The local entropy along Ricci flow---Part A: the no-local-collapsing theorems
aa r X i v : . [ m a t h . DG ] J un The local entropy along Ricci flow —Part A: the no-local-collapsing theoremsBing Wang
Abstract
We localize the entropy functionals of G. Perelman and generalize his no-local-collapsingtheorem and pseudo-locality theorem. Our generalization is technically inspired by furtherdevelopment of Li-Yau estimate along the Ricci flow. It can be used to show the Gromov-Hausdor ff convergence of the K¨ahler Ricci flow on each minimal projective manifold of gen-eral type. Contents ff ective monotonicity formulas for local functionals 336 General reduced distance and volume density function 407 A generalization of the no-local-collapsing theorem 508 K¨ahler Ricci flow on minimal models of general type 57 A Ricci flow solution { ( M m , g ( t )) , t ∈ I ⊂ R } is a smooth family of metrics satisfying the evolutionequation ∂∂ t g = − Rc , (1.1)1here M m is a complete manifold of dimension m . For simplicity of our discussion, we alsoassume that sup M | Rm | g ( t ) < ∞ for each time t ∈ I . This condition holds automatically if M is aclosed manifold. It is very often to put an extra term on the right hand side of (1.1) to obtain thefollowing rescaled Ricci flow ∂∂ t g = − { Rc + λ ( t ) g } , (1.2)where λ ( t ) is a function depending only on time. Typically, λ ( t ) is chosen as the average of thescalar curvature, i.e., m > Rdv or some fixed constant independent of time. In the case that M isclosed and λ ( t ) = m > Rdv , the flow (1.2) is also called the normalized Ricci flow.The Ricci flow equations (1.1) and (1.2) were introduced by R. Hamilton in his seminal pa-per [21]. Starting from a positive Ricci curvature metric on a 3-manifold, he showed that thenormalized Ricci flow exists forever and converges to a space form metric. Hamilton developedthe maximum principle for tensors to study the Ricci flow initiated from some metric with positivecurvature conditions. Along this direction, there are various convergence theorems of the flow(1.2), proved by G. Huisken [25], R. Hamilton [22], Bohm-Wilking [1], etc. Such developmentsfinally lead to the sphere theorem of Brendle-Schoen [2], which asserts that starting from a man-ifold whose Riemannian curvature is quater-pinched, the normalized Ricci flow (1.2) convergesto a round metric. For metrics without positive curvature condition, the study of Ricci flow wasprofoundly a ff ected by the celebrated work of G. Perelman [33]. He introduced new tools, i.e., theentropy functionals µ , ν , the reduced distance and the reduced volume, to investigate the behav-ior of the Ricci flow. Perelman’s new input enabled him to revive Hamilton’s program of Ricciflow with surgery, leading to solutions of the Poincar´e conjecture and Thurston’s geometrizationconjecture(c.f. [33], [34], [35]).In the general theory of the Ricci flow developed by Perelman in [33], the entropy functionals µ and ν are of essential importance. Perelman discovered the monotonicity of such functionals andapplied them to prove the no-local-collapsing theorem(c.f. Theorem 4.1 of [33]), which removesthe stumbling block for Hamilton’s program of Ricci flow with surgery. By delicately using suchmonotonicity, he further proved the pseudo-locality theorem(c.f. Theorem 10.1 and Theorem 10.3of [33]), which claims that the Ricci flow can not quickly turn an almost Euclidean region intoa very curved one, no matter what happens far away. Besides the functionals, Perelman alsointroduced the reduced distance and reduced volume. In terms of them, the Ricci flow space-timeadmits a remarkable comparison geometry picture(c.f. Section 6 and Section 7 of [33]), which isthe foundation of his “local”-version of the no-local-collapsing theorem(c.f. Theorem 8.2 of [33]).Each of the tools has its own advantages and shortcomings. The functionals µ and ν have theadvantage that their definitions only require the information for each time slice ( M , g ( t )) of theflow. However, they are global invariants of the underlying manifold ( M , g ( t )). It is not convenientto apply them to study the local behavior around a given point x . Correspondingly, the reducedvolume and the reduced distance reflect the natural comparison geometry picture of the space-time.Around a base point ( x , t ), the reduced volume and the reduced distance are closely related to the“local” geometry of ( x , t ). Unfortunately, it is the space-time “local”, rather than the Riemanniangeometry “local” that is concerned by the reduced volume and reduced geodesic. In order toapply them, some extra conditions of the space-time neighborhood of ( x , t ) are usually required.However, such strong requirement of space-time is hard to fulfill. Therefore, it is desirable to2ave some new tools to balance the advantages of the reduced volume, the reduced distance andthe entropy functionals. In this paper, we localize the functionals µ and ν for this purpose(c.f.Section 2). On one hand, our local functionals enjoy similar geometric pictures of the reduceddistance and the reduced volume. On the other hand, they only require local information of singletime-slices of the underlying flow. It turns out that the localized functionals are convenient tools.We shall apply them to generalize the no-local-collapsing theorem and the pseudo-locality theoremof Perelman [33] in this paper and the forthcoming paper [50].Our study is motivated by the comparison geometry picture of the Ricci flow space-time. Let( M m , g ) be a complete Ricci-flat manifold, x is a point on M such that d ( x , x ) < A . Supposethe ball B ( x , r ) is A − − non-collapsed, i.e., r − m | B ( x , r ) | ≥ A − , can we obtain uniform non-collapsing for the ball B ( x , r ), whenever 0 < r < r and d ( x , x ) < Ar ? This question can beanswered easily by applying triangle inequalities and Bishop-Gromov volume comparison theo-rems. In particular, there exists a κ = κ ( m , A ) ≥ − m A − m − (c.f. Remark 7.4) such that B ( x , r ) is κ -non-collapsed, i.e., r − m | B ( x , r ) | ≥ κ . Consequently, there is an estimate of propagation speed ofnon-collapsing constant on the manifold M . This is easily illustrated by Figure 1. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) B ( x , r ) B ( x , r )Figure 1: Propagation of non-collapsing on Ricci-flat manifoldsSimilar to the discussions in Section 2.7 of Chen-Wang [7], we now regard ( M , g ) as a trivialspace-time { ( M , g ( t )) , −∞ < t < ∞} such that g ( t ) ≡ g . Clearly, g ( t ) is a static Ricci flow solutionby the Ricci-flatness of g . Then the above estimate can be explained as the propagation of volumenon-collapsing constant on the space-time(c.f. Figure 2). However, in a more intrinsic way, itcan also be interpreted as the propagation of non-collapsing constant of Perelman’s reduced vol-ume(c.f. Section 7 of Perelman [33] or Section 6 of the current paper for a brief discussion of thereduced volume and the reduced distance). Recall that on the Ricci flat space-time, Perelman’sreduced volume(c.f. equations (2.85) of Chen-Wang [7]) has a special formula V (( x , t ) , r ) = (4 π ) − m r − m Z M e − d y , x )4 r dv y , B g ( t ) ( x , r ). On a general Ricci flow solution, the reduced vol-ume is also well-defined and has monotonicity with respect to the parameter r , if one replace d ( y , x )4 r in the above formula by the reduced distance l (( x , t ) , ( y , t − r )). Therefore, via the com-parison geometry of Bishop-Gromov type, one can regard a Ricci-flow as an “intrinsic-Ricci-flat”space-time. Consequently, the above reduced volume interpretation of non-collapsing propagationcan be easily generalized to general Ricci flows, as done by Perelman in Section 8 of [33]. B ( x , r ) B ( x , r ) Mt x xt = r t = x , t ), ratherthan the scalar curvature estimate of a single time slice t . We shall show that such strong require-ment of space-time geometry is not necessary by the following version of no-local-collapsingtheorem(c.f. Figure 3). Theorem 1.1 ( Improved version of no-local-collapsing ) . For every A > there exists κ = κ ( m , A ) > with the following property. Suppose { ( M m , g ( t )) , ≤ t ≤ r } is a solution of Ricciflow (1.1) such thatr | Rm | ( x , t ) ≤ m − , ∀ x ∈ B g (0) ( x , r ) , ≤ t ≤ r ; r − m (cid:12)(cid:12)(cid:12) B g (0) ( x , r ) (cid:12)(cid:12)(cid:12) dv g (0) ≥ A − . (1.3) Then we have r − m (cid:12)(cid:12)(cid:12) B g ( t ) ( x , r ) (cid:12)(cid:12)(cid:12) dv g ( t ) ≥ κ (1.4) whenever A − r ≤ t ≤ r , < r ≤ r , and B g ( t ) ( x , r ) ⊂ B g ( t ) ( x , Ar ) is a geodesic ball satisfyingr R ( · , t ) ≤ . In Theorem 1.1, we replace the requirement of space-time condition by a time-slice conditionaround ( x , t ). Namely, we only need R ( · , t ) ≤ r − in the ball B g ( t ) ( x , r ) to conclude the non-collapsing of B g ( t ) ( x , r ) whenever ( x , t ) is not very far away from ( x , x , g (0) should be more natural. The quest of such a natural conditionleads us to develop the pseudo-locality theorems, which unite and improve the pseudo-localitytheorems(c.f. Theorem 10.1 and Corollary 10.3 of [33]) of Perelman and a similar pseudo-locality4heorem(c.f. Proposition 3.1 and Theorem 3.1 of [44]) of Tian and the author. The details of thepseudo-locality theorems will be discussed in the second paper of this series [50]. 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0) ( x , t )Figure 3: Propagation of non-collapsing when base neighborhood has bounded geometryIn order to prove Theorem 1.1, we need to revisit the work of Perelman [33] and reorganize itfrom its starting point: the functionals µ and ν . We start from localizing them to each boundeddomain Ω ⊂ M and study the properties of the minimizer functions. Recall that on a Riemannianmanifold ( M , g ), for each positive number τ , the functional µ ( g , τ ) is defined as the infimum of W ( g , ϕ, τ ) = − m − m πτ ) + Z Ω n τ (cid:16) R ϕ + |∇ ϕ | (cid:17) − ϕ log ϕ o dv among all smooth nonnegative functions ϕ satisfying R M ϕ dv =
