( z, t ), z ≤ − √− t , and t ≤ t ∗ .For each of the functions above, we will define a function V in the usual rescaledcoordinates. For scaling reasons, it is convenient to define the functions labeled by V to be the square-root of the corresponding functions labeled by U . As usual,define coordinates τ and ρ by the identities t = − e − τ and r = e − τ ρ . Then wedefine: V ( ρ, τ ) := q U ( e − τ ρ, − e − τ ) ,V − ( ρ, τ ) := q U − ( e − τ ρ, − e − τ ) ,V ( ρ, τ ) := q U ( e − τ ρ, − e − τ ) ,V − ( ρ, τ ) := q U − ( e − τ ρ, − e − τ ) ,V βγ ( ρ, τ ) := q U βγ ( e − τ ρ, − e − τ ) ,V βγ − ( ρ, τ ) := q U βγ − ( e − τ ρ, − e − τ ) . By solving − e − ˜ τ = e − γ ( − e − τ − β ) and e − ˜ τ/ ˜ ρ = e − γ/ e − τ/ ρ for ˜ τ and ˜ ρ , you canconfirm that V βγ ( ρ, τ ) = V (cid:16) ρ √ βe τ , τ + γ − log(1 + βe τ ) (cid:17) ,V βγ − ( ρ, τ ) = V − (cid:16) ρ √ βe τ , τ + γ − log(1 + βe τ ) (cid:17) . In the following proposition, we recover a version of the estimates establishedin Proposition 4.21 and Corollary 4.23 for the modified profile functions V βγ and V βγ − . Proposition 6.4.
Fix a small number η > . Then we can find a small number θ ∈ (0 , η ) (depending on η ) and a small number ε > (depending on θ and η ) withthe following property. If the triplet ( α, β, γ ) is ε -admissible with respect to time t ∗ = − e − τ ∗ and − τ ∗ is sufficiently large, then | V βγ ( ρ, τ ) − − Φ(( − τ ) ρ ) − | ≤ η ( V βγ ( ρ, τ ) − − for ρ ≤ θ and τ ≤ τ ∗ , and (cid:12)(cid:12)(cid:12) ∂∂τ V βγ ( ρ, τ ) − (cid:12)(cid:12)(cid:12) ≤ η ρ − ( V βγ ( ρ, τ ) − − for ρ ≤ and τ ≤ τ ∗ . Here, Φ denotes the profile of the Bryant soliton. Proof.
The proof is identical to the proof of Proposition 6.4 in [13].We next consider the difference between the two solutions near each of the tips: W βγ + ( ρ, τ ) := V ( ρ, τ ) − V βγ ( ρ, τ ) ,W βγ − ( ρ, τ ) := V − ( ρ, τ ) − V βγ − ( ρ, τ ) . For each τ , we have W βγ + ( ρ, τ ) = O ( ρ ) and W βγ − ( ρ, τ ) = O ( ρ ) as ρ →
0. More-over, let µ + ( ρ, τ ) and µ − ( ρ, τ ) denote the weights associated with the solution( S n , g ( t )). The following proposition is the first of four key estimates. Proposition 6.5.
We can choose θ > and ε > sufficiently small so that thefollowing holds. If − τ ∗ is sufficiently large (depending on θ ) and the triplet ( α, β, γ ) is ε -admissible with respect to time t ∗ = − e − τ ∗ , then sup τ ≤ τ ∗ ( − τ ) − Z ττ − Z θ V − ( W βγ + ) e µ + ≤ C ( θ ) ( − τ ∗ ) − sup τ ≤ τ ∗ ( − τ ) − Z ττ − Z θθ V − ( W βγ + ) e µ + . An analogous estimate holds for W βγ − . We will give the proof of Proposition 6.5 in Section 7.From this point on, we fix θ small enough so that the conclusion of Proposition6.5 holds. Let χ C denote a smooth, even cutoff function satisfying χ C = 1 on[0 , q − θ n − ] and χ C = 0 on [ q − θ n − , ∞ ). Having fixed θ as in Proposition6.5, the factor of ( n −
2) in higher dimensions ensures there is overlap betweenthe tip region and the collar region where estimates can be played off one another.Moreover, we may assume that χ C is monotone decreasing on [0 , ∞ ).We define the rescaled profile functions G ( ξ, τ ) := e τ F ( e − τ ξ, − e − τ ) − p n − ,G ( ξ, τ ) := e τ F ( e − τ ξ, − e − τ ) − p n − ,G αβγ ( ξ, τ ) := e τ F αβγ ( e − τ ξ, − e − τ ) − p n − . Then we consider the difference of the rescaled profile functions in the collar regionvia H αβγ ( ξ, τ ) := G ( ξ, τ ) − G αβγ ( ξ, τ )and H αβγ C ( ξ, τ ) := χ C (( − τ ) − ξ ) H αβγ ( ξ, τ ) . NIQUENESS OF COMPACT ANCIENT SOLUTIONS 33
Using the PDEs for G and G αβγ , we can derive a PDE for the function H αβγ . Asin three dimensions, the leading term in that PDE is given by the operator L f := f ξξ − ξ f ξ + f. We will analyze this operator as in [13]. We consider the Hilbert space H = L ( R , e − ξ dξ ) and recall that the Hilbert space H has a natural direct sum decom-position H = H + ⊕ H ⊕ H − . Furthermore, we recall that H + is a two-dimensionalsubspace spanned by the functions 1 and ξ ; H is a one-dimensional subspacespanned by the function ξ −
2; and H − is the orthogonal complement of H + ⊕ H .Finally, let P + , P , and P − denote the projection operators associated to the directsum decomposition H = H + ⊕ H ⊕ H − .With these conventions, we write P H αβγ C ( ξ, τ ) = p n − a αβγ ( τ ) ( ξ − , where a αβγ ( τ ) := 116 p n − π Z R e − ξ ( ξ − H αβγ C ( ξ, τ ) dξ. Moreover, we let ˆ H αβγ C = P + H αβγ C + P − H αβγ C denote the sum of projections ontothe spaces of positive and negative modes.In the following proposition, we use our freedom of choice in the parameters( α, β, γ ) to ensure the projections our solution P H αβγ C and P + H αβγ C (i.e. theprojections onto the spaces of non-decaying modes of the operator L ) vanish at aparticular time τ ∗ . Proposition 6.6.
Fix θ > and ε > small enough so that the conclusion ofProposition 6.5 holds. Let δ ∈ (0 , ε ) be given. If − τ ∗ is sufficiently large (dependingon δ ), then we can find a triplet ( α, β, γ ) (depending on τ ∗ ) such that P + H αβγ C = 0 and P H αβγ C = 0 at time τ ∗ . Moreover, if − τ ∗ is sufficiently large (depending on δ ), then the triplet ( α, β, γ ) is δ -admissible with respect to time t ∗ = − e − τ ∗ . Proof.
