On the noncollapsedness of positively curved Type I ancient Ricci flows
aa r X i v : . [ m a t h . DG ] F e b On the noncollapsedness of positively curved Type I ancientRicci flows
Liang Cheng and Yongjia ZhangFebruary 9, 2021
Abstract
In this article, we study complete Type I ancient Ricci flows with positivesectional curvature. Our main results are as follows: in the complete and noncompactcase, all such ancient solutions must be noncollapsed on all scales; in the closed case,if the dimension is even, then all such ancient solutions must be noncollapsed on allscales. This furthermore gives a complete classification for three-dimensionalnoncompact Type I ancient solutions without assuming the noncollapsing condition.
The study of ancient solutions to the Ricci flow, ever since Hamilton had published hisprogram [12], has been an important topic in the field of Ricci flow. Ancient solutionsare Ricci flows whose existing intervals extend to negative infinity. They are of greatimportance because they usually arise as blow-up limits at finite-time singularities of theRicci flow, and to this kind of blow-up limits, a term not improper, “singularity models”,is assigned.Perelman [21] proved that a Ricci flow on a closed manifold cannot become locallycollapsed within finite time. Subsequently one may conclude that every singularity modelmust be κ -noncollapsed on all scales. This precisely means the following. Definition 1.1 ( κ -noncollapsing) . A Ricci flow ( M n , g ( t )) is called κ -noncollapsed on allscales, where κ is a positive number, if for any point ( x, t ) in space-time and any positivescale r , it holds that Vol g ( t ) (cid:0) B g ( t ) ( x, r ) (cid:1) ≥ κr n whenever R ≤ r − on B g ( t ) ( x, r ). Here R stands for the scalar curvatureThe noncollapsing notion defined above is sometimes called the strong noncollapsing inthe literature. The weak noncollapsing notion is defined similarly with only the “whenever R ≤ r − on B g ( t ) ( x, r )” statement replaced by “whenever | Rm | ≤ r − on B g ( t ) ( x, r ) × [ t − r , t ]”. It is known that for an ancient solution with bounded and nonnegative curvatureoperator, the weak noncollapsing is equivalent to the strong noncollapsing condition, with1 INTRODUCTION TO THE MAIN RESULTS κ . The noncollapsing condition which we use throughout this paperis the strong one. We remark that these two notions are not equivalent in general. Forinstance, a closed nonflat and Ricci-flat (static) Ricci flow is weakly noncollapsed but notstrongly noncollapsed.Since, according to Hamilton [12], ancient solutions are critical to the understandingof the singularity formation in the Ricci flow (see, for instance, Perelman’s proof of thecanonical neighborhood theorem [21]), it makes sense to assume the noncollapsing conditionwhen studying ancient solutions. With this assumption, many groundbreaking works aredone, and the most outstanding one is of Perelman [21]. See also [3], [4], [5], and [16], etc.,to list but a few.It is well-known that not all ancient solutions are noncollpased. But what if somefurther conditions are added? Concerning this a question is proposed in [8]: Problem 9.41.
Are nonflat Type I ancient solutions with nonnegativecurvature operator κ -solutions? Recall that an ancient solution (
M, g ( t )) t ∈ ( −∞ ,w ) is called Type I iflim sup t →−∞ | t | (cid:12)(cid:12) Rm g ( t ) (cid:12)(cid:12) < ∞ . Without any further qualification, the answer to the above question is obviously “no”, eitherin the closed case or in the complete and noncompact case. One may immediately think of S n − × S or S n − × R × S as counterexamples. There are also much more sophisticatedcounterexamples constructed. For instance, Fateev [10] constructed an ancient solutionon the Hopf fiber bundle, and Bakas-Kong-Ni [1] generalized this construction to all odd-dimensional spheres; all these ancient solutions are Type I, collapsed, and with positivecurvature operator.In this article, we give a relatively satisfactory answer to Problem 9.41 in [8] as quotedabove. First of all, to rule out the possibility of a compact flat factor, with which theancient solution is always collapsed, we would like to assume that the sectional curvatureis strictly positive. This condition also largely simplifies the geometry in the complete andnoncompact case, since the underlying manifold must be diffeomorphic to the Euclideanspace by the Gromoll-Meyer theorem. Theorem 1.2.