1. Now we let the infimum beachieved among all the ϕ ’s satisfying an extra condition ϕ ∈ C ∞ c ( Ω ). The infimum is denoted by µ ( Ω , g , τ ), which is a functional of the domain Ω . Then we set ν ( Ω , g , τ ) = inf < s ≤ τ µ ( Ω , g , s ). Wecall µ ( Ω , g , τ ) and ν ( Ω , g , τ ) as the local functionals, or the functionals localized on Ω . Althoughnot written down explicitly, it is implied by the work of Perelman(c.f. [33]) that both µ ( Ω , g , τ ) and ν ( Ω , g , τ ) reveal information of the underlying geometry of ( Ω , g ). In particular, if Ω is a geodesicball of radius r and the scalar curvature satisfies R ≤ r − in Ω , then the uniform lower bound of ν ( Ω , g , r ) implies a uniform lower bound of the volume ratio of Ω (c.f. Theorem 3.3, Remark 3.4and Remark 5.8). Our new observation is that the minimizer function ϕ of µ ( Ω , g , τ ) carries morecomplete information than that of the value µ ( Ω , g , τ ). We could study the local geometry viathe study of the the local minimizer functions, in particular under the Ricci flow evolution. Forexample, in the Ricci flow space-time, we focus our attention on a bounded domain Ω and time T ,and choose ϕ ( T ) as the minimizer function of µ ( Ω , g ( T ) , τ T ) for some τ T >
0. Let u ( T ) = ϕ ( T )and u be the solution of the conjugate heat equation (cid:3) ∗ u = ( ∂ t − ∆ + R ) u =
0. Similar to theHarnack inequality of Perelman, we have (c.f. Theorem 4.2 for full details) v = { ( τ T + T − t )( R + ∆ f − |∇ f | ) + f − m − µ } u ≤ , (1.5)where µ is the value of µ ( Ω , g ( T ) , τ T ) and f = − log u − m log { π ( τ T + T − t ) } . The proof of (1.5)follows from Section 9 of Perelman [33] intuitively. However, we need to deal with extra technicaldi ffi culties and regularity issues caused by ∂ Ω . By careful heat kernel estimates, we show that theintuitive argument can actually be made rigorous. It is not hard to see(c.f. Remark 4.10) that (1.5)is a generalization of Perelman’s Harnack inequality.5ote that (1.5) provides a bridge between di ff erent domains at di ff erent time slices, via thestudy of the evolution of u . In fact, for each domain Ω ⊂ M , we can study the restriction of u on Ω to obtain the relationship between µ ( Ω , g (0) , τ T + T ) and µ ( Ω , g ( T ) , τ T ). Suppose Ω ′ ⋐ Ω and h is a cuto ff function which vanishes outside Ω and equals 1 inside Ω ′ , the relationship canbe calculated explicitly(c.f. (5.3) in Theorem 5.1): µ ( Ω , g (0) , τ T + T ) − µ ( Ω , g ( T ) , τ T ) ≤ (cid:16) τ C h + e − (cid:17) · R Ω \ Ω ′ u R Ω ′ u (1.6)where C h = sup Ω (cid:12)(cid:12)(cid:12) ∇ √ h (cid:12)(cid:12)(cid:12) . It can be chosen as 4 r − if Ω is a ball of radius 2 r . As R Ω u ≤ R Ω ′ u . Once we obtainuniform lower bound of R Ω ′ u which we denote by c u , (1.6) implies (c.f. (5.10) in Theorem 5.2)that ν ( Ω , g (0) , τ T + T ) − ν ( Ω , g ( T ) , τ T ) ≤ (cid:16) τ C h + e − (cid:17) (cid:16) c − u − (cid:17) . (1.7)Therefore, ν ( Ω , g ( T ) , τ T ) can be bounded from below by some number determined by c u and ν ( Ω , g (0) , τ T + T ).Now the proof of Theorem 1.1 is clear. Without loss of generality, we set t = r = r ∈ (0 ,
1) and τ = r . We set Ω = B g (1) ( x , r ) and Ω = B g (0) ( x , . u ≥ (4 π [ T − t ]) − m e − l , we obtain auniform lower bound of c u . On the other hand, the uniformly bounded local geometry around x provides a uniform lower bound of ν ( Ω , g (0) , τ + ν ( Ω , g (1) , τ ). However, the lower bound of ν ( Ω , g (1) , τ ) explicitly implies a non-collapsingconstant(c.f. Theorem 3.3) if R ≤ r − inside Ω . This finishes the proof.Using the same idea, we indeed have a formula much more precise than the one stated in The-orem 1.1, under much weaker conditions(c.f. (7.3) in Theorem 7.2). Translating the informationcontained in ν to volume ratios, we obtain an explicit formula(c.f. (7.13) in Theorem 7.3 andRemark 7.4) of the propagation speed of the non-collapsing constant. We believe such precise for-mulas will be useful in the further study of the Ricci flow(collapsing case in particular), althoughthe rough estimate in Theorem 1.1 is enough for many of our applications.If one would like to sacrifice preciseness, there is an alternative shorter proof of Theorem 1.1.Actually, it follows from (1.5) that µ ≥ ( τ T + T − t )( R + ∆ f − |∇ f | ) + f − m . (1.8)The uniformly bounded geometry around ( x ,
0) and the uniform lower bound of u implies a uni-form two sided bound of u around ( x , . x , .
1) are bounded by Shi’s estimate(c.f. [39], or chapter 6 of [9]).Then the relationship u = (4 π [ τ T + T − t ]) − m e − f implies that the right hand side of (1.8) is uniformlybounded from below.Although not natural in general Riemannian setting, the conditions (1.3) in Theorem 1.1 areavailable whenever we study specific types of K¨ahler Ricci flow, up to an elementary parabolic6escaling. In fact, (1.3) can often be obtained by regularity theory of parabolic Monge-Amp`ereequation, which is deeply a ff ected the fundamental work of Yau [52]. Therefore, it seems reason-able to believe that Theorem 1.1 will be useful in the study of general K¨ahler Ricci flow. As anevidence, we apply Theorem 1.1 to show the following convergence theorem. Theorem 1.2 ( Convergence of the K¨ahler Ricci flow ) . Let X be a minimal projective manifoldof general type. Starting from a K ¨ahler metric g , the flow solution of ∂ t g = − { Rc + g } converges to the unique singular K ¨ahler-Einstein metric ω KE on the canonical model X can in theGromov-Hausdor ff topology as t → ∞ . Theorem 1.2 confirms a long-standing conjecture(c.f. Conjecture 8.1), whose low dimensionalcases were confirmed by Guo-Song-Weinkove [19] in dimension 2, and Tian-Zhang [46] in di-mension ≤ ff erent methods. The key for the proof of Theorem 1.2 is to develop a uniform κ -non-collapsing estimate and a uniform diameter bound along the flow. Similar estimates alongthe Fano K¨ahler Ricci flow were discovered by Perelman(c.f. Remark 8.3). It is interesting toobserve that the statement of Theorem 1.2 mirrors that of the Fano K¨ahler Ricci flow(c.f. Re-mark 8.4). We provide the proof of Theorem 1.2 and necessary background and references inSection 8.This paper is organized as follows. In Section 2, we localize Perelman’s functionals µ and ν , together with other closely related functionals ¯ µ , ¯ ν . We discuss the basic properties of the lo-calized functionals and the minimizer functions. In Section 3, we study the relationships among ν , ¯ ν and the volume ratios. In Section 4, we generalize the Harnack inequality of Perelman andprovide an alternative approach to understand the meaning of Perelman’s reduced distance, via Li-Yau’s Harnack estimate. In Section 5, we derive e ff ective monotonicity formulas for local µ and ν -functionals and consequently deduce one version of no-local-collapsing theorem. In Section 6,we generalize the reduced distance and the reduced volume density functions to be defined froma probability measure and develop e ff ective lower bound of the reduced volume density function.In Section 7, by combining the generalized reduced volume density function estimate with thegeneralized Harnack inequality, we can estimate the propagation speed of the local ν -functionals.Such an estimate in turn implies a strong version of the no-local-collapsing theorem, i.e. Theo-rem 1.1. Finally, in Section 8, we show the uniform κ -non-collapsing estimate and the uniformdiameter bound along each K¨ahler Ricci flow on a minimal projective manifold of general typeand consequently prove Theorem 1.2. Acknowledgements : This paper is partially supported by NSF grant DMS-1510401. Theauthor would also like to acknowledge the invitation to MSRI Berkeley in spring 2016 sup-ported by NSF grant DMS-1440140. Part of this work was done while the author was visitingAMSS(Academy of Mathematics and Systems Science) in Beijing and USTC(University of Sci-ence and Technology of China) in Hefei, during the summer of 2016. He wishes to thank AMSSand USTC for their hospitality. He would also like to thank Mikhail Feldman, Je ff Viaclovsky, LuWang and Shaosai Huang for helpful discussions.7
Localization of Perelman’s functionals
Let ( M , g ) be a complete Riemannian manifold of dimension m , and Ω be a connected, open subsetof M with smooth boundary. Then we can regard ( Ω , ∂ Ω , g ) as a smooth manifold with boundary.Let a be a smooth function on ¯ Ω , and τ be a positive constant. Then we define S ( Ω ) ≔ ( ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϕ ∈ W , ( Ω ) , ϕ ≥ , Z Ω ϕ dv = ) , (2.1) W ( a ) ( Ω , g , ϕ, τ ) ≔ − m − m πτ ) + Z Ω n τ (cid:16) a ϕ + |∇ ϕ | (cid:17) − ϕ log ϕ o dv , (2.2) µ ( a ) ( Ω , g , τ ) ≔ inf ϕ ∈ S ( Ω ) W ( a ) ( Ω , g , ϕ, τ ) , (2.3) ν ( a ) ( Ω , g , τ ) ≔ inf s ∈ (0 ,τ ] µ ( a ) ( Ω , g , s ) , (2.4) ν ( a ) ( Ω , g ) ≔ inf τ ∈ (0 , ∞ ) µ ( a ) ( Ω , g , τ ) . (2.5)By the result of O. Rothaus(c.f. [36]), we know that for each smooth function a and positivenumber τ > µ ( a ) ( Ω , g , τ ) is achieved by a function ϕ ∈ W , ( Ω ) whenever Ω is bounded.Moreover, ϕ is positive and smooth in Ω , and ϕ satisfies the following Euler-Lagrangian equation − τ ∆ ϕ + τ a ϕ − ϕ log ϕ − (cid:18) µ ( a ) + m + m πτ ) (cid:19) ϕ = . (2.6)We call ϕ as the minimizer function of µ ( a ) ( Ω , g , τ ). Since in our setting, ∂ Ω is smooth, we cansay more about the boundary behavior of ϕ . Note that if a = R , Ω = M and let ϕ = (4 πτ ) − m e − f ,then we have W ( a ) ( Ω , g , ϕ, τ ) = Z M n τ ( R + |∇ f | ) + f − m o (4 πτ ) − m e − f dv , which is the functional introduced by Perelman(c.f. (3.1) of [33]). If a = Ω = M , the corre-sponding functionals are the ones studied by L. Ni(c.f. (1.2) and (1.7) of [32]).The functional µ ( a ) ( Ω , g , τ ) reveals the information of the Riemannian geometry of ( Ω , ∂ Ω , g ),by choosing di ff erent a . If a =
0, then µ (0) ( Ω , g , τ ) is exactly the classical Logarithmic Sobolevconstant(c.f. [20], [12]). In [33], Perelman choose a as the scalar curvature function R and discov-ered the monotonicity of the functional µ ( R ) ( M , g ( t ) , T − t ), which plays a foundational role in hiscelebrated resolution of Poincar´e conjecture and the geometrization conjecture(c.f. [33], [34], [35]).The cases a = R are the most important cases for the application of µ ( a ) in the studyof the Ricci flow. However, most discussion of µ ( a ) ( Ω , g , τ ) in the current literature of Ricci flowfocuses on the closed manifold case. In this paper, we shall pay our attention to manifolds withboundary. For simplicity of notation, we define µ ≔ µ ( R ) , ν ≔ ν ( R ) ; (2.7)¯ µ ≔ µ (0) , ¯ ν ≔ ν (0) . (2.8)We list some elementary properties of the functionals µ ( a ) , ν ( a ) . For simplicity, the metric g willnot appear explicitly when it is clear in the context.8 roposition 2.1 ( Monotonicity induced by inclusion ) . Suppose Ω , Ω are bounded domains ofM satisfying Ω ( Ω . Then we have µ ( a ) ( Ω , τ ) > µ ( a ) ( Ω , τ ) , (2.9) ν ( a ) ( Ω , τ ) ≥ ν ( a ) ( Ω , τ ) . (2.10) Proof.