By definition, s αβγ ( t ∗ ) = α . Hence F αβγ ( z, t ∗ ) = e γ F ( e − γ ( z + α ) , e − γ ( t ∗ − β )) . It follows by a straightforward computation that G αβγ ( ξ, τ ∗ ) = p βe τ ∗ G (cid:16) ξ + αe τ ∗ √ βe τ ∗ , τ ∗ + γ − log(1 + βe τ ∗ ) (cid:17) + p n −
2) ( p βe τ ∗ − . The proof of Proposition 6.6 now proceeds as in [3]. This argument relies only onthe asymptotics of our solution in the cylindrical region. Since the asymptoticsof our ancient solutions to Ricci flow in the cylindrical region are very similar tothe cylindrical region asymptotics of ancient solutions to mean curvature flow, theproof of Proposition 6.6 is identical to the proof of the corresponding Proposition4.1 in [3].From this point on, we assume that the triplet ( α, β, γ ) is chosen as in Proposi-tion 6.6, pending our choice of τ ∗ (which we have not yet fixed). In particular, this will ensure that a αβγ ( τ ∗ ) = 0.We can now state the remaining three key estimates used in completing the proofof Theorem 1.3. The first estimate is an estimate for the difference of the solutionsin the cylindrical region. Proposition 6.7.
Fix θ > small enough so that the conclusion of Proposition 6.5holds. Suppose that − τ ∗ is sufficiently large, and that the triplet ( α, β, γ ) is chosenas in Proposition 6.6. Then ( − τ ∗ ) sup τ ≤ τ ∗ Z ττ − Z R e − ξ ( ˆ H αβγ C ,ξ ( ξ, τ ′ ) + ˆ H αβγ C ( ξ, τ ′ ) ) dξ dτ ′ ≤ C ( θ ) sup τ ≤ τ ∗ Z ττ − a αβγ ( τ ′ ) dτ ′ + C ( θ ) sup τ ≤ τ ∗ Z ττ − Z { q − θ n − ( − τ ′ ) ≤| ξ |≤ q − θ n − ( − τ ′ ) } e − ξ H αβγ ( ξ, τ ′ ) dξ dτ ′ . We will give the proof of Proposition 6.7 in Section 8.Next, by combining Proposition 6.5 and Proposition 6.7, we can show that inthe cylindrical region the norm of P H αβγ C dominates over the norm of ˆ H αβγ C . Moreprecisely, we have the following result: Proposition 6.8.
Fix θ > small enough so that the conclusion of Proposition 6.5holds. Suppose that − τ ∗ is sufficiently large, and that the triplet ( α, β, γ ) is chosenas in Proposition 6.6. Then ( − τ ∗ ) sup τ ≤ τ ∗ Z ττ − Z R e − ξ ( ˆ H αβγ C ,ξ ( ξ, τ ′ ) + ˆ H αβγ C ( ξ, τ ′ ) ) dξ dτ ′ ≤ C ( θ ) sup τ ≤ τ ∗ Z ττ − a αβγ ( τ ′ ) dτ ′ . The proof of Proposition 6.8 will be given in Section 9.Using Proposition 6.8, we are able to derive an ODE for the function a αβγ ( τ ): Proposition 6.9.
Fix θ > small enough so that the conclusion of Proposition6.5 holds. Let δ > be given. Suppose that − τ ∗ is sufficiently large (dependingon δ ), and the triplet ( α, β, γ ) is chosen as in Proposition 6.6. Let Q αβγ ( τ ) := ddτ a αβγ ( τ ) − − τ ) − a αβγ ( τ ) . Then sup τ ≤ τ ∗ ( − τ ) Z ττ − | Q αβγ ( τ ′ ) | dτ ′ ≤ δ sup τ ≤ τ ∗ (cid:18) Z ττ − a αβγ ( τ ′ ) dτ ′ (cid:19) . The proof of Proposition 6.9 will be given in Section 10.We can now finish the proof of Theorem 1.3, exactly as in [13] in dimensionthree. For the convenience of the reader, we include a copy of the proof here.Using the ODE ddτ a αβγ ( τ ) = 2 ( − τ ) − a αβγ ( τ ) + Q αβγ ( τ ) together with the factthat a αβγ ( τ ∗ ) = 0, we obtain( − τ ) a αβγ ( τ ) = − Z τ ∗ τ ( − τ ′ ) Q αβγ ( τ ′ ) dτ ′ . NIQUENESS OF COMPACT ANCIENT SOLUTIONS 35
This implies( − τ ) | a αβγ ( τ ) | ≤ Z τ ∗ τ ( − τ ′ ) | Q αβγ ( τ ′ ) | dτ ′ ≤ [ τ ∗ − τ ] X j =0 Z τ ∗ − jτ ∗ − j − ( − τ ′ ) | Q αβγ ( τ ′ ) | dτ ′ ≤ ( − τ ) max ≤ j ≤ [ τ ∗ − τ ] Z τ ∗ − jτ ∗ − j − ( − τ ′ ) | Q αβγ ( τ ′ ) | dτ ′ . We now divide by − τ , and take the supremum over all τ ≤ τ ∗ . This impliessup τ ≤ τ ∗ | a αβγ ( τ ) | ≤ sup τ ≤ τ ∗ Z ττ − ( − τ ′ ) | Q αβγ ( τ ′ ) | dτ ′ . On the other hand, Proposition 6.9 gives the following estimate for Q αβγ :sup τ ≤ τ ∗ ( − τ ) Z ττ − | Q αβγ ( τ ′ ) | dτ ′ ≤ δ sup τ ≤ τ ∗ | a αβγ ( τ ) | . Hence, if we choose δ sufficiently small, and − τ ∗ sufficiently large (depending on δ ), then sup τ ≤ τ ∗ | a αβγ ( τ ) | = 0. Thus, a αβγ ( τ ) = 0 for all τ ≤ τ ∗ . Proposition6.8 then implies ˆ H αβγ C ( ξ, τ ) = 0 for all τ ≤ τ ∗ . Putting these facts together, weobtain H αβγ C ( ξ, τ ) = 0 for all τ ≤ τ ∗ . From this, we deduce that W βγ + ( ρ, τ ) = 0for ρ ∈ [ θ, θ ] and τ ≤ τ ∗ . Proposition 6.5 yields W βγ + ( ρ, τ ) = 0 for ρ ∈ [0 , θ ] and τ ≤ τ ∗ . Thus, we conclude that F ( z, t ) = F αβγ ( z, t ) for all t ≤ t ∗ = − e − τ ∗ . Inother words, the two ancient solutions coincide for t ≤ t ∗ .7. Energy estimates in the tip region and proof of Proposition 6.5
In this section, we give the proof of Proposition 6.5. Let ω T denote a nonnegativesmooth cutoff function satisfying ω T ( ρ ) = 1 for ρ ≤ θ and ω T ( ρ ) = 0 for ρ ≥ θ .We define W βγT + ( ρ, τ ) := ω T ( ρ ) W βγ + ( ρ, τ ) . To simplify the notation, we will write W + and W T + instead of W βγ + and W βγT + . Proposition 7.1.