Let ( M n , g ( t )) t ∈ ( −∞ ,w ) , where < w ≤ ∞ , be a complete and noncompactType I ancient solution with positive sectional curvature. Then ( M n , g ( t )) t ∈ ( −∞ , is κ -noncollapsed on all scales for some κ > . The Type I, closed, and collapsed examples constructed in [1] are only in odddimensions, while the cases of even dimensions are yet open. The following theoremshows that there are no such collapsed examples in even dimensions.
INTRODUCTION TO THE MAIN RESULTS Theorem 1.3.
Let ( M n , g ( t )) t ∈ ( −∞ ,w ) , where n is an even number and < w ≤ ∞ , be aclosed Type I ancient solution with positive sectional curvature. Then ( M n , g ( t )) t ∈ ( −∞ , is κ -noncollapsed on all scales for some κ > . The critical observations applied to the proofs of Theorem 1.2 and Theorem 1.3 are someinjectivity radius estimates resulted from the Gromoll-Meyer theorem and the Klingenbergtheorem. These classical theorems imply that an ancient solution as described in Theorem1.2 or Theorem 1.3 has a Type I injectivity radius lower bound. This is sufficient to concludethe existence of an asymptotic shrinker, which in turn implies noncollapsedness; the authorsused a similar idea in [7] to prove the noncollapsedness of a more general type of ancientRicci flows—the locally uniformly Type I ancient solutions. Here we emphasize that theType I injectivity radius lower bound itself does not directly imply the noncollapsedness;see section 2 for more details concerning this point.An immediate application of Theorem 1.2 is the following classification ofthree-dimensional Type I ancient solutions without assuming the noncollapsing condition.This is a generalization of [23] (or [15]). In consequence, all noncompactthree-dimensional Type I ancient solutions must be noncollapsed.
Corollary 1.4.
A three-dimensional noncompact Type I ancient solution must be thestandard cylinder S × R , ( S × R ) / Z , or R P × R . Hence it must also be κ -noncollapsedon all scales for some κ > . In [8], it is asked in Problem 9.40 whether a Type I ancient solution with positivecurvature operator is closed. While we are not yet able to give an answer to this question,the following Corollary rules out one possibility of its asymptotic shrinker—the standardcylinder. This also means that if such an ancient solution did exist, then its geometrycould not be very simple (though topologically it is the Euclidean space).
Corollary 1.5.
A Type I complete and noncompact ancient Ricci flow with nonnegativecurvature operator and positive sectional curvature cannot have the standard cylinder S n − × R as its asymptotic shrinker. Ni [19] classified all closed Type I ancient and noncollapsed Ricci flows withnonnegative curvature operator. It is noted in [1] that, because of the examples therein,the noncollapsing condition in the classification of [19] cannot be dropped. Nonetheless,because the examples in [1] are only in odd dimensions, this conclusion is not decisive ineven dimensions. Indeed, the noncollapsing condition can be dropped in even dimension,at least for the strictly positive curvature operator case.
Corollary 1.6.
An even-dimensional closed Type I ancient Ricci flow satisfying the strict
PIC − curvature condition (and in particular, with positive curvature operator) must be astandard shrinking round space form. THE INJECTIVITY RADIUS S × S m +1 , where the S factor is staticand the S m +1 factor is the standard shrinking sphere.This paper is organized as follows. In section 2 we use the Gromoll-Meyer theoremand the Klingenberg theorem to derive the Type I injectivity radius lower bound for theancient solutions in question. In section 3 we review the fact that Perelman’s entropy andthe Nash entropy converge to the entropy of the asymptotic shrinker. In section 4 we showthat the existence of asymptotic shrinker implies noncollapsedness. In section 5 we proveall the corollaries. Acknowledgment.
The second author would like to thank Professor Jiaping Wang,Professor Lei Ni, and Professor Bennett Chow for many helpful discussions.
The injectivity radius estimates are provided by the following classical theorems of Gromoll-Meyer and Klingenberg.
Proposition 2.1 (Gromoll-Meyer; c.f. Theorem 1.168 in [8]) . Let ( M n , g ) be a completeand noncompact Riemannian manifold satisfying < sec ≤ K , where K is a positivenumber. Then the injectivity radius of ( M n , g ) satisfies inj( g ) ≥ π √ K .