Let ϕ be the minimizer function of µ ( a ) ( Ω , τ ). Then ϕ is positive in Ω and ϕ ∈ W , ( Ω )(c.f. Theorem on page 116 of Rothaus [36]). It follows from the definition that µ ( a ) ( Ω , τ ) = W ( a ) ( Ω , ϕ , τ ) . On the other hand, every minimizer function of µ ( a ) ( Ω , τ ) is positive on Ω . Since ϕ is supportedon Ω , a strict subdomain of Ω , we know ϕ cannot be a minimizer function of µ ( a ) ( Ω , τ ).Therefore, we have W ( a ) ( Ω , ϕ , τ ) > µ ( a ) ( Ω , τ ) . Therefore, (2.9) follows from the combination of the above two inequalities. Clearly, (2.10) isimplied by (2.9) for each s ∈ (0 , τ ] and the definition equation (2.4). (cid:3) Proposition 2.2 ( Non-positivity of ν ( a ) ) . For each bounded domain Ω ⊂ M, we have ν ( a ) ( Ω , τ ) ≤ . (2.11) Proof.
Recall that ν ( a ) ( Ω , τ ) = inf s ∈ (0 ,τ ] µ ( a ) ( Ω , τ ). For (2.11), it su ffi ces to show thatlim s → + µ ( a ) ( Ω , τ ) ≤ . Fix x as an interior point of Ω . We choose ǫ > d ( x , ∂ Ω ) > ǫ . Let η be a cuto ff function such that η ≡ B ( x , ǫ ) and vanishes outside B ( x , ǫ ). Motivated by thestandard heat kernel expression on Euclidean space, we define ϕ s ≔ a s · η · (4 π s ) − m e − d s , where a s is a normalization constant such that R Ω ϕ s dv =
1. Clearly, we have a s → s → + .Then it follows from the definition and direct calculation thatlim s → + µ ( a ) ( Ω , τ ) ≤ lim s → + W ( a ) ( Ω , g , ϕ s , s ) = , which implies (2.11). (cid:3)
9n contrast to (2.11) of Proposition 2.2, µ ( a ) could be positive. For example, we can let a ≡ Ω , g ) be the unit ball in the standard Euclidean space ( R m , g Euc ). For each τ >
0, it followsfrom Proposition 2.1 that µ ( a ) ( Ω , τ ) > µ ( a ) ( R m , τ ) = . Since the above inequality holds for each positive τ , the above inequality implies the non-negativityof ν ( a ) ( Ω , τ ), which together with (2.11) yields that ν ( a ) ( Ω , τ ) = Ω and positive number τ . In particular, letting Ω and Ω be the balls in R m centered at the origin and have radii 1 and 2 respectively, we have ν ( a ) ( Ω , τ ) = ν ( a ) ( Ω , τ ) = . Therefore, the inequality (2.10) in Proposition 2.1 cannot be improved to a strict one in general.This example also shows that there may exist many Ω ’s such that ν ( a ) ( Ω , τ ) =
0. However, if Ω isallowed to be a complete manifold and a = R , then there are more rigidities(c.f. Proposition 4.9).For unbounded domains, we have the following result. Proposition 2.3 ( Continuity of µ ( a ) ) . Suppose D is a possibly unbounded domain of M with anexhaustion D = ∪ ∞ i = Ω i by bounded domains. In other words, we have Ω ⊂ Ω ⊂ · · · ⊂ Ω k ⊂ · · · ⊂ Dand each Ω i is a bounded domain. Then for each τ > we have µ ( a ) ( D , τ ) = lim i →∞ µ ( a ) ( Ω i , τ ) . (2.12) Proof.
Since Ω i ⊂ D for each i , it follows from the definition that µ ( a ) ( D , τ ) ≤ µ ( a ) ( Ω i , τ ) for each i . Moreover, n µ ( a ) ( Ω i , τ ) o ∞ i = is a decreasing sequence. Consequently, we have µ ( a ) ( D , τ ) ≤ lim i →∞ µ ( a ) ( Ω i , τ ) . (2.13)On the other hand, we can find a sequence of smooth functions ϕ i with compact support in D suchthat µ ( a ) ( D , τ ) = lim i →∞ W ( a ) ( D , ϕ i , τ ) . As D = ∪ ∞ i = Ω i , we can assume the support of ϕ i is contained in Ω k i for some k i . Therefore, wehave W ( a ) ( D , ϕ i , τ ) = W ( a ) ( Ω k i , ϕ i , τ ) ≥ µ ( a ) ( Ω k i , τ ) . It follows from the combination of the previous two steps that µ ( a ) ( D , τ ) ≥ lim i →∞ µ ( a ) ( Ω k i , τ ) = lim i →∞ µ ( a ) ( Ω i , τ ) . (2.14)Combining (2.13) and (2.14), we obtain (2.12). (cid:3) D is the whole manifold M . Then wehave µ ( a ) ( M , τ ) = lim i →∞ µ ( a ) ( B ( x , r i ) , τ ) (2.15)for a fixed point x ∈ M and radii r i → ∞ . However, unless M satisfies some well-behavedgeometry condition, the minimizer of µ ( a ) ( M , τ ) does not exist in general(c.f. Q. Zhang [54]).Now we discuss some fundamental properties of the minimizer functions. Proposition 2.4 ( Boundary regularity of minimizers ) . Suppose ( M m , g ) is a smooth Riemannianmanifold, Ω is a bounded open set in M such that ∂ Ω is smooth. Suppose a is a smooth functionon ¯ Ω and τ is a positive number. Suppose ϕ is a minimizer for the functional µ ( a ) ( Ω , g , τ ) . Thenwe have ϕ ∈ C ,α ( ¯ Ω ) for each α ∈ (0 , . In particular, we have ϕ ( x ) + |∇ ϕ ( x ) | d ( x ) ≤ Cd ( x ) , ∀ x ∈ Ω (2.16) where d ( x ) = d ( x , ∂ Ω ) , C depends on Ω and a .Proof. As a minimizer, ϕ satisfies the following Euler-Lagrange equation (cid:18) − τ ∆ + τ a − µ ( a ) − m − m πτ ) (cid:19) ϕ = ϕ log ϕ. (2.17)Recall that ϕ is a positive function in Ω and ϕ ≡ ∂ Ω . Also we have ϕ ∈ W , ( Ω )(c.f.Rothaus [36]). Using Moser iteration, it is not hard to obtain that ϕ is bounded(c.f. inequality (25)of Tian-Wang [44]), which implies that 2 ϕ log ϕ is a bounded function on Ω . Rewriting (2.17) as4 τ ∆ ϕ = (cid:18) τ a − µ ( a ) − m − m πτ ) (cid:19) ϕ − ϕ log ϕ. Let L be 4 τ ∆ and h be the right hand side of the above equation. Then ϕ satisfies the equation L ϕ = h and ϕ | ∂ Ω =
0. Since L is uniformly elliptic and h is bounded, it follows(c.f. Theorem 8.34of Gilbarg-Trudinger [16]) from the smoothness of ∂ Ω that ϕ ∈ C , ( ¯ Ω ). Note that the function2 x log x is in C α for each α ∈ (0 , ϕ log ϕ ∈ C α ( ¯ Ω ) for each α ∈ (0 , ϕ ∈ C ,α ( ¯ Ω ). (cid:3) Proposition 2.4 is useful in the study of convergence of µ along the C ∞ -Cheeger-Gromovconvergence. Suppose ( M mi , x i , g i ) is a sequence of pointed smooth Riemannian manifolds and( M m ∞ , x ∞ , g ∞ ) is also a pointed smooth Riemannian manifold. We say that( M mi , x i , g i ) C ∞ − Cheeger − Gromov −−−−−−−−−−−−−−−−−→ ( M ∞ , x ∞ , g ∞ ) (2.18)if there exists an exhaustion ∪ ∞ k = K k of M ∞ by compact sets K k ∋ x ∞ such that for each k , thereexist di ff eomorphisms ψ i , k from K k to their images in M i such that ψ ∗ i , k ( g i ) C ∞ −−−→ g ∞ , on K k . Let Ω ∞ be a bounded set in M ∞ , we say that Ω i is a sequence of sets in M i converging to Ω ∞ alongthe convergence (2.18) if we can find a big k such that Ω ∞ ⊂ K k and Ω i = ψ i , k ( Ω ∞ ). With theseterminologies, we can discuss the following continuity property of the local- µ -functional.11 orollary 2.5 ( Continuity of local- µ -functional under C ∞ -Cheeger-Gromov convergence ) . Suppose ( M mi , x i , g i ) is a sequence of pointed Riemannian manifolds such that ( M mi , x i , g i ) C ∞ − Cheeger − Gromov −−−−−−−−−−−−−−−−−→ ( M m ∞ , x ∞ , g ∞ ) . (2.19) Suppose Ω ∞ is a bounded open set in M ∞ with smooth boundary, Ω i ⊂ M i are the sets convergingto Ω ∞ along the convergence (2.19). Then for each τ > we have lim i →∞ µ ( Ω i , g i , τ ) = µ ( Ω ∞ , g ∞ , τ ) . (2.20) Proof.