The function W + ( ρ, τ ) satisfies the equation V − (cid:16) ∂W + ∂τ + ρ ∂W + ∂ρ (cid:17) = ∂ W + ∂ρ + ∂∂ρ (cid:16) ρ − ( n −
2) ( V − − W + (cid:17) (1) + ( n − ρ − ∂W + ∂ρ − n − ρ − W + + V − B + W + . (2) where B + := ( n − ρ − (cid:0) − V ( V βγ ) − (cid:1) + ( n − ρ − (cid:16) V − ∂V ∂ρ − ( V βγ ) − ( V + V βγ ) ∂V βγ ∂ρ (cid:17) + ( V βγ ) − ( V + V βγ ) (cid:16) ∂V βγ ∂τ + ρ ∂V βγ ∂ρ (cid:17) . Proof.
The functions U ( r, t ), U − ( r, t ), U βγ ( r, t ), and U βγ − ( r, t ) all satisfy thesame PDE: U − ∂U∂t = ∂ U∂r − U − (cid:16) ∂U∂r (cid:17) + ( n − r ( U − − (cid:16) r ∂U∂r + 2 U (cid:17) + ( n − r ∂U∂r . Consequently, the functions V ( ρ, τ ), V − ( ρ, τ ), V βγ ( ρ, τ ), and V βγ − ( ρ, τ ) satisfythe following PDE: V − (cid:16) ∂V∂τ + ρ ∂V∂ρ (cid:17) = ∂ V∂ρ + ( n − ρ ( V − − (cid:16) ρ ∂V∂ρ + V (cid:17) + ( n − ρ ∂V∂ρ . The assertion now follows by a straightforward calculation.
Proposition 7.2.
The function W T + ( ρ, τ ) satisfies ∂∂τ (cid:0) V − W T + e µ + (cid:1) − ∂∂ρ h(cid:16) ∂W T + ∂ρ + ( n − ρ − (cid:0) V − − (cid:1) W T + (cid:17) W T + e µ + i + ∂∂ρ (cid:0) W ω ′ T ω T e µ + (cid:1) − ∂∂ρ (cid:0) ( n − ρ − W T + e µ + (cid:1) ≤ − (cid:16) ∂W T + ∂ρ + ∂µ + ∂ρ W T + (cid:17) e µ + −
12 (3 n − ρ − W T + e µ + + V − (cid:16) ∂µ + ∂τ − V − ∂V ∂τ + ρ ∂µ + ∂ρ + B + (cid:17) W T + e µ + + 12 (cid:16) ∂µ + ∂ρ − ( n − ρ − (cid:0) V − − (cid:1) − ρ V − (cid:17) W T + e µ + + (cid:16) ∂µ + ∂ρ − ( n − ρ − (cid:0) V − − (cid:1) + ρ V − − ( n − ρ − (cid:17) W ω ′ T ω T e µ + + ( ω ′ T ) W e µ + −
12 ( n − ρ − W T + ∂µ + ∂ρ e µ + Proof.
Using Proposition 7.1, we obtain V − (cid:16) ∂W T + ∂τ + ρ ∂W T + ∂ρ (cid:17) = ∂ W T + ∂ρ + ∂∂ρ (cid:16) ρ − ( n − V − − W T + (cid:17) − n − ρ − W T + + V − B + W T + + (cid:16) − ω ′ T ∂W + ∂ρ − ω ′′ T W + − ( n − ρ − ( V − − W + ω ′ T + ρ ω ′ T V − W + (cid:17) + ( n − ρ − (cid:16) ∂W T + ∂ρ − W + ω ′ T (cid:17) . NIQUENESS OF COMPACT ANCIENT SOLUTIONS 37
We next bring in the weight µ + ( ρ, τ ). A straightforward calculation gives12 ∂∂τ (cid:0) V − W T + e µ + (cid:1) − ∂∂ρ h(cid:16) ∂W T + ∂ρ + ρ − ( n − V − − W T + (cid:17) W T + e µ + i + ∂∂ρ (cid:0) W ω ′ T ω T e µ + (cid:1) − ∂∂ρ (cid:0) ( n − ρ − W T + e µ + (cid:1) = − (cid:16) ∂W T + ∂ρ + ∂µ + ∂ρ W T + (cid:17) e µ + −
12 (3 n − ρ − W T + e µ + + V − (cid:16) ∂µ + ∂τ − V − ∂V ∂τ + ρ ∂µ + ∂ρ + B + (cid:17) W T + e µ + + (cid:16) ∂µ + ∂ρ − ( n − ρ − ( V − − − ρ V − (cid:17)(cid:16) ∂W T + ∂ρ + ∂µ + ∂ρ W T + (cid:17) W T + e µ + + (cid:16) ∂µ + ∂ρ − ( n − ρ − ( V − −
1) + ρ V − (cid:17) W ω ′ T ω T e µ + + ( ω ′ T ) W e µ + − ( n − ρ − W ω T ω ′ T e µ + −
12 ( n − ρ − W T + ∂µ + ∂ρ e µ + . The assertion follows now from Young’s inequality and combining terms.
Corollary 7.3.
Fix a small number η > . Then we can find a small number θ ∈ (0 , η ) and a small number ε ∈ (0 , η ) (both depending on η ) with the followingproperty. If − τ ∗ sufficiently large (depending on η and θ ) and the triplet ( α, β, γ ) is ε -admissible with respect to time t ∗ = − e − τ ∗ , then we have ∂∂τ (cid:0) V − W T + e µ + (cid:1) − ∂∂ρ h(cid:16) ∂W T + ∂ρ + ρ − ( n − (cid:0) V − − (cid:1) W T + (cid:17) W T + e µ + i + ∂∂ρ (cid:0) W ω ′ T ω T e µ + (cid:1) − ∂∂ρ (cid:0) ( n − ρ − W T + e µ + (cid:1) ≤ − (cid:16) ∂W T + ∂ρ + ∂µ + ∂ρ W T + (cid:17) e µ + −
12 (3 n − ρ − W T + e µ + + η ρ − V − W T + e µ + + η ρ − V − W e µ + { θ ≤ ρ ≤ θ } . for ρ ≤ θ and τ ≤ τ ∗ . Proof.
By Proposition 4.21, Proposition 4.22, and Proposition 6.4, we canchoose θ ∈ (0 , η ) (depending on η ) sufficiently small and − τ ∗ sufficiently large(depending on η and θ ) such that |B + | ≤ η ρ − V − for ρ ≤ θ and τ ≤ τ ∗ . By Corollary 4.23, Lemma 5.2, and Lemma 5.4, we canchoose θ ∈ (0 , η ) sufficiently small (depending on η ) and − τ ∗ sufficiently large(depending on η and θ ) such that (cid:12)(cid:12)(cid:12) ∂µ + ∂τ − V − ∂V ∂τ + ρ ∂µ + ∂ρ (cid:12)(cid:12)(cid:12) ≤ η ρ − V − , (cid:12)(cid:12)(cid:12) ∂µ + ∂ρ − ( n − ρ − ( V − − − ρ V − (cid:12)(cid:12)(cid:12) ≤ η ρ − V − , (cid:12)(cid:12)(cid:12) ∂µ + ∂ρ − ( n − ρ − ( V − −
1) + ρ V − (cid:12)(cid:12)(cid:12) ≤ η ρ − V − for ρ ≤ θ and τ ≤ τ ∗ . Note also that for any η > θ ∈ (0 , η ), so that | ( n − ρ − ω ′ T ω T | ≤ η ρ − V − { θ ≤ ρ ≤ θ } . Finally, recall by Lemma 5.2 ∂µ + ∂ρ ≥
0, so −
12 ( n − ρ − W T + ∂µ + ∂ρ e µ + ≤ . Hence, the assertion follows from Proposition 7.2.We now finalize our choice of θ . Proposition 7.4.