Moreover, ( M n , g ) is diffeomorphic to the standard Euclidean space. Proposition 2.2 (Klingenberg; c.f. Theorem 1.115 in [8]) . Let ( M n , g ) be aneven-dimensional closed orientable manifold satisfying < sec ≤ K , where K is a positivenumber. Then the injectivity radius of ( M n , g ) satisfies inj( g ) ≥ π √ K .
The results above imply that the ancient solutions in question have Type I injectivityradii lower bounds.
Lemma 2.3.
Let ( M n , g ( t )) t ∈ ( −∞ ,w ) be an ancient solution as described in either Theorem1.2 or Theorem 1.3. Then there exists a constant c > , such that the injectivity radius of g ( t ) satisfies inj( g ( t )) ≥ c p | t | , for all t ∈ ( −∞ , . THE ASMPTOTIC SHRINKER Proof.
By the Type I and the positive sectional curvature conditions, we have that thereexists
C >
0, such that 0 < sec( g ( t )) ≤ C | t | , for all t ∈ ( −∞ , κ -noncollapsedness on all scales, since at some point on the manifold, the curvature coulddecay faster than Type I. Nevertheless, this estimate is sufficient for the existence of anasymptotic shrinker, which in turn implies κ -noncollapsedness on all scales. Perelman [21] and Naber [18] proved the existence of the asymptotic shrinker for ancientsolutions under the assumptions of nonnegative curvature operator and of Type I curvaturebound, respectively. They both also assumed the noncollapsing condition. However, itturns out that the only place where they applied this condition was to obtain an injectivityradius lower bound for a blow-down sequence, and they need this injectivity radius lowerbound only at base points to conclude the convergence. This, of course, can be covered byLemma 2.3. A more important fact is that Perelman’s entropy and the Nash entropy onthe ancient solution converge to the entropy of the asymptotic shrinker; see Proposition3.1 below. Let us first review these notions of entropy.Let (
M, g ( t )) t ∈ ( −∞ ,w ) be an ancient solution, where 0 < w ≤ ∞ . Let ( x , t ) ∈ M × ( −∞ , w ) be a fixed point in space-time and u : M × ( −∞ , t ) → R + the fundamentalsolution to the conjugate heat equation − ∂ t u − ∆ u + Ru = 0 based at ( x , t ). We write u as u := (4 πτ ) − n e − f , where τ = t − t ∈ (0 , ∞ ). Then Perelman’s entropy and the Nash entropy based at ( x , t )are respectively defined as W x ,t ( τ ) := Z M (cid:16) τ (cid:0) |∇ f | + R (cid:1) + f − n (cid:17) udg t , N x ,t ( τ ) := Z M f udg t − n . THE ASMPTOTIC SHRINKER W x ,t ( τ ) and N x ,t ( τ ) are increasing in t (or decreasingin τ ). Furthermore, we havelim τ → W x ,t ( τ ) = lim τ → N x ,t ( τ ) = 0 . The following result is already well-established in literature.
Proposition 3.1.
Let ( M, g ( t )) t ∈ ( −∞ ,w ) , where < w ≤ ∞ , be an ancient solution asdescribed in Theorem 1.2 or Theorem 1.3. Let ( x , t ) ∈ M × ( −∞ , w ) be an arbitrarilyfixed point and u := (4 πτ ) − n e − f the conjugate heat kernel based at ( x , t ) . Let τ i ր ∞ be an increasing sequence of positive numbers. Then, the following sequence of tuples n(cid:0) M, g i ( t ) , ( x , − , f i (cid:1) t ∈ ( −∞ , o ∞ i =1 converges, possibly after passing to a subsequence, to the canonical form of a shrinkinggradient Ricci soliton, called the asymptotic shrinker (cid:0) M ∞ , g ∞ ( t ) , ( x ∞ , − , f ∞ (cid:1) t ∈ ( −∞ , , where f ∞ is the potential function, satisfying Ric g ∞ + ∇ f ∞ = 1 − t g ∞ . The convergence g i → g ∞ is in the pointed Cheeger-Gromov-Hamilton [13] sense, andthe convergence f i → f ∞ is in the locally smooth sense. Here g i and f i are obtained bytime-shifting and parabolic scaling as follows g i ( t ) := τ − i g ( τ i t + t ) ,f i ( · , t ) := f ( · , τ i t + t ) . Furthermore, Perelman’s entropy and the Nash entropy converge to the entropy of theasymptotic shrinker. By this we mean Z M (4 π | t | ) − n e − f ∞ dg ∞ = 1 , (3.1)lim τ →∞ W x ,t ( τ ) = lim τ →∞ N x ,t ( τ ) = µ ∞ , where µ ∞ = Z M (cid:16) | t | ( |∇ f ∞ | + R g ∞ ) + f ∞ − n (cid:17) (4 π | t | ) − n e − f ∞ dg ∞ (3.2)= Z M f ∞ (4 π | t | ) − n e − f ∞ dg ∞ − n is a negative constant independent of time t , which we call the entropy of the asymptoticshrinker . THE NASH ENTROPY AND NONCOLLAPSING Proof.