By taking further subsequence if necessary, we can assume that there exist di ff eomor-phisms ψ i from a compact set K ⊂ M ∞ to its image on M i such that ψ ∗ i g i C ∞ −−−→ g ∞ , on K ⊃ Ω ∞ . (2.21)Then Ω i is ψ i ( Ω ∞ ). Let ϕ i be a minimizer of the functional µ ( Ω i , g i , τ ). Clearly, ϕ i ◦ ψ i is aminimizer of µ (cid:16) Ω ∞ , ψ ∗ i ( g i ) , τ (cid:17) . For simplicity of notation, we denote ψ ∗ i g i and ϕ i ◦ ψ i by ˜ g i and ˜ ϕ i respectively. Recall that ˜ ϕ i satisfies Euler-Lagrange equation (2.17) for a = R (˜ g i ). Then it followsfrom Proposition 2.4 and the uniform equivalence condition (2.21) that ˜ ϕ i have uniform C ,α ( ¯ Ω ∞ )bounds, with respect to the metric g ∞ . It follows that˜ ϕ i C ,α ′ −−−−→ ˜ ϕ ∞ , on ¯ Ω ∞ , (2.22)for some α ′ ∈ (0 , α ). It is also clear that (2.21) implies that µ i = µ (cid:16) Ω ∞ , ψ ∗ i ( g i ) , τ (cid:17) are uniformlybounded. By taking subsequence if necessary, we assume that µ i converges to µ ∞ . Recall that(2.22) guarantees the convergence of the Euler-Lagrange equation (2.17). So we have (cid:18) − τ ∆ + τ R − µ ∞ − m − m πτ ) (cid:19) ˜ ϕ ∞ = ϕ ∞ log ˜ ϕ ∞ . It is also clear from (2.22) that ˜ ϕ ∞ satisfies the normalization condition R Ω ∞ ˜ ϕ ∞ dv = W , ( Ω ∞ , g ∞ ). Therefore, it follows from the definition that µ ( Ω ∞ , g ∞ , τ ) ≤ W ( R ) ( Ω ∞ , g ∞ , ˜ ϕ ∞ , τ ) = µ ∞ . (2.23)On the other hand, let ϕ be a minimizer of µ ( Ω ∞ , g ∞ , τ ). In view of (2.21) and (2.22), we know λ i ϕ satisfies the normalization condition R Ω ∞ ( λ i ϕ ) dv ˜ g i = λ i →
1. Itfollows that µ i = µ ( Ω ∞ , ˜ g i , τ ) ≤ W ( R ) ( Ω ∞ , ˜ g i , λ i ϕ, τ ) , whose limit reads as µ ∞ ≤ W ( R ) ( Ω ∞ , g ∞ , ϕ, τ ) = µ ( Ω ∞ , g ∞ , τ ) . (2.24)Consequently, the combination of (2.23) and (2.24) implies that µ ( Ω ∞ , g ∞ , τ ) = µ ∞ , which isnothing but (2.20). (cid:3)
12n applications, the smooth boundary condition cannot always be satisfied. Therefore, we oftenneed to slightly perturb the domain in study to have better boundary regularity. Such perturbationis guaranteed by the following lemma.
Lemma 2.6 ( Approximate smooth boundary condition ) . Suppose Ω is a connected, boundedopen set of M. For each small ǫ , there exists a set Ω ǫ satisfying • Ω ⊂ Ω ǫ ⊂ the ǫ -neighborhood of Ω ; • ∂ Ω ǫ is smooth.Proof. Fix ǫ > ǫ -neighborhood of Ω by Ω ′ . Then { M \ ¯ Ω , Ω ′ } is a covering of M .By partition of unity, there exists a smooth function u supported on Ω ′ and u ≡ Ω . By Sardtheorem(c.f. the book of J. Lee [29]), the function u has a regular value λ ∈ (0 ,
1) such that u − ( λ )is a smooth manifold of dimension m −
1. Clearly, we have u − ([ λ, ⊃ ¯ Ω . There exists exactlyone connected component of u − ([ λ, Ω . We denote this component by Ω ǫ . It isclear that ¯ Ω ⊂ Ω ǫ ⊂ u − ([ λ, ⊂ Ω ′ . Moreover, ∂ Ω ǫ = u − ( λ ) is a smooth manifold. Therefore, Ω ǫ satisfies all the desired propertiesand the proof of Lemma 2.6 is complete. (cid:3) Remark 2.7.
The estimate (2.16) in Proposition 2.4 is needed to estimate the lower bound of µ ( Ω , τ ) along the Ricci flow whenever Ω has a smooth boundary. However, smooth boundarycondition is not satisfied by many domains, e.g., geodesic balls. Lemma 2.6 is used to drop thesmooth boundary condition of Ω . Actually, by Proposition 2.1, we have µ ( Ω , τ ) ≥ µ ( Ω ǫ , τ ) . Therefore, the lower bound of µ ( Ω , τ ) can be derived from the lower bound of µ ( Ω ǫ , τ ) for ǫ su ffi ciently small. Therefore, for simplicity, we may always assume that ∂ Ω is smooth wheneverwe want to develop the lower bound of µ ( Ω , τ ) . An alternative way to achieve this purpose is toapproximate Ω by smaller sets with smooth boundary, say Ω − ǫ ’s, and then apply Proposition 2.3. Remark 2.8.
Similar to the discussion in this section, one can also localize the steady soliton func-tional λ of Perelman [33], and the expanding soliton functional µ + and ν + of Feldman-Ilmanen-Ni [15]. For each Ω ⊂ M bounded domain with smooth boundary, it is not hard to see that ¯ ν ( Ω ) is nothingbut the optimal uniform logarithmic Sobolev constant(c.f. [20], [12]). It is known in the literaturethat the bound of ¯ ν ( Ω ) is equivalent to the bound of many other quantities, like Sobolev constantbound, Faber-Krahn constant, Nash constant, heat kernel on-diagonal upper bound, heat kernelo ff -diagonal Gaussian upper bound, etc(e.g. see Section 6.1 of [17] for a survey). It is also known13hat ¯ ν ( Ω , τ ) is enough to bound many quantities(e.g., see p. 320 of [11]). In this section, we shallonly investigate some elementary estimates of ¯ ν ( Ω , τ ) and its relationship with ν ( Ω , τ ) and thevolume ratios, whenever some scalar or Ricci curvature conditions are satisfied. For simplicity ofnotations, the following conditions are assumed by default in all the discussion in this section. − Λ ≤ R ( x ) ≤ ¯ Λ , Rc ( x ) ≥ − ( m − K , ∀ x ∈ Ω , (3.1)where Λ , ¯ Λ and K are nonnegative constants.We first note that there are elementary relationships among the local functionals. Lemma 3.1.
For each τ > we have µ ( Ω , τ ) − ¯ Λ τ ≤ ¯ µ ( Ω , τ ) ≤ µ ( Ω , τ ) + Λ τ, (3.2) ν ( Ω , τ ) − ¯ Λ τ ≤ ¯ ν ( Ω , τ ) ≤ ν ( Ω , τ ) + Λ τ. (3.3) Proof.
For each function ϕ ∈ S ( Ω ) (c.f. equations (2.1) -(2.4)) and each τ >
0, we have − ¯ Λ τ ≤ W (0) ( Ω , ϕ, τ ) − W ( R ) ( Ω , ϕ, τ ) = − Z Ω τ R ϕ dv ≤ Λ τ Z Ω ϕ dv = Λ τ. (3.4)If we choose ϕ as the minimizer of µ (0) ( Ω , ϕ, τ ), then the first part of (3.4) implies that µ (0) ( Ω , τ ) = W (0) ( Ω , ϕ, τ ) ≥ W ( R ) ( Ω , ϕ, τ ) − ¯ Λ τ ≥ µ ( R ) ( Ω , τ ) − ¯ Λ τ. (3.5)However, if we choose ϕ as the minimizer of µ ( R ) ( Ω , ϕ, τ ), then the last part of (3.4) implies that µ ( R ) ( Ω , τ ) = W ( R ) ( Ω , ϕ, τ ) ≥ W (0) ( Ω , ϕ, τ ) − Λ τ ≥ µ (0) ( Ω , τ ) − Λ τ. (3.6)Combining (3.5) and (3.6) gives us µ ( R ) ( Ω , τ ) − ¯ Λ τ ≤ µ (0) ( Ω , τ ) ≤ µ ( R ) ( Ω , τ ) + Λ τ, (3.7)which is nothing but (3.2) by the choice of our notion in (2.7) and (2.8). The proof of (3.2) iscomplete.We proceed to prove (3.3). The second inequality in (3.7) implies that for each s ∈ (0 , τ ), wehave µ (0) ( Ω , s ) ≤ µ ( R ) ( Ω , s ) + Λ s ≤ µ ( R ) ( Ω , s ) + Λ τ. Taking infimum of the above inequality yields that ν (0) ( Ω , τ ) ≤ ν ( R ) ( Ω , τ ) + Λ τ. Similarly, we can analyze the first inequality in (3.7) and obtain that ν ( R ) ( Ω , τ ) ≤ ν (0) ( Ω , τ ) + ¯ Λ τ. Consequently, (3.3) follows from the combination of the previous two inequalities. (cid:3)
Proposition 3.2.