We can find sufficiently small numbers θ > , λ > , and ε > with the following property. If − τ ∗ is sufficiently large (depending on θ ) and thetriplet ( α, β, γ ) is ε -admissible with respect to time t ∗ = − e − τ ∗ , then ddτ (cid:18) Z θ V − W T + e µ + dρ (cid:19) ≤ − λ ( − τ ) Z θ V − W T + e µ + dρ + Z θθ ρ − V − W e µ + dρ for τ ≤ τ ∗ . Proof.
Fix a small number η >
0. In the following, we choose θ and ε sufficientlysmall (depending on η ), and we choose − τ ∗ sufficiently large (depending on η and θ ). Recall W T + (2 θ, τ ) = 0 and, for each τ , W + ( ρ, τ ) = O ( ρ ) as ρ →
0, hence W T + ( ρ, τ ) = O ( ρ ) as ρ →
0. Moreover, we can see from the definition of µ + that µ + ( ρ, τ ) is bounded as ρ →
0. In particular, if we integrate the differentialinequality of Corollary 7.3, the divergence terms vanish. Using Corollary 7.3, weobtain12 ddτ (cid:18) Z θ V − W T + e µ + dρ (cid:19) ≤ − Z θ (cid:16) ∂W T + ∂ρ + ∂µ + ∂ρ W T + (cid:17) e µ + dρ −
12 (3 n − Z θ ρ − W T + e µ + dρ + η Z θ ρ − V − W T + e µ + dρ + η Z θθ ρ − V − W e µ + dρ, for τ ≤ τ ∗ .We will next estimate the terms on the right hand side of the above inequalityto deduce the statement of the Proposition. First, applying Proposition 5.5 to thefunction f := e µ + W T + gives0 ≤ Z θ (cid:16) ∂W T + ∂ρ + ∂µ + ∂ρ W T + (cid:17) e µ + dρ + K ∗ Z θ ρ − W T + e µ + dρ − Z θ (cid:16) ∂µ + ∂ρ (cid:17) W T + e µ + dρ for τ ≤ τ ∗ . Using Lemma 5.2, we obtain ( ∂µ + ∂ρ ) ≥ ρ − ( V − − for ρ ≤ θ ,hence 0 ≤ η Z θ (cid:16) ∂W T + ∂ρ + ∂µ + ∂ρ W T + (cid:17) e µ + dρ + 16 ηK ∗ Z θ ρ − W T + e µ + dρ − η Z θ ρ − ( V − − W T + e µ + dρ NIQUENESS OF COMPACT ANCIENT SOLUTIONS 39 for τ ≤ τ ∗ .Adding the two inequalities above, we obtain12 ddτ (cid:18) Z θ V − W T + e µ + dρ (cid:19) ≤ − (cid:16) − η (cid:17) Z θ (cid:16) ∂W T + ∂ρ + ∂µ + ∂ρ W T + (cid:17) e µ + dρ − (cid:16)
12 (3 n − − η − ηK ∗ (cid:17) Z θ ρ − W T + e µ + dρ − η Z θ ρ − (4( V − − + 4 − V − ) W T + e µ + dρ + η Z θθ ρ − V − W e µ + dρ for τ ≤ τ ∗ . We now choose η > − η > n − − η − ηK ∗ >
0. (Here, it is crucial that the constant K ∗ in the weightedPoincar´e inequality does not depend on θ .) This ensures that the first two terms onthe right hand side of the last estimate have a favorable sign. To estimate the thirdterm on the right hand side, we observe that ρ − [4( V − − + 4 − V − ] ≥ ρ − V − .Finally, in view of Proposition 4.21, we can bound ρ − V − from below by a smallpositive multiple of ( − τ ) V − . This completes the proof of Proposition 7.4.We now complete the proof of Proposition 6.5. Let θ , λ , and ε be chosen as inProposition 7.4. Let I ( τ ) := Z ττ − Z θ V − W T + e µ + and J ( τ ) := Z ττ − Z θθ V − W e µ + . If we choose − τ ∗ sufficiently large, then Proposition 7.4 gives12 I ′ ( τ ) + λ ( − τ ) I ( τ ) ≤ θ − J ( τ ) , hence ddτ ( e − λτ I ( τ )) ≤ θ − e − λτ J ( τ )for τ ≤ τ ∗ . Clearly, lim τ →−∞ e − λτ I ( τ ) = 0. Consequently, e − λτ I ( τ ) ≤ θ − Z τ −∞ e − λτ ′ J ( τ ′ ) dτ ′ ≤ θ − (cid:16) sup τ ′ ≤ τ ( − τ ′ ) − J ( τ ′ ) (cid:17) Z τ −∞ e − λτ ′ ( − τ ′ ) dτ ′ ≤ θ − λ − e − λτ sup τ ′ ≤ τ ( − τ ′ ) − J ( τ ′ ) for τ ≤ τ ∗ . This finally gives( − τ ) − I ( τ ) ≤ θ − λ − ( − τ ) − sup τ ′ ≤ τ ( − τ ′ ) − J ( τ ′ ) ≤ θ − λ − ( − τ ) − sup τ ′ ≤ τ ( − τ ′ ) − J ( τ ′ )for τ ≤ τ ∗ . Taking the supremum over τ ≤ τ ∗ givessup τ ≤ τ ∗ ( − τ ) − I ( τ ) ≤ θ − λ − ( − τ ∗ ) − sup τ ≤ τ ∗ ( − τ ) − J ( τ ) . From this, the conclusion of Proposition 6.5 follows immediately.8.
Energy estimates in the cylindrical region and proof ofProposition 6.7
In this section, we give the proof of Proposition 6.7. Throughout this section,we assume that θ is chosen as in Proposition 6.5. To simplify the notation, we willwrite H , H C , ˆ H C , and a instead of H αβγ , H αβγ C , ˆ H αβγ C , and a αβγ .Our goal is to study the evolution equation satisfied by the function H . Thelinearized operator L f := f ξξ − ξ f ξ + f is the same as in [3], and hence the linear theory from [3] carries over to the Ricciflow case as well. In order for this article to be self-contained, we will state theresults from [3] that we will use later, but for the proofs of the same we refer thereader to [3].As in [3], we consider the Hilbert space H = L ( R , e − ξ dξ ). The norm on H isgiven by k f k H := Z R e − ξ f ( ξ ) dξ. Moreover, we denote by
D ⊂ H the Hilbert space of all functions f such that f ∈ H and f ′ ∈ H . The norm on D is given by k f k D := Z R e − ξ ( f ′ ( ξ ) + f ( ξ ) ) dξ. Let D ∗ denote the dual space of D . Clearly, the dual space H ∗ is a subspace of D ∗ .After identifying H ∗ with H in the standard way, we can view H as a subspace of D ∗ . The restriction of k · k D ∗ to H is given by k f k D ∗ := sup (cid:26) Z R e − ξ f ( ξ ) g ( ξ ) dξ : k g k D ≤ (cid:27) . For later reference, we collect some basic facts from [3].