The proof of this proposition can be modified from, for instance, [6] or [22]. Firstof all, the Type I condition implies that there exists a positive number C , such that | Rm g ( t ) | ≤ C | t | (3.3)for all t ∈ ( −∞ , x, t ) ∈ M × ( −∞ , g ( t ) (cid:0) B g ( t ) ( x, p | t | ) (cid:1) ≥ c | t | n , (3.4)where c > { ( x i , t i ) } ∞ i =1 ⊂ M × ( −∞ , M, g i ( t )) t ∈ ( −∞ , , where g i ( t ) := | t i | − g i ( t | t i | ), all satisfy (3.3), butVol g i ( − (cid:0) B g i ( − ( x i , (cid:1) → . (3.5)By Lemma 2.3, we have inj( g i ( − , x i ) ≥ c > . Hence, by [13], the sequence of Ricci flows (cid:8) ( M, g i ( t ) , ( x i , − t ∈ ( −∞ , (cid:9) ∞ i =1 converges,possibly after passing to a subsequence, to a smooth ancient solution( M ∞ , g ∞ ( t ) , ( x ∞ , − t ∈ ( −∞ , . In particular, we haveVol g ∞ ( − (cid:0) B g ∞ ( − ( x ∞ , (cid:1) > , and this contradicts (3.5).One may then follow the proofs in [6] or [22] to conclude this Proposition. Obviously,the noncollapsing condition in these proofs can be replaced by (3.4). It turns out that from Proposition 3.1 it is sufficient to conclude that the ancient solution isnoncollapsed on all scales everywhere. This follows from an observation made in [20]. Let(
M, g ( t )) t ∈ ( −∞ ,w ) , where 0 < w ≤ ∞ , be an ancient solution as describe in Theorem 1.2or Theorem 1.3. One may generally regard ( −∞ , w ) as the maximum existing interval of g ( t ), in which case t = w is the singular time (whether it is infinity or not). The followinglemma says g ( t ) has bounded geometry as long as it is regular. Lemma 4.1.
For all t ∈ ( −∞ , w ) , it holds that sup M (cid:12)(cid:12) Rm g ( t ) (cid:12)(cid:12) < ∞ and inf x ∈ M Vol g ( t ) (cid:0) B g ( t ) ( x, (cid:1) > . THE NASH ENTROPY AND NONCOLLAPSING Proof. If w < ∞ , then by the definition of finite singular time, this is the first instance atwhich g ( t ) has unbounded curvature. If w = ∞ , then this means that g ( t ) has boundedcurvature for all t . The volume lower bound for unit balls follows from a straightforwardvolume distortion estimate.From this time-wise geometry bound, we may conclude the following proposition; thisis a combination of Proposition 3.1 above and Proposition 4.6 in [20], where the secondauthor together with Zilu Ma proved (as a consequence of Corollary 5.11 in [2]) that on anancient solution with bounded geometry on each time-slice, Perelman’s entropies and theNash entropies based at all points converge to the same number as the time approachesnegative infinity. Proposition 4.2.