Suppose < τ < τ , then ¯ ν ( Ω , τ ) ≤ ¯ ν ( Ω , τ ) ≤ ¯ ν ( Ω , τ ) + m τ τ . (3.8) ν ( Ω , τ ) ≤ ν ( Ω , τ ) ≤ ν ( Ω , τ ) + m τ τ + ¯ Λ τ + Λ τ . (3.9) Proof.
The first inequality of (3.8) follows trivially from the definition equation (2.4). We onlyneed to prove the second part of (3.8). We may assume ¯ ν ( Ω , τ ) <
0, for otherwise we havenothing to prove by the non-positivity of ¯ ν (c.f. Proposition 2.2). Therefore, there exists some τ ′ ∈ (0 , τ ] such that ¯ ν ( Ω , τ ) = ¯ µ ( Ω , τ ′ ). If τ ′ ≤ τ , then we have nothing to prove. So weassume τ ′ ∈ ( τ , τ ]. Let ϕ be the minimizer function of ¯ µ ( Ω , τ ′ ). Then the Euler-Lagrangeequation (2.6) reads as (cid:18) ¯ µ ( Ω , τ ′ ) + m + m πτ ′ ) (cid:19) ϕ = − τ ′ ∆ ϕ − ϕ log ϕ. Multiplying both sides of the above equation by ϕ and integrating on Ω , we obtain¯ µ ( Ω , τ ′ ) + m + m πτ ′ ) = τ ′ Z Ω |∇ ϕ | dv − Z Ω ϕ log ϕ dv ≥ τ Z Ω |∇ ϕ | dv − Z Ω ϕ log ϕ dv ≥ ¯ µ ( Ω , τ ) + m + m πτ ) , where we used the definition equation (2.3) in the last step. Consequently, it follows from (2.4)that ¯ µ ( Ω , τ ′ ) ≥ ¯ µ ( Ω , τ ) + m τ τ ′ ≥ ¯ µ ( Ω , τ ) + m τ τ ≥ ¯ ν ( Ω , τ ) + m τ τ , whence we obtain the second part of (3.8) since ¯ ν ( Ω , τ ) = ¯ µ ( Ω , τ ′ ) by our choice of τ ′ .Now we focus on the proof of (3.9). Again, we only need to prove the second part of (3.9). Itfollows from the first part of (3.3) of Lemma 3.1 that ν ( Ω , τ ) ≤ ¯ ν ( Ω , τ ) + ¯ Λ τ . (3.10)Plugging the second part of (3.8) into the above inequality, we arrive at ν ( Ω , τ ) ≤ ¯ ν ( Ω , τ ) + m τ τ + ¯ Λ τ . Then we apply the second part of (3.3) of Lemma 3.1 to obtain¯ ν ( Ω , τ ) ≤ ν ( Ω , τ ) + Λ τ . ν ( Ω , τ ) ≤ ν ( Ω , τ ) + m τ τ + ¯ Λ τ + Λ τ , whence we derive (3.9). (cid:3) Theorem 3.3 ( Lower bound of volume ratio in terms of ν and scalar curvature ) . SupposeB = B ( x , r ) ⊂ Ω is a geodesic ball, then we have | B | ω m r m ≥ e ¯ ν − m + ≥ e ν − m + − ¯ Λ r (3.11) where ¯ ν = ¯ ν ( B , r ) , ν = ν ( B , r ) .Proof. The second inequality in (3.11) follows directly from (3.3) in Lemma 3.1. It su ffi ces toprove the first inequality in (3.11). Let q be the volume ratio function: q ( ρ ) ≔ | B ( x ,ρ ) | ω m ρ m , if 0 < ρ ≤ r ;1 , if ρ = . Clearly, q is continuous on [0 , r ]. Suppose the minimum value of q is achieved at ρ . If ρ = | B | ω m r m ≥ . (3.12)Now we assume ρ >
0. It follows from the definition of ρ that | B ( x , ρ ) | ω m ( ρ ) m ≥ | B ( x , ρ ) | ω m ρ m = q ( ρ ) , ⇒ | B ( x , ρ ) | ≤ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) B (cid:18) x , ρ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (3.13)Take a cuto ff function η which equals 1 on ( −∞ , ], decreases 0 on [ ,
1] and equals 0 on [1 , ∞ ).Furthermore, we have | η ′ | ≤
4. Set d = d ( · , x ) and define L ≔ Z M η d ρ ! dv , ϕ ≔ L − η d ρ ! . For simplicity of notations, denote B ( x , ρ ) by B ′ and B ( x , ρ ) by B ′ . It follows from (3.13)and the definition of ϕ and L that | B ′ | ≥ L ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) B ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ − m | B ′ | . (3.14)Then we have¯ µ ( B ′ , ρ ) + m + m πρ ) ≤ W (0) ( B ′ , ϕ, ρ ) + m + m πρ ) = Z B ′ − ϕ log ϕ dv + Z B ′ \ B ′ ρ |∇ ϕ | dv = log L + R B ′ n − η log η o dvL + R B ′ \ B ′ | η ′ | dvL . − η log η ≤ e − and | η ′ | ≤ B ′ . Therefore, the last two terms in the aboveinequality can be bounded. Z B ′ n − η log η o dv ≤ e − | B ′ | , Z B ′ \ B ′ | η ′ | dv ≤ | B ′ | − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) B ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)! ≤ − − m ) | B ′ | , where we used the conditions (3.14) in the last step. Combining all the above inequalities, weobtain ¯ µ ( B ′ , ρ ) + m + m πρ ) ≤ log L + (cid:16) e − + − − m ) (cid:17) | B ′ | L ≤ log L + (cid:16) e − + − − m ) (cid:17) m < log | B ′ | + · m . It follows that log | B ′ | ρ m > ¯ µ ( B ′ , ρ ) + m + m π − · m . Recall that r ≥ ρ > | B | r m ≥ | B ′ | ρ m and ¯ µ ( B ′ , ρ ) ≥ ¯ µ ( B , ρ ) ≥ ¯ ν ( B , r ) = ¯ ν . Recall also the explicitformula of unit ball volume in Euclidean space R m : ω m = π m Γ ( m +
1) (3.15)where Γ is the traditional Γ -function. We then obtain | B | ω m r m > e ¯ ν − log ω m + m + m log 4 π − · m = e ¯ ν + { log Γ ( m + + m + m log 2 } − · m . (3.16)Since m ≥
3, it is easy to check that log Γ ( m + + m + m log 2 < m . Notice that ¯ ν ≤ e ¯ ν + { log Γ ( m + + m + m log 2 − m } − m + < e − m + << . Combining the cases that ρ = ρ ∈ (0 , r ], i.e., (3.12) and (3.16), we obtain | B | ω m r m > e ¯ ν + { log Γ ( m + + m + m log 2 } − · m > e ¯ ν − m + (3.17)which is exactly the first part of (3.11). The proof of Theorem 3.3 is complete. (cid:3) Remark 3.4.
The volume ratio lower bound determined by Theorem 3.3 are far away from beingsharp. For example, on the Euclidean space, applying Theorem 3.3 on unit ball B, and noting that ¯ ν ( B , g E , = ν ( B , g E , = , we obtain an explicit lower bound of the volume of each unit balle − m + · ω m which is a very small number compared to the actual value ω m . ν is nothing but the optimal uniform Logarithmic Sobolev constant. It is well-known(c.f. Section 6.1 of [17]) that the Logarithmic Sobolev inequality is dominated by the L -Sobolev inequality constant C S and consequently the L -Sobolev inequality constant C I . Itis also well-known(c.f. section 3.1 of [37]) that C I is equivalent to I − where I is the isoperimetricconstant I ( Ω ) ≔ inf D ⋐ Ω | ∂ D || D | m − m . (3.18)Combing the previous steps, it is clear that ¯ ν can be bounded by I ( Ω ). However, there are toomany intermediate steps in the above deduction where errors occur. So the result obtained in thisway cannot be sharp. In order to reduce the errors, we shall develop a direct estimate of ¯ ν by I ( Ω ),in terms of Logarithmic eigenfunctions. For simplicity of notation, we define I m ≔ I ( R m , g E ) . (3.19)Clearly, the best isoperimetric constant in the Euclidean space ( R m , g E ) is achieved by standardballs. Consequently, I m can be calculated explicitly as I m = m ω m ω m − m m = m ω m m . (3.20) Lemma 3.5 ( Estimate of functionals by isoperimetric constant and scalar lower bound ) . Sup-pose Ω is a bounded domain in ( M m , g ) , ˜ Ω is a ball in ( R m , g E ) such that | ˜ Ω | = | Ω | . Define λ ≔ I ( Ω ) I m . (3.21) Then we have ¯ µ ( Ω , g , τ ) ≥ ¯ µ (cid:16) ˜ Ω , g E , τλ (cid:17) + m log λ. (3.22) Consequently, we have ¯ ν ( Ω , g , τ ) ≥ m log λ, (3.23) µ ( Ω , g , τ ) ≥ ¯ µ (cid:16) ˜ Ω , g E , τλ (cid:17) + m log λ − Λ τ, (3.24) ν ( Ω , g , τ ) ≥ m log λ − Λ τ. (3.25) Proof.