Proposition 8.1.
The following statements hold: (i)
The operators f ξ f , f f ′ , f
7→ − f ′ + ξ f are bounded from D to H . (ii) The operators f ξ f , f f ′ , f
7→ − f ′ + ξ f are bounded from H to D ∗ . (iii) The operators f ξ f , f ξ f ′ , f f ′′ are bounded from D to D ∗ . (iv) The operator f R ξ f is bounded from H to D . NIQUENESS OF COMPACT ANCIENT SOLUTIONS 41
Proof.
Statements (i), (ii), and (iii) were proved in [3]. To prove statement(iv), let us consider a function f ∈ H , and let g ( ξ ) := R ξ f ( ξ ′ ) dξ ′ . Then g ( ξ ) ≤ ξ R ξ f ( ξ ′ ) dξ ′ for ξ ≥
0. Using Fubini’s theorem, we obtain Z ∞ e − ξ g ( ξ ) dξ ≤ Z ∞ e − ξ ξ (cid:18) Z ξ f ( ξ ′ ) dξ ′ (cid:19) dξ = Z ∞ (cid:18) Z ∞ ξ ′ e − ξ ξ dξ (cid:19) f ( ξ ′ ) dξ ′ = 2 Z ∞ e − ξ ′ f ( ξ ′ ) dξ ′ . An analogous argument gives R −∞ e − ξ g ( ξ ) dξ ≤ R −∞ e − ξ ′ f ( ξ ′ ) dξ ′ . There-fore, k g k H ≤ C k f k H . Since g ′ = f , it follows that k g k D ≤ C k f k H , as claimed.For a time-dependent function f , we introduce the following norms: k f k H , ∞ ,τ ∗ := sup τ ≤ τ ∗ Z ττ − k f ( · , τ ′ ) k H dτ ′ , k f k D , ∞ ,τ ∗ := sup τ ≤ τ ∗ Z ττ − k f ( · , τ ′ ) k D dτ ′ , k f k D ∗ , ∞ ,τ ∗ := sup τ ≤ τ ∗ Z ττ − k f ( · , τ ′ ) k D ∗ dτ ′ . The following energy estimate was proved in [3]:
Proposition 8.2.
Let g : ( −∞ , τ ∗ ] → D ∗ be a bounded function. Let f : ( −∞ , τ ∗ ] →D be a bounded function which satisfies the linear equation ∂∂τ f ( τ ) − L f ( τ ) = g ( τ ) . Then the function ˆ f := P + f + P − f satisfies the estimate sup τ ≤ τ ∗ k ˆ f ( τ ) k H + Λ − k ˆ f k D , ∞ ,τ ∗ ≤ k P + f ( τ ∗ ) k H + Λ k g k D ∗ , ∞ ,τ ∗ , where Λ is a universal constant. Proof.
See [3], Lemma 6.6.We now continue with the proof of Proposition 6.7. The functions G ( ξ, τ ) and G αβγ ( ξ, τ ) satisfy the equation G τ ( ξ, τ ) = G ξξ ( ξ, τ ) − ξ G ξ ( ξ, τ )+ 12 ( p n −
2) + G ( ξ, τ )) − ( n −
2) ( p n −
2) + G ( ξ, τ )) − − ( p n −
2) + G ( ξ, τ )) − G ξ ( ξ, τ ) + ( n − G ξ ( ξ, τ ) (cid:20) G ξ (0 , τ ) p n −
2) + G (0 , τ ) − Z ξ G ξ ( ξ ′ , τ ) ( p n −
2) + G ( ξ ′ , τ )) dξ ′ (cid:21) . Note that the two terms on the second line above can be written12 ( p n −
2) + G ) − ( n −
2) ( p n −
2) + G ) − = G −
12 ( p n −
2) + G ) − G . Consequently, the difference H ( ξ, τ ) = G ( ξ, τ ) − G αβγ ( ξ, τ ) satisfies H τ ( ξ, τ ) = H ξξ ( ξ, τ ) − ξ H ξ ( ξ, τ ) + H ( ξ, τ ) + X k =1 E k ( ξ, τ ) , where E ( ξ, τ ) = h ( n −
2) ( p n −
2) + G ( ξ, τ )) − ( p n −
2) + G αβγ ( ξ, τ )) − − i H ( ξ, τ ) E ( ξ, τ ) = ( p n −
2) + G ( ξ, τ )) − ( p n −
2) + G αβγ ( ξ, τ )) − G ξ ( ξ, τ ) H ( ξ, τ ) ,E ( ξ, τ ) = − ( p n −
2) + G αβγ ( ξ, τ )) − ( G ξ ( ξ, τ ) + G αβγ ξ ( ξ, τ )) H ξ ( ξ, τ ) E ( ξ, τ ) = ( n − (cid:20) G ξ (0 , τ ) p n −
2) + G (0 , τ ) − Z ξ G ξ ( ξ ′ , τ ) ( p n −
2) + G ( ξ ′ , τ )) dξ ′ (cid:21) H ξ ( ξ, τ ) ,E ( ξ, τ ) = ( n − G αβγ ξ ( ξ, τ ) H ξ (0 , τ ) p n −
2) + G (0 , τ ) − ( n − G αβγ ξ ( ξ, τ ) G αβγ ξ (0 , τ ) H (0 , τ )( p n −
2) + G (0))( p n −
2) + G αβγ (0 , τ )) ,E ( ξ, τ ) = ( n − G αβγ ξ ( ξ, τ ) (cid:20) − Z ξ ( G ξ ( ξ ′ , τ ) + G αβγ ξ ( ξ ′ , τ )) H ξ ( ξ ′ , τ )( p n −
2) + G αβγ ( ξ ′ , τ )) dξ ′ + Z ξ (2 p n −
2) + G ( ξ ′ , τ ) + G αβγ ( ξ ′ , τ )) H ( ξ ′ , τ ) G ξ ( ξ ′ , τ ) ( p n −
2) + G ( ξ ′ , τ )) ( p n −
2) + G αβγ ( ξ ′ , τ )) dξ ′ (cid:21) . Consequently, the function H C ( ξ, τ ) = χ C (( − τ ) − ξ ) H ( ξ, τ ) satisfies H C ,τ ( ξ, τ ) = H C ,ξξ ( ξ, τ ) − ξ H C ,ξ ( ξ, τ ) + H C ( ξ, τ ) + X k =1 E C ,k ( ξ, τ ) , NIQUENESS OF COMPACT ANCIENT SOLUTIONS 43 where E C , ( ξ, τ ) = h ( n − p n −
2) + G ( ξ, τ )) − ( p n −
2) + G αβγ ( ξ, τ )) − − i H C ( ξ, τ ) ,E C , ( ξ, τ ) = ( p n −
2) + G ( ξ, τ )) − ( p n −