Let ( M, g ( t )) t ∈ ( −∞ ,w ) , where < w ≤ ∞ , be an ancient solution asdescribe in Theorem 1.2 or Theorem 1.3. Then for all ( x, t ) ∈ M × ( −∞ , w ) , the followingholds lim τ →∞ W x,t ( τ ) = lim τ →∞ N x,t ( τ ) = µ ∞ , where µ ∞ is the entropy of any one of asymptotic shrinkers based at any point (for theirentropies are all equal). In particular, we have N x,t ( τ ) ≥ µ ∞ (4.1) for all ( x, t ) ∈ M × ( −∞ , w ) and for all τ > . Though in the original proof of Perelman [21], he uses the bound of the µ functionalto show the no local noncollapsing theorem, yet the second author showed that theboundedness of the Nash entropy could also be used to prove the noncollapsedness at itsbase point; this is the following Proposition. Proposition 4.3 (Theorem 6.1 in [2]) . Let ( M, g ( t )) be a Ricci flow and ( x, t ) a point inthe space time. Let r be a positive scale such that [ t − r , t ] is in the existing interval and R ≤ r − on B g ( t ) ( x, r ) . Then, it holds that Vol g ( t ) (cid:0) B g ( t ) ( x, r ) (cid:1) ≥ c exp (cid:0) N x,t ( r ) (cid:1) r n . Here c is a positive constant depending only on the dimension.Remark: Though Bamler [2] proved the above result for Ricci flows on closed manifolds,yet one may check the proof of Theorem 6.1 in [2] and easily verify its validity for Ricciflows with bounded geometry on each time-slice; one may need to apply Theorem 4.4of [20] in this verification. Fortunately, all the Ricci flows we work with in this papersatisfy this condition. On the other hand, the second author proved that bounded Nashentropy implies weak noncollapsing, and this proof does not need bounded geometry oneach time-slice; see Proposition 3.3 in [23].
APPLICATIONS Proof of Theorem 1.2 and Theorem 1.3 .
Let (
M, g ( t )) t ∈ ( −∞ ,w ) be an ancient solution asdescribed in either Theorem 1.2 or Theorem 1.3. Let ( x, t ) ∈ M × ( −∞ , w ) be an arbitraryspace-time point, and r any scale that satisfies R ≤ r − on B g ( t ) ( x, r ) . Since, by Proposition 4.2, we have N x,t ( r ) ≥ µ ∞ ∈ ( −∞ , , where µ ∞ is the entropy of one of the asymptotic shrinkers, it then follows from Proposition4.3 that Vol g ( t ) (cid:0) B g ( t ) ( x, r ) (cid:1) ≥ ce µ ∞ r n ;this finishes the proof. In this section, we prove all the corollaries proposed in the introduction section.
Proof of Corollary 1.4.
Let ( M , g ( t )) be a three-dimensional noncompact Type I ancientsolution. By Chen [9], g ( t ) has nonnegative sectional curvature everywhere. If its sectionalcurvature is strictly positive, then, by Theorem 1.2, it is also noncollapsed. It follows from[24] (or [15]) that such ancient solution does not exist.If g ( t ) ever attains zero sectional curvature somewhere, then by the strong maximumprinciple of Hamilton [11], ( M , g ( t )) splits locally and hence its universal cover must bethe standard shrinking cylinder S × R . Furthermore, the only noncompact quotients of S × R are the Z quotients. The reason is that the projection of a group action on the R factor can only be either the reflection or the identity—if it is ever a translation, thenthis action will generate an infinity group action on S × R , and the quotient space mustbe compact. This finishes the proof of the corollary. Proof of Corollary 1.5.
We argue by contradiction. Assume one of the asymptoticshrinkers is the standard cylinder S n − × R , then all the results obtained in section 3 of[17] can be applied to this ancient solution. In particular, it satisfies a canonicalneighborhood theorem and hence always has a non-neck-like region at each time (c.f.Theorem 1.3 in [17]), it always splits as S n − × R at space infinity (c.f. Proposition 3.9 in[17]), it satisfies the neck stability theorem of Kleiner-Lott [14] (c.f. Theorem 3.11 in[17]), and all such ancient solutions form a compact space (c.f. Theorem 1.2 in [17]).Knowing all these facts, one may follow the arguments in [24] line by line to concludethat such an ancient solution does not exist. EFERENCES Proof of Corollary 1.6.