We shall partly follow the argument in the proof of Proposition 4.1 of L. Ni [32], whichwas suggested by Perelman at the end of the proof of Theorem 10.1 of [33]. The new ingredienthere is to only estimate the smooth eigenfunctions.Let ϕ be a minimizer function of ¯ µ ( Ω , g , τ ). Clearly, ϕ > Ω and ϕ = ∂ Ω . Define Ω t ≔ { x ∈ Ω | ϕ ( x ) ≥ t } , F ( t ) ≔ | Ω t | . h = h ( | y | ) be a radial symmetric function on ˜ Ω such that |{ h ( | y | ) ≥ t }| = F ( t ) , h | ∂ B = . We can similarly define ˜ Ω t ≔ { x ∈ B | h ≥ t } . Then the above equations imply | Ω t | = F ( t ) = | ˜ Ω t | . (3.26)It follows from the definition of the isoperimetric constant that | ∂ Ω t | ≥ I | Ω t | m − m = I | ˜ Ω t | m − m = II m | ∂ ˜ Ω t | = λ | ∂ ˜ Ω t | , (3.27)where we used the facts that I m is achieved by balls and ˜ Ω t is a ball for each t by its construction.Recall that the integration by parts implies that Z ˜ Ω χ ( h ) d ˜ v = Z ∞ χ ′ ( s ) F ( s ) ds = Z Ω χ ( ϕ ) dv for any Lipschitz function χ ( t ) with χ (0) =
0. Let χ ( t ) = t log t , we obtain Z ˜ Ω h log h d ˜ v = Z Ω ϕ log ϕ dv . (3.28)Similarly, we have Z ˜ Ω h d ˜ v = Z Ω ϕ dv = . (3.29)On the other hand, by the co-area formula, we have Z ∞ t Z ϕ = s |∇ ϕ | d σ ds = | Ω t | = F ( t ) = | ˜ Ω t | = Z ∞ t Z h = s |∇ h | d ˜ σ ds , ∀ t > , which implies that Z ∂ Ω t |∇ ϕ | d σ = Z ∂ ˜ Ω t |∇ h | d ˜ σ, ∀ t > . (3.30)Note that |∇ h | is a constant on ∂ ˜ Ω t . Therefore, in light of (3.27) and (3.30), we can apply theH ¨older inequality to obtain that Z ∂ Ω t |∇ ϕ | d σ ≥ | ∂ Ω t | R ∂ Ω t |∇ ϕ | d σ ≥ λ | ∂ ˜ Ω t | R ∂ Ω t |∇ ϕ | d σ = λ R ∂ ˜ Ω t |∇ h | d ˜ σ R ∂ ˜ Ω t |∇ h | d ˜ σ R ∂ Ω t |∇ ϕ | d σ = λ Z ∂ ˜ Ω t |∇ h | d ˜ σ. Using the co-area formula again, we have Z Ω |∇ ϕ | dv = Z ∞ Z ∂ Ω t |∇ ϕ | d σ dt ≥ λ Z ∞ Z ∂ ˜ Ω t |∇ h | d ˜ σ dt = λ Z ˜ Ω |∇ h | d ˜ v . (3.31)19ecall that ϕ is a minimizer for ¯ µ ( Ω , g , τ ) and that h satisfies the normalization condition (3.29).Consequently, it follows from (3.28) and (3.31) that¯ µ ( Ω , g , τ ) + (cid:18) m πτ ) + m (cid:19) = Z Ω n τ |∇ ϕ | − ϕ log ϕ o dv ≥ Z ˜ Ω n τλ |∇ h | − h log h o d ˜ v ≥ ¯ µ (cid:16) ˜ Ω , g E , τλ (cid:17) + (cid:18) m πτλ ) + m (cid:19) , which yields (3.22) directly. Since ˜ Ω ⊂ R m , it is clear(c.f. Proposition 2.1) that¯ µ (cid:16) ˜ Ω , g E , τλ (cid:17) ≥ ¯ µ (cid:16) R m , g E , τλ (cid:17) = . Plugging the above inequality into (3.22) and taking infimum over s ∈ (0 , τ ], we obtain (3.23).Then (3.24) follows from the combination of (3.22) and (3.2). Similarly, (3.25) follows from thecombination of (3.23) and (3.3). The proof of Lemma 3.5 is complete. (cid:3) Up to now, we only used the information from scalar curvature bound. In the next theorem,the Ricci lower bound start to play an important role, because of the comparison geometry. Weremind the reader that we have the condition (3.1) holds by default.
Theorem 3.6 ( Equivalence of volume ratio and functionals ) . SupposeB = B ( x , r ) ⊂ ˜ B = B ( x , r ) ⊂ Ω . Let ¯ ν = ¯ ν ( B , r ) and ν = ν ( B , r ) . Then we have − m (1 + Kr ) + ( m +
1) log | B | ω m r m ≤ ¯ ν ≤ log | B | ω m r m + m + , (3.32) − n m (1 + Kr ) + Λ r o + ( m +
1) log | B | ω m r m ≤ ν ≤ log | B | ω m r m + n m + + ¯ Λ r o . (3.33) Proof.
Clearly, (3.33) follows from the combination of (3.32) and (3.3) in Lemma 3.1. Therefore,it su ffi ces to show (3.32) only.By the scaling invariance of ¯ ν , the volume ratios, the numbers Kr , Λ r and ¯ Λ r , we can assume r = p ∈ B = B ( x , B = B ( x , ⊂ B ( p , U p be the directions ~ v in the unit sphere of T p M suchthat the unit geodesic staring from ~ v is always the shortest geodesic before it hits the boundary of˜ B . Then calculation in the unit bundle U M implies that(c.f. Lemma 4.2 of [49]):˜ ω ≔ inf p ∈ B | ˜ U p | ≥ | ˜ B \ B | m ω m R (cid:16) sinh KrK (cid:17) m − dr . (3.34)Consequently, the isoperimetric constant I ( B ) can be bounded by(c.f. Lemma 4.1 of [49]):20 m ( B ) ≥ m − · ( m ω m ) m { ( m + ω m + } m − · ˜ ω m + . Recall that I m = m ω m m by (3.20). It follows from the definition(c.f. (3.21)) that λ m = ( I ( B ) I m ) m ≥ ω m ( m + ω m + ! m − ˜ ω m + . (3.35)We now focus on the estimate of ˜ ω defined in (3.34). For each positive number t , it is clear that1 < sinh tt < e t . Therefore, we have6 m m < Z sinh KrK ! m − dr < e m − K Z r m − dr = e m − K · m m . (3.36)Since ∂ Ω , ∅ , we can choose a shortest unit-speed geodesic γ connecting x to some point on ∂ Ω satisfying | γ | ≥
5. Let y = γ (2). By triangle inequalities, it is clear that B = B ( x , ⊂ B ( y , ⊂ ˜ B = B ( x , ⊂ Ω . Note that B ( y , ⊂ ˜ B \ B . In view of the Bishop-Gromov volume comparison, we obtain | ˜ B \ B | ≥ | B ( y , | ≥ | B ( y , | · R (cid:16) sinh KrK (cid:17) m − dr R (cid:16) sinh KrK (cid:17) m − dr ≥ | B | · − m · e − m − K , (3.37)where we used an inequality similar to (3.36) in the last step. Combining (3.37) and (3.36) with(3.34), we arrive at the estimate of ˜ ω :˜ ω ≥ | B | ω m ! · − m · e − m − K . (3.38)Then we estimate the term in the parenthesis of the right hand side of (3.35). In view of the formula(3.15), we calculate ω m + ω m = √ π · Γ ( m + Γ ( m + + ≤ √ π, where we used the fact that m ≥ Γ -function is increasing on [2 , ∞ ). It follows that2 ω m ( m + ω m + ≥ √ π ( m + ≥ m (3.39)whenever m ≥
3. Plugging (3.39) and (3.38) into (3.35) and taking logarithm on both sides, weobtain that m log λ ≥ − ( m −
1) log 2 m + ( m + ( log | B | ω m − m log 18 − m − K ) > − m − m K + ( m +
1) log | B | ω m = − m (1 + K ) + ( m +
1) log | B | ω m , whence we obtain the first inequality of (3.32), up to rescaling and application of (3.23). The sec-ond inequality of (3.32) follows from inequality (3.11) in Theorem 3.3. The proof of Theorem 3.6is complete. (cid:3) Li-Yau-Hamilton-Perelman type Harnack inequality
The purpose of this section is to improve the Harnack inequality along the Ricci flow.