2) + G αβγ ( ξ, τ )) − G ξ ( ξ, τ ) H C ( ξ, τ ) ,E C , ( ξ, τ ) = − ( p n −
2) + G αβγ ( ξ, τ )) − ( G ξ ( ξ, τ ) + G αβγ ξ ( ξ, τ )) H C ,ξ ( ξ, τ ) E C , ( ξ, τ ) = ( n − (cid:20) G ξ (0 , τ ) p n −
2) + G (0 , τ ) − Z ξ G ξ ( ξ ′ , τ ) ( p n −
2) + G ( ξ ′ , τ )) dξ ′ (cid:21) H C ,ξ ( ξ, τ ) ,E C , ( ξ, τ ) = ( n − χ C (( − τ ) − ξ ) G αβγ ξ ( ξ, τ ) H ξ (0 , τ ) p n −
2) + G (0 , τ ) − ( n − χ C (( − τ ) − ξ ) G αβγ ξ ( ξ, τ ) G αβγ ξ (0 , τ ) H (0 , τ )( p n −
2) + G (0))( p n −
2) + G αβγ (0 , τ )) ,E C , ( ξ, τ ) = ( n − χ C (( − τ ) − ξ ) G αβγ ξ ( ξ, τ ) · (cid:20) − Z ξ ( G ξ ( ξ ′ , τ ) + G αβγ ξ ( ξ ′ , τ )) H ξ ( ξ ′ , τ )( p n −
2) + G αβγ ( ξ ′ , τ )) dξ ′ + Z ξ (2 p n −
2) + G ( ξ ′ , τ ) + G αβγ ( ξ ′ , τ )) H ( ξ ′ , τ ) G ξ ( ξ ′ , τ ) ( p n −
2) + G ( ξ ′ , τ )) ( p n −
2) + G αβγ ( ξ ′ , τ )) dξ ′ (cid:21) ,E C , ( ξ, τ ) = ( p n −
2) + G αβγ ( ξ, τ )) − ( G ξ ( ξ, τ ) + G αβγ ξ ( ξ, τ )) · ( − τ ) − χ ′C (( − τ ) − ξ ) H ( ξ, τ ) ,E C , ( ξ, τ ) = − ( n − (cid:20) G ξ (0 , τ ) p n −
2) + G (0 , τ ) − Z ξ G ξ ( ξ ′ , τ ) ( p n −
2) + G ( ξ ′ , τ )) dξ ′ (cid:21) · ( − τ ) − χ ′C (( − τ ) − ξ ) H ( ξ, τ ) ,E C , ( ξ, τ ) = ( − τ ) − χ ′′C (( − τ ) − ξ ) H ( ξ, τ )+ 12 ( − τ ) − ξ χ ′C (( − τ ) − ξ ) H ( ξ, τ ) ,E C , ( ξ, τ ) = − − τ ) − ∂∂ξ (cid:2) χ ′C (( − τ ) − ξ ) H ( ξ, τ ) (cid:3) + 12 ( − τ ) − ξ χ ′C (( − τ ) − ξ ) H ( ξ, τ ) . In the following, we will estimate the terms P k =1 k E C ,k k H , ∞ ,τ ∗ and P k =7 k E C ,k k D ∗ , ∞ ,τ ∗ .To that end, we need the following estimates for the functions G ( ξ, τ ) and G αβγ ( ξ, τ ): Proposition 8.3.
Fix a small number θ > and a small number η > . Then thereexists a small number ε > (depending on θ and η ) with the following property.If the triplet ( α, β, γ ) is ε -admissible with respect to time t ∗ = − e − τ ∗ and − τ ∗ issufficiently large, then (cid:12)(cid:12)(cid:12) ( p n −
2) + G ( ξ, τ )) − n −
2) + ( n − ξ − − τ ) (cid:12)(cid:12)(cid:12) ≤ η ξ + 1( − τ ) , (cid:12)(cid:12)(cid:12) ( p n −
2) + G αβγ ( ξ, τ )) − n −
2) + ( n − ξ − − τ ) (cid:12)(cid:12)(cid:12) ≤ η ξ + 1( − τ ) and (cid:12)(cid:12)(cid:12) ( p n −
2) + G ( ξ, τ )) G ξ ( ξ, τ ) + ( n − ξ − τ ) (cid:12)(cid:12)(cid:12) ≤ η | ξ | + 1( − τ ) , (cid:12)(cid:12)(cid:12) ( p n −
2) + G αβγ ( ξ, τ )) G αβγ ξ ( ξ, τ ) + ( n − ξ − τ ) (cid:12)(cid:12)(cid:12) ≤ η | ξ | + 1( − τ ) for | ξ | ≤ q − θ n − ( − τ ) and τ ≤ τ ∗ . Proof.
This follows directly from Proposition 4.12, Proposition 4.13, and Propo-sition 6.3.In order to estimate the term k E C , k H , ∞ ,τ ∗ , we need the following pointwiseestimate: Lemma 8.4.
We have | E C , ( ξ, τ ) | ≤ C ( θ ) ( − τ ) − | G αβγ ξ ( ξ, τ ) | (cid:12)(cid:12)(cid:12)(cid:12) Z ξ | H C ( ξ ′ , τ ) | dξ ′ (cid:12)(cid:12)(cid:12)(cid:12) + C ( θ ) ( − τ ) − | G αβγ ξ ( ξ, τ ) | ( | H C ( ξ, τ ) | + | H (0 , τ ) | ) . Proof.
The proof is analogous to the proof of Lemma 8.4 in [13].In order to estimate the term k E C , k H , ∞ ,τ ∗ , we need the following estimate for H ξ (0 , τ ): Lemma 8.5.
We have sup τ ≤ τ ∗ (cid:18) Z ττ − H ξ (0 , τ ′ ) dτ ′ (cid:19) ≤ C k H C k H , ∞ ,τ ∗ + C X k =1 k E C ,k k H , ∞ ,τ ∗ . Proof.
In the region {| ξ | ≤ } , we have ∂∂τ H C = L H C + P k =1 E C ,k . Usingstandard interior estimates for linear parabolic equations and the embedding of theSobolev space H ([ − , C ([ − , τ ≤ τ ∗ (cid:18) Z ττ − H C ,ξ (0 , τ ′ ) dτ ′ (cid:19) ≤ C k H C k H , ∞ ,τ ∗ + C X k =1 k E C ,k k H , ∞ ,τ ∗ . Since H C ,ξ (0 , τ ) = H ξ (0 , τ ), the assertion follows. Lemma 8.6.
We have X k =1 k E C ,k k H , ∞ ,τ ∗ ≤ C ( θ ) ( − τ ∗ ) − k H C k D , ∞ ,τ ∗ . Proof.
The proof is analogous to the proof of Lemma 8.6 in [13].
Lemma 8.7.