The PIC − κ -noncollapsed on all scales for some κ >
0. The conclusion then follows from Corollary0.4 in [19].
References [1] I. Bakas, S. Kong and L. Ni,
Ancient solutions of Ricci flow on spheres and generalizedHopf fibrations.
J. Reine Angew. Math., 663 (2012), 209-248.[2] R. Bamler,
Entropy and heat kernel bounds on a Ricci flow background . arXiv preprintarXiv:2008.07093, 2020.[3] S. Brendle,
Rotational symmetry of self-similar solutions to the Ricci flow . InventionesMathematicae, 2013, 194(3): 731-764.[4] S. Brendle,
Rotational symmetry of Ricci solitons in higher dimensions . Journal ofDifferential Geometry, 2014, 97(2): 191-214.[5] S. Brendle,
Ancient solutions to the Ricci flow in dimension
3. Acta Mathematica,2020, 225(1): 1-102.[6] X. Cao, Q S. Zhang,
The conjugate heat equation and ancient solutions of the Ricciflow . Advances in Mathematics, 2011, 228(5): 2891-2919.[7] L. Cheng, Y. Zhang,
Perelman-type no breather theorem for noncompact Ricci flows .arXiv preprint arXiv:2011.14973, 2020.[8] B. Chow, P. Lu, L. Ni,
Hamilton’s Ricci flow . American Mathematical Soc., 2006.[9] B-L. Chen,
Strong uniqueness of the Ricci flow . J. Differential Geom. 82 (2009), no.2, 363-382.[10] V. A. Fateev,
The sigma model (dual) representation for a two-parameter family ofintegrable quantum field theories.
Nuclear Phys. B 473(1996), no. 3, 509-538.[11] R. Hamilton,
Four-manifolds with positive curvature operator . Journal of DifferentialGeometry, 1986, 24(2): 153-179.[12] R. Hamilton,
The formations of singularities in the Ricci Flow . Surveys in DifferentialGeometry. 1993, 2(1): 7-136.[13] R. Hamilton,
A compactness property for solutions of the Ricci flow . American journalof mathematics, 1995, 117(3): 545-572.
EFERENCES
Singular Ricci flows I . Acta Mathematica, 2017, 219(1): 65-134.[15] M. Hallgren,
The Nonexistence of Noncompact Type-I Ancient 3-d κ -Solutions of RicciFlow with Positive Curvature . arXiv preprint arXiv:1801.08643, 2018.[16] Y. Li, Ancient solutions to the K¨ahler Ricci flow. arXiv preprint arXiv:2008.06951,2020.[17] X. Li, Y. Zhang,
Ancient solutions to the Ricci flow in higher dimensions .Communications in Geometry and Analysis, to appear.[18] A. Naber,
Noncompact shrinking four solitons with nonnegative curvature . Journal f¨urdie reine und angewandte Mathematik, 2010, 2010(645): 125-153.[19] L. Ni,
Closed type I ancient solutions to Ricci flow . Recent Advances in GeometricAnalysis, ALM, 2009, 11: 147-150.[20] Z. Ma, Y. Zhang,
Perelman’s entropy on ancient Ricci flows . arXiv preprintarXiv:2101.01233, 2021.[21] G. Perelman,
The entropy formula for the Ricci flow and its geometric applications. arXiv preprint math/0211159, 2002.[22] G. Xu,
An equation linking W -entropy with reduced volume . Journal f¨ur die reine undangewandte Mathematik, 2017, 2017(727): 49-67.[23] Y. Zhang, Entropy, noncollapsing, and a gap theorem for ancient solutions to the Ricciflow . Communications in Geometry and Analysis, to appear.[24] Y. Zhang,
On three-dimensional type I κ -solutions to the Ricci flow . Proceedings ofthe American Mathematical Society, 2018, 146(11): 4899-4903.[25] Y. Zhang, On the equivalence between noncollapsing and bounded entropy for ancientsolutions to the Ricci flow . Journal f¨ur die reine und angewandte Mathematik, 2020,2020(762): 35-51.School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences,Central China Normal University, Wuhan, 430079, P.R.ChinaE-mail address: [email protected]