Theorem 4.1 (Corollary 9.3 of Perelman [33]) . Suppose { ( M m , g ( t )) , ≤ t ≤ T } is a Ricci flowsolution. Let u be a conjugate heat solution, i.e., (cid:3) ∗ u = ( − ∂ t − ∆ + R ) u = , starting from a δ -function from ( x , T ) for some point x ∈ M. Then we havev = n ( T − t )(2 ∆ f − |∇ f | + R ) + f − m o u ≤ , on M × [0 , T ) , (4.1) where f = − log u − m log { π ( T − t ) } . The inequality (4.1) was discovered by Perelman [33]. If one regard the Ricci-flat space asa trivial Ricci flow solution, then the conjugate heat equation is nothing but the backward heatequation since R =
0. Then the classical Li-Yau estimate(c.f. [30]) reads as2( T − t ) ∆ f − m ≤ , (4.2)which is similar(c.f. Section 3.3 of M¨uller [31]) to (4.1) and holds for all positive backwardheat solutions. For evolving space-times, Hamilton used his matrix-Harnack inequality to studythe backward heat solution starting from fundamental solutions in [23]. Li-Yau estimate (4.2)suggests that (4.1) holds for more conjugate heat solutions. We shall generalize (4.1) to the casethat u is a positive conjugate heat solution starting from a local minimizer function, rather than a δ -function. Theorem 4.2 ( Li-Yau-Hamilton-Perelman type Harnack inequality ) . Suppose { ( M m , g ( t )) , ≤ t ≤ T } is a Ricci flow solution, Ω is a bounded domain of M with smooth boundary. Fix τ T > ,let ϕ T be the minimizer function of µ ( Ω , g ( T ) , τ T ) for some τ T > , u T = ϕ T . Starting from u T attime t = T , let u solve the conjugate heat equation (cid:3) ∗ u = ( − ∂ t − ∆ + R ) u = . (4.3) Define τ ≔ τ T + T − t , (4.4) f ≔ − m πτ ) − log u , (4.5) v ≔ n τ (2 ∆ f − |∇ f | + R ) + f − m − µ o u , (4.6) where µ = µ ( Ω , g ( T ) , τ T ) . Then we have v ≤ on M × [0 , T ) . Moreover, if v = at some point ( x , s ) ∈ M × [0 , T ) , then v ≡ on M × [0 , T ] and Ω =
M, the flow is induced by a gradient shrinking Ricci soliton metric. ff ers from Theorem 4.1 by the choice of u and τ . In Theorem 4.1, u is chosen as a conjugate heat solution coming out of a δ -function at time t = T and τ = T − t .However, in Theorem 4.2, u arises from a minimizer function ϕ T for µ ( Ω , g ( T ) , τ T ) and τ = τ T + T − t . Since ∂ Ω is a non-empty smooth manifold, some extra technical di ffi culties appearfor the analysis of u and v . Fortunately, the extra e ff orts needed to prove Theorem 4.2 are justifiedby the consequence that Theorem 4.2 is more general (c.f. Remark 4.10) and provides moreinformation for later applications.Formally, Theorem 4.2 can be “proved” in the same way as Theorem 4.1. Actually, it followsfrom the Euler-Lagrangian equation satisfied by ϕ T that v ( x , T ) = , ∀ x ∈ Ω . (4.8)By extending v as zero on M \ ¯ Ω , we can regard v as a zero function defined on M . By the calcula-tion of Perelman(c.f. Proposition 9.1 of [33]), we have (cid:3) ∗ v = − τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R i j + f i j − g i j τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ , on M × [0 , T ) . Therefore, at least intuitively, one can apply maximum principle to show that v ≤ M × [0 , T ). However, due to the fact that v may not be a continuous function on M × [0 , T ], thereexist technical di ffi culties in applying such argument directly. We shall show the same conclusion,using a di ff erent approach which requires less regularity of v around t = T .Theorem 4.2 is proved by carefully analyzing the minimizer function ϕ T and the conjugateheat solution u . Before we delve into the details of the proof of Theorem 4.2, we first setup severallemmas. For simplicity of notation, from now on, we always assume that T = , τ = τ T + T = τ + . (4.9) Lemma 4.3 ( Precise gradient estimate of conjugate heat solutions ) . Then there is a constant Cdepending on Ω and the flow such that (cid:12)(cid:12)(cid:12) ∇ √ u (cid:12)(cid:12)(cid:12) ≤ C , on M × [0 , . (4.10) Proof.
Since Ω is a bounded domain, (2.16) clearly yields that |∇ ϕ | ≤ C . This in turn means that |∇ u | ≤ Cu (4.11)at time t =
1. Since u satisfies (cid:3) ∗ u =
0, direct calculation shows that (cid:3) ∗ |∇ u | = ( − ∂ t − ∆ + R ) |∇ u | = − Rc ( ∇ u , ∇ u ) − h∇ u , ∇ u t i − |∇∇ u | − Rc ( ∇ u , ∇ u ) − h∇ u , ∇ ∆ u i + R |∇ u | = − Rc ( ∇ u , ∇ u ) − R |∇ u | − u h∇ u , ∇ R i − |∇∇ u | . (4.12)Note that u is a positive function on M × [0 , − u h∇ u , ∇ R i ≤ u ( |∇ u | + |∇ R | ) ≤ u |∇ u | + K u K is some constant depending on the geometry of the flow. Plugging the above inequalityinto (4.12) implies that (cid:3) ∗ |∇ u | ≤ K ( |∇ u | + u )for some constant K depending on the flow. Consequently, we have (cid:3) ∗ ( |∇ u | + u ) = (cid:3) ∗ |∇ u | ≤ K ( |∇ u | + u ) , which yields that (cid:3) ∗ n e Kt ( u + |∇ u | ) o ≤ . (4.13)At time t =
1, note that |∇ u | = ϕ |∇ ϕ | . By (2.16), it is clear that u is at least in C , ( M ).Therefore, |∇ u | is at least a continuous function on M × [0 , t =
1, wehave |∇ u | ≤ Cu . Therefore, we can choose a large constant L such that u + |∇ u | ≤ Lu at time t =
1. Combining (4.3) and (4.13), we obtain (cid:3) ∗ n e Kt ( u + |∇ u | ) − Lu o ≤ . It follows from the maximum principle that u + |∇ u | ≤ Le − Kt u , ⇒ |∇ u | ≤ Cu , on M × [0 , . (4.14)Since u > M × [0 , (cid:3) It seems tempting to apply the maximum principle directly to ∇ √ u by calculating (cid:3) ∗ (cid:12)(cid:12)(cid:12) ∇ √ u (cid:12)(cid:12)(cid:12) .However, for this purpose, one needs better boundary regularity of √ u around ∂ Ω at time t = Lemma 4.4 ( Heat kernel and gradient estimate , c.f. Chau-Tam-Yu [4]) . Let p be the heat kernelfunction on M × [0 , . There exists a constant depending on the flow and Ω such thatC − | t − | − m e − d x , y )3 | t − | < p ( y , x , t ) < C | t − | − m e − d x , y )5 | t − | , ∀ x ∈ M , y ∈ Ω , t ∈ [0 . , − t ) (cid:12)(cid:12)(cid:12) ∇ √ u (cid:12)(cid:12)(cid:12) ≤ C ( u log k u k C ( M ) u + u ) , ∀ t ∈ [0 . , . (4.16) Proof.
Note that on M × [0 . , u is a nonnegativesolution of (cid:3) ∗ u =
0, the maximum principle implies that 0 < u ≤ k u k C ( M ) . Therefore, (4.16)follows directly from Lemma 6.3 of Chau-Tam-Yu [4]. (cid:3) Lemma 4.5 ( Continuity of integral of gradients ) . Suppose that h is a smooth positive functionon M × [1 − η, for some small η > , then we have lim t → − Z M (cid:12)(cid:12)(cid:12) ∇ √ u (cid:12)(cid:12)(cid:12) hdv = Z Ω (cid:12)(cid:12)(cid:12) ∇ √ u (cid:12)(cid:12)(cid:12) hdv . (4.17)24 roof. Since all metrics g ( t ) and measures dv g ( t ) are equivalent on M × [0 . ,
1] by the uniformbound of | Rm | . In the following discussion, we shall assume g (1) as the default metric and dv g (1) as the default measure.Fix δ >
0. Define Ω − δ ≔ { x ∈ M | d ( x , Ω ) ≤ δ } , Ω δ ≔ { x ∈ M | d ( x , Ω c ) ≥ δ } . Choose y ∈ Ω and L as a large number such that Ω ⊂ B ( y , L ). See Figure 4. y Ω δ Ω Ω − δ B ( y , L ) B ( y , L )Figure 4: Separating integrals into several domainsBy triangle inequality, for each x ∈ M \ Ω − δ and y ∈ Ω , we have d ( x , y ) ≤ ( d ( x , y ) + d ( y , y )) ≤ d ( x , y ) + L . Recall that u ( x , t ) = R Ω u ( y ) p ( y , x , t ) dv y . By the heat kernel estimate, i.e. (4.15), for each t ∈ [0 . , C | t − | − m e − d x , y + L | t − | ≤ u ( x , t ) ≤ C | t − | − m e − d x , Ω )5 | t − | , ∀ x ∈ M \ Ω − δ . (4.18)For simplicity of notation, we denote d ( x , y ) by r = r ( x ). If r ( x ) < L , then (4.18) yields that C | t − | − m e − L | t − | ≤ u ( x , t ) ≤ C | t − | − m e − δ | t − | , ∀ x ∈ B ( y , L ) \ Ω − δ . (4.19)Now we suppose r ( x ) > L . Let z be a point in ¯ Ω such that d ( x , Ω ) is achieved at z . Then thetriangle inequality implies that d ( x , Ω ) = d ( x , z ) > r ( x ) − d ( z , y ) > r ( x ) − L > . r ( x ) . C | t − | − m e − r | t − | ≤ u ( x , t ) ≤ C | t − | − m e − r | t − | , ∀ x ∈ M \ B ( y , L ) . (4.20)We first estimate the integral of (cid:12)(cid:12)(cid:12) ∇ √ u (cid:12)(cid:12)(cid:12) on B ( y , L ) \ Ω − δ . In this case, combining (4.19) withthe gradient estimate (4.16) implies that (cid:12)(cid:12)(cid:12) ∇ √ u (cid:12)(cid:12)(cid:12) ( x , t ) ≤ C | t − | − m − e − δ | t − | ( + m | t − | + L | t − | ) ≤ Ce − δ | t − | , where C depends on m , L , δ, Ω and the given flow. It follows thatlim t → − Z B ( y , L ) \ Ω − δ (cid:12)(cid:12)(cid:12) ∇ √ u (cid:12)(cid:12)(cid:12) dv ≤ C Vol( B ( y , L )) lim t → − e − δ | t − | = . (4.21)Next we estimate the integral of (cid:12)(cid:12)(cid:12) ∇ √ u (cid:12)(cid:12)(cid:12) on M \ B ( y , L ). Now the combination of (4.20) and(4.16) gives us (cid:12)(cid:12)(cid:12) ∇ √ u (cid:12)(cid:12)(cid:12) ( x , t ) ≤ C | t − | − m − e − r | t − | ( + m | t − | + r | t − | ) ≤ Ce − r | t − | , for some C depending on m , L , Ω and the flow. Since | Rm | is bounded for space-time M × [0 . , K such that | B ( y , r ) | ≤ e Kr for each r > L . Therefore, by calculating integralson annular parts B ( y , i + L \ B ( y , i L )), we have Z M \ B ( y , L ) (cid:12)(cid:12)(cid:12) ∇ √ u (cid:12)(cid:12)(cid:12) dv ≤ C ∞ X i = e KL i e − iL | t − | ≤ C ∞ X i = e − iL | t − | when | t − | < KL . Note that the series on the right hand side converges faster than the geometricseries when | t − | very small. It follows that Z M \ B ( y , L ) (cid:12)(cid:12)(cid:12) ∇ √ u (cid:12)(cid:12)(cid:12) dv ≤ Ce − L | t − | . Consequently, we obtain lim t → − Z M \ B ( y , L ) (cid:12)(cid:12)(cid:12) ∇ √ u (cid:12)(cid:12)(cid:12) dv ≤ C lim t → − e − L | t − | = , which together with (4.21) implies thatlim t → − Z M \ Ω − δ (cid:12)(cid:12)(cid:12) ∇ √ u (cid:12)(cid:12)(cid:12) dv = . (4.22)On the other hand, note that u is a C -function at time t =