We have X k =7 k E C ,k k H , ∞ ,τ ∗ + k E C , k D ∗ , ∞ ,τ ∗ ≤ C ( θ ) ( − τ ∗ ) − (cid:13)(cid:13)(cid:13) H { q − θ n − ( − τ ) ≤| ξ |≤ q − θ n − ( − τ ) } (cid:13)(cid:13)(cid:13) H , ∞ ,τ ∗ . NIQUENESS OF COMPACT ANCIENT SOLUTIONS 45
Proof.
Using Proposition 8.1, we obtain k E C , k D ∗ , ∞ ,τ ∗ ≤ C ( − τ ∗ ) − k χ ′C (( − τ ) − ξ ) H k H , ∞ ,τ ∗ . This gives the desired estimate for E C , . The estimates for E C , , E C , , and E C , fol-low directly from the respective definitions. This completes the proof of Lemma 8.7.We now complete the proof of Proposition 6.7. To that end, we apply Proposition8.2 to the function H C . Since P + H C ( τ ∗ ) = 0, we obtainsup τ ≤ τ ∗ k ˆ H C ( τ ) k H + Λ − k ˆ H C k D , ∞ ,τ ∗ ≤ Λ X k =1 k E C ,k ( ξ, τ ) k D ∗ , ∞ ,τ ∗ by Proposition 6.7. We use Lemma 8.6 and Lemma 8.7 to estimate the terms onthe right hand side. This givessup τ ≤ τ ∗ k ˆ H C ( τ ) k H + Λ − k ˆ H C k D , ∞ ,τ ∗ ≤ C ( θ ) ( − τ ∗ ) − k ˆ H C k D , ∞ ,τ ∗ + C ( θ ) ( − τ ∗ ) − k P H C k D , ∞ ,τ ∗ + C ( θ ) ( − τ ∗ ) − (cid:13)(cid:13)(cid:13) H { q − θ n − ( − τ ) ≤| ξ |≤ q − θ n − ( − τ ) } (cid:13)(cid:13)(cid:13) H , ∞ ,τ ∗ . If − τ ∗ is sufficiently large, the first term on the right hand side can be absorbedinto the left hand side. This completes the proof of Proposition 6.7.9. Analysis of the overlap region and proof of Proposition 6.8
In this section, we give the proof of Proposition 6.8. We remind the reader that θ is chosen as in Proposition 6.5. We also recall that χ C is a smooth cutoff, whichsatisfies χ C = 1 on [0 , q − θ n − ] and χ C = 0 on [ q − θ n − , ∞ ). We alsoassume χ C is monotone decreasing on [0 , ∞ ). As before, we write H , H C , ˆ H C , and a instead of H αβγ , H αβγ C , ˆ H αβγ C , and a αβγ . We begin by recalling the followingelementary lemma from [13]: Lemma 9.1 (Lemma 9.1 in [13]) . Assume that ≤ L < L < L . Then L Z { L ≤ ξ ≤ L } e − ξ f ( ξ ) dξ ≤ C Z { L ≤ ξ ≤ L } e − ξ f ′ ( ξ ) dξ + C ( L − L ) − Z { L ≤ ξ ≤ L } e − ξ f ( ξ ) dξ, where C is a numerical constant that is independent of L , L , L , and f . The following lemma relates the function H ( ξ, τ ) to the function W + ( ρ, τ ): Lemma 9.2.
If we choose − τ ∗ sufficiently large (depending on θ ), then (cid:12)(cid:12) H ξ ( ξ, τ ) + W + ( p n −
2) + G ( ξ, τ ) , τ ) (cid:12)(cid:12) ≤ C ( θ ) | H ( ξ, τ ) | provided that q − θ n − ( − τ ) ≤ ξ ≤ q − θ n − ( − τ ) and τ ≤ τ ∗ . Proof.
The proof of this lemma is analogous to the proof of Lemma 9.2 in [13].Replacing θ by θ/ √ n − Lemma 9.3.
We have ( − τ ) Z { q − θ n − ( − τ ) ≤ ξ ≤ q − θ n − ( − τ ) } e − ξ H ( ξ, τ ) dξ ≤ C ( θ ) ( − τ ) − Z θ θ V ( ρ, τ ) − W + ( ρ, τ ) e µ + ( ρ,τ ) dρ + C ( θ ) Z { q − θ n − ( − τ ) ≤ ξ ≤ q − θ n − ( − τ ) } e − ξ H ( ξ, τ ) dξ, provided that τ ≤ τ ∗ and − τ ∗ is sufficiently large. Proof.
The proof of this lemma is analogous to the proof of Lemma 9.3 in[13]. The first step, as in [13] is to apply Lemma 9.1 with L = q − θ n − ( − τ ) , L = q − θ n − ( − τ ) , L = q − θ n − ( − τ ) , and f ( ξ ) = H ( ξ, τ ). Besidesreplacing θ by θ/ √ n −
2, the remainder of the proof goes through unchanged.
Lemma 9.4.
We have ( − τ ) − Z θθ V ( ρ, τ ) − W + ( ρ, τ ) e µ + ( ρ,τ ) dρ ≤ C ( θ ) Z { q − θ n − ( − τ ) ≤ ξ ≤ q − θ n − ( − τ ) } e − ξ ( H ξ ( ξ, τ ) + H ( ξ, τ ) ) dξ provided that τ ≤ τ ∗ and − τ ∗ is sufficiently large. Proof.
The proof is analogous to the proof of Lemma 9.4 in [13].
Proposition 9.5.
We have sup τ ≤ τ ∗ ( − τ ) Z ττ − Z { q − θ n − ( − τ ′ ) ≤| ξ |≤ q − θ n − ( − τ ′ ) } e − ξ H ( ξ, τ ′ ) dξ dτ ′ ≤ C ( θ ) sup τ ≤ τ ∗ Z ττ − Z R e − ξ ( H C ,ξ ( ξ, τ ′ ) + H C ( ξ, τ ′ ) ) dξ dτ ′ . Proof.
The proof is analogous to the proof of Proposition 9.5 in [13].After these preparations, we now finish the proof of Proposition 6.8. UsingProposition 9.5, we obtainsup τ ≤ τ ∗ ( − τ ) Z ττ − Z { q − θ n − ( − τ ′ ) ≤| ξ |≤ q − θ n − ( − τ ′ ) } e − ξ H ( ξ, τ ′ ) dξ dτ ′ ≤ C ( θ ) sup τ ≤ τ ∗ Z ττ − a ( τ ′ ) dτ ′ + C ( θ ) sup τ ≤ τ ∗ Z ττ − Z R e − ξ ( ˆ H C ,ξ ( ξ, τ ′ ) + ˆ H C ( ξ, τ ′ ) ) dξ dτ ′ . NIQUENESS OF COMPACT ANCIENT SOLUTIONS 47
Combining this estimate with Proposition 6.7 gives( − τ ∗ ) sup τ ≤ τ ∗ Z ττ − Z R e − ξ ( ˆ H C ,ξ ( ξ, τ ′ ) + ˆ H C ( ξ, τ ′ ) ) dξ dτ ′ ≤ C ( θ ) sup τ ≤ τ ∗ Z ττ − a ( τ ′ ) dτ ′ if − τ ∗ is chosen sufficiently large. This completes the proof of Proposition 6.8.10. Analysis of the neutral mode and proof of Proposition 6.9
In this final section, we give the proof of Proposition 6.9.
Lemma 10.1.
We have sup τ ≤ τ ∗ ( − τ ) Z ττ − Z { q − θ n − ( − τ ′ ) ≤| ξ |≤ q − θ n − ( − τ ′ ) } e − ξ H ( ξ, τ ′ ) dξ dτ ′ ≤ C ( θ ) sup τ ≤ τ ∗ Z ττ − a ( τ ′ ) dτ ′ . Proof.
This follows by combining Proposition 6.8 and Proposition 9.5.We next establish an improved version of Lemma 8.5:
Lemma 10.2.
We have ( − τ ∗ ) sup τ ≤ τ ∗ Z ττ − H ξ (0 , τ ′ ) dτ ′ ≤ C ( θ ) sup τ ≤ τ ∗ Z ττ − a ( τ ′ ) dτ ′ . Proof.
The proof of this lemma is analogous to the proof of Lemma 10.2 in [13].After these preparations, we now study the evolution of the function a ( τ ). Usingthe evolution equation ∂∂τ H C = L H C + P k =1 E C ,k , we obtain ddτ a ( τ ) = X k =1 I k ( τ ) , where I k ( τ ) = 116 p n − π Z R e − ξ ( ξ − E C ,k ( ξ, τ ) dξ. In the remainder of this section, we estimate the terms I k ( τ ). Lemma 10.3.
Let δ > be given. If − τ ∗ is sufficiently large (depending on δ ),then sup τ ≤ τ ∗ ( − τ ) Z ττ − | I ( τ ′ ) − ( − τ ′ ) − a ( τ ′ ) | dτ ′ ≤ δ sup τ ≤ τ ∗ (cid:18) Z ττ − a ( τ ′ ) dτ ′ (cid:19) . Proof.
The proof of this lemma is analogous to the proof of Lemma 10.3 in [13].
Lemma 10.4.
Let δ > be given. If − τ ∗ is sufficiently large (depending on δ ),then sup τ ≤ τ ∗ ( − τ ) Z ττ − | I ( τ ′ ) | dτ ′ ≤ δ sup τ ≤ τ ∗ (cid:18) Z ττ − a ( τ ′ ) dτ ′ (cid:19) . Proof.
The proof of this lemma is analogous to the proof of Lemma 10.4 in [13].
Lemma 10.5.
Let δ > be given. If − τ ∗ is sufficiently large (depending on δ ),then sup τ ≤ τ ∗ ( − τ ) Z ττ − | I ( τ ′ ) − ( − τ ′ ) − a ( τ ′ ) | dτ ′ ≤ δ sup τ ≤ τ ∗ (cid:18) Z ττ − a ( τ ′ ) dτ ′ (cid:19) . Proof.
The proof of this lemma is analogous to the proof of Lemma 10.5 in [13].
Lemma 10.6.
Let δ > be given. If − τ ∗ is sufficiently large (depending on δ ),then sup τ ≤ τ ∗ ( − τ ) Z ττ − | I ( τ ′ ) | dτ ′ ≤ δ sup τ ≤ τ ∗ (cid:18) Z ττ − a ( τ ′ ) dτ ′ (cid:19) . Proof.
The proof of this lemma is analogous to the proof of Lemma 10.6 in [13].
Lemma 10.7.
Let δ > be given. If − τ ∗ is sufficiently large (depending on δ ),then sup τ ≤ τ ∗ ( − τ ) Z ττ − | I ( τ ′ ) | dτ ′ ≤ δ sup τ ≤ τ ∗ (cid:18) Z ττ − a ( τ ′ ) dτ ′ (cid:19) . Proof.
The proof of this lemma is analogous to the proof of Lemma 10.7 in [13].
Lemma 10.8.
Let δ > be given. If − τ ∗ is sufficiently large (depending on δ ),then sup τ ≤ τ ∗ ( − τ ) Z ττ − | I ( τ ′ ) | dτ ′ ≤ δ sup τ ≤ τ ∗ (cid:18) Z ττ − a ( τ ′ ) dτ ′ (cid:19) . Proof.
The proof of this lemma is analogous to the proof of Lemma 10.8 in [13].
Lemma 10.9.
Let δ > be given. If − τ ∗ is sufficiently large (depending on δ ),then sup τ ≤ τ ∗ ( − τ ) Z ττ − X k =7 | I k ( τ ′ ) | dτ ′ ≤ δ sup τ ≤ τ ∗ (cid:18) Z ττ − a ( τ ′ ) dτ ′ (cid:19) . Proof.
The proof of this lemma is analogous to the proof of Lemma 10.9 in [13].Proposition 6.9 follows immediately from Lemma 10.3 – Lemma 10.9 togetherwith the identity ddτ a ( τ ) = P k =1 I k ( τ ). NIQUENESS OF COMPACT ANCIENT SOLUTIONS 49
Appendix A. The Bryant soliton
In [16] Bryant showed that up to constant multiples, there is only one complete,steady, rotationally symmetric soliton in dimension three that is not flat. It haspositive sectional curvature. The maximum scalar curvature is equal to 1, and isattained at the center of rotation. The complete soliton can be written in the form g = dz ⊗ dz + B ( z ) g S n , where z is the distance from the center of rotation. Forlarge z , the metric has the following asymptotics: the aperature B ( z ) has leadingorder term p n − z , the orbital sectional curvature K orb has leading order term n − z , and the radial sectional curvature K rad has leading order term z .Sometimes it is more convenient to write the metric in the form Φ( r ) − dr + r g S n , where the function Φ( r ) is defined by Φ( B ( z )) = (cid:0) ddz B ( z ) (cid:1) . The functionΦ( r ) is known to satisfy the equationΦ( r )Φ ′′ ( r ) −
12 Φ ′ ( r ) + n − − Φ( r ) r Φ ′ ( r ) + 2( n − r Φ( r )(1 − Φ( r )) = 0 . The orbital and radial sectional curvatures are given by K orb = r (1 − Φ( r )) and K rad = − r Φ ′ ( r ). It is known that Φ( r ) has the following asymptotics. Near r = 0,Φ is smooth and has the asymptotic expansionΦ( r ) = 1 + b r + o ( r ) , where b is a negative constant (since the curvature is positive). As, r → ∞ , Φ issmooth and has the asymptotic expansionΦ( r ) = c r − + 5 − nn − c r − + o ( r − ) , where c is a positive constant.We will next find (for the convenience of the reader) the exact values of theconstants b and c in the above asymptotics for the Bryant soliton of maximalscalar curvature one .Recall that the scalar curvature is given by R = ( n − n − K orb +2( n − K rad .The maximal scalar curvature is attained at z = 0, at which point K orb = K rad .The maximal scalar curvature being equal to 1 is equivalent to K orb = K rad = n ( n − at z = 0. On the other hand, the asymptotic expansion of Φ( r ) gives K orb = r (1 − Φ( r )) = − b + o (1) as r →
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