1. It follows from standard parabolicequation theory that |∇ u | is a continuous function on M × [0 , u is a continuous functionon M × [0 ,
1] and u > Ω δ . It follows that u > Ω δ × [1 − ξ,
1] for some small positive26umber ξ . Consequently, |∇ √ u | = |∇ u | u is a continuous function on Ω δ × [1 − ξ, h isa smooth function on M × [1 − η, ⊃ Ω δ × [1 − ξ, t → − Z Ω δ (cid:12)(cid:12)(cid:12) ∇ √ u (cid:12)(cid:12)(cid:12) hdv = Z Ω δ (cid:12)(cid:12)(cid:12) ∇ √ u (cid:12)(cid:12)(cid:12) hdv . (4.23)Combining (4.22) and (4.23), we havelim t → − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z M (cid:12)(cid:12)(cid:12) ∇ √ u (cid:12)(cid:12)(cid:12) hdv − Z Ω (cid:12)(cid:12)(cid:12) ∇ √ u (cid:12)(cid:12)(cid:12) hdv (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ lim t → − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Ω − δ \ Ω δ (cid:12)(cid:12)(cid:12) ∇ √ u (cid:12)(cid:12)(cid:12) hdv (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Ω − δ \ Ω δ (cid:12)(cid:12)(cid:12) ∇ √ u (cid:12)(cid:12)(cid:12) hdv (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C h ( lim t → − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Ω − δ \ Ω δ (cid:12)(cid:12)(cid:12) ∇ √ u (cid:12)(cid:12)(cid:12) dv (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Ω − δ \ Ω δ (cid:12)(cid:12)(cid:12) ∇ √ u (cid:12)(cid:12)(cid:12) dv (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)) ≤ C δ, where we used (4.10) and (4.11) in the last step. Since the choice of δ can be arbitrarily small, weobtain (4.17) from the above inequality. (cid:3) Lemma 4.6 ( Interplay between v and positive smooth functions ) . For every smooth positivefunction h defined on M × [0 , , we have lim t → − Z M hv = , (4.24) where the integral is with respect to the classical volume element induced by g ( t ) .Proof. For simplicity, we omit the default volume element, which is the classical one induced by g ( t ).At each time t <
1, everything is smooth. Hence integration by parts implies that Z M hv = Z M n τ ( R + |∇ f | ) + f − m − µ o uh − τ Z M u ∆ h . Note that u is a nonnegative continuous function on M × [0 , u log u and hence u f are continuous functions on M × [0 , h is a smooth positive function, we can take limit onboth sides of the above inequality to obtain thatlim s → − Z M hv (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = s = lim s → − Z M τ |∇ f | uh (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = s + Z M ( τ R + f − m − µ ) uh (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = − τ Z M u ∆ h (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = . (4.25)Splitting the right hand side of the above inequality into three terms, we shall show thatlim s → − Z M τ |∇ f | uh (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = s = Z Ω τ |∇ f | uh (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = , (4.26) Z M ( τ R + f − m − µ ) uh (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = = Z Ω ( τ R + f − m − µ ) uh (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = , (4.27) Z M u ∆ h (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = = Z Ω h ∆ u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = . (4.28)27ctually, it follows from the definition of f , i.e., equation (4.5), that u |∇ f | = |∇ √ u | . Therefore,(4.26) is nothing but (4.17). The equation (4.27) holds since u and u f are supported on Ω at time t =
1. Recall that ∂ Ω is smooth and ϕ ∈ C ,α ( ¯ Ω ) for each α ∈ (0 , t =
1, note that u = ϕ satisfies ∂ u ∂~ n ( x ) = ϕ ( x ) ∂ϕ ∂~ n ( x ) = , ∀ x ∈ ∂ Ω where ~ n is the outward normal vector field on ∂ Ω . Therefore, it follows from Green’s formula that Z M u ∆ h = Z Ω u ∆ h = Z Ω h ∆ u + Z ∂ Ω u ∇ h · ~ n − h ∇ u · ~ n = Z Ω h ∆ u , which is exactly (4.28). Therefore, we have finished the proof of (4.26), (4.27) and (4.28). Plug-ging them into (4.25), we obtainlim s → − Z M hv (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = s = Z Ω n τ ( R + ∆ f − |∇ f | ) + f − m − µ o uh (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = = , which is nothing but (4.24). The proof of Lemma 4.6 is complete. (cid:3) Now we are ready to prove Theorem 4.2.
Proof of Theorem 4.2.
Recall that v = n τ (cid:16) ∆ f − |∇ f | + R (cid:17) + f − m − µ o u is a smooth functionon M × [0 ,
1) satisfying (4.24) in Lemma 4.6. Direct calculation(c.f. Section of Perelman [33])shows that (cid:3) ∗ v = − τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R i j + f i j − g i j τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u ≤ . (4.29)Fix an arbitrary point ( x , s ) ∈ M × [0 , w be a heat solution, i.e., (cid:3) w = ( ∂ t − ∆ ) w = δ -function at ( x , s ). Because of (4.29), for each s ∈ ( s , v ( x , s ) = Z M wv (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = s = Z M wv (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = s + Z ss Z M w (cid:3) ∗ v = Z M wv (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = s − Z ss Z M τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R i j + f i j − g i j τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) wu . Letting s → − , it follows from (4.24) that v ( x , s ) = − Z s Z M τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R i j + f i j − g i j τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) wu ≤ . (4.30)Therefore, by the arbitrary choice of ( x , s ), we obtain v ≤ M × [0 ,
1) and we finish the proofof (4.7).Suppose v ( x , s ) =
0. By strong maximum principle, it is clear that v ≡ M × [ s , τ (2 ∆ f − |∇ f | + R ) + f − m − µ ≡ . (4.31)28t follows from (4.30) that R i j + f i j − g i j τ ≡ M × [ s , Ω = M . Actually, taking trace of the shrinking Ricci solitonequation (4.32) implies that τ ( R + ∆ f ) − m = , which combined with (4.31) implies that τ ( R + |∇ f | ) − f + µ = . (4.33)Since (cid:3) ∗ u =
0, it follows from the definition (4.6) that f t = − ∆ f + |∇ f | − R + m τ = |∇ f | . On a shrinking Ricci soliton, it is well known(c.f. B.L. Chen [5]) that R ≥
0. Then we have f t = |∇ f | ≤ R + |∇ f | = f − µ τ , where we applied (4.33) in the last step. The above inequality can be written as( τ f ) t ≤ − µ . Note that u ( · , s ) > M . Fix an arbitrary point x ∈ M , integrating the aboveinequality implies that τ f ( x , ≤ ( τ + − s ) f ( x , s ) − µ (1 − s ) < ∞ . Since τ >
0, this means f ( x , < ∞ and u ( x , >
0. By the arbitrary choice of x , we knowthe support of u = u ( · ,
1) is M . However, the support of u is Ω according to our assumption.Therefore, we obtain M = Ω .We already know that the space-time M × [ s ,
1] is induced by a gradient shrinking Ricci solitonmetric. It follows from the backward uniqueness of the Ricci flow solution(c.f. Kotschwar [28])that the whole space-time M × [0 ,
1] is induced by a gradient shrinking Ricci soliton metric. Theproof of Theorem 4.2 is complete. (cid:3)
Following the proof of Theorem 4.2, we can obtain several rigidity results.
Proposition 4.7 ( Strong maximum principle in terms of v ) . Same conditions as in Theorem 4.1.If v = at some point ( x , s ) ∈ M × [0 , T ) , then v ≡ on M × [0 , T ] and the flow is induced bythe Gaussian shrinking soliton metric on R m .Proof. By the exact same argument as in the proof of Theorem 4.2, we know that v ≡ M × [ s , T ). Note that (4.32) now reads as R i j + f i j − g i j T − t ) = . (4.34)29n other words, M × [ s , T ) is induced by a gradient shrinking Ricci soliton metric with terminationtime T . We shall show that this soliton must be the Gaussian shrinking soliton on R m . Actually,by the self-similar property of the shrinking soliton, we have Q ( t )( T − t ) ≡ T Q (0) , where Q ( t ) = sup x ∈ M | Rm | ( x , t ). By our hypothesis, the flow { ( M m , x , g ( t )) , ≤ t ≤ T } has boundedcurvature. So we have C ≥ lim sup t → T − Q ( t ) = Q (0) lim t → T − TT − t , which forces that Q (0) = Q ( t ) ≡ t ∈ [0 , T ]. In particular, Rc ≡ M × [0 , T ]. At time t ∈ [ s , T ), (4.34) becomes f i j = g i j T − t ) , which implies that ( M , g ( t )) is a metric cone. However, a flat manifold which is also a metriccone can only be the Euclidean space ( R m , g E ) and f = d ( · , y )4( T − t ) for some point y ∈ R m . There-fore, { ( M m , x , g ( t )) , s ≤ t ≤ T } is the flat Ricci flow induced by the Gaussian soliton metric on( R m , g E ). It then follows from backward uniqueness of Ricci flow again(c.f. [28]) that the wholespace-time M × [0 ,
1) is induced by the Gaussian soliton metric. (cid:3)
Proposition 4.8 ( Rigidity of Ricci flows in terms of µ ) . Let { ( M m , x , g ( t )) , ≤ t ≤ T } be a Ricciflow solution. Then we have µ ( M , g (0) , T ) ≤ . (4.35) Moreover, the equality in (4.35) holds if and only if the flow is the static flow on Euclidean space ( R m , g E ) .Proof. Let u solve the conjugate heat equation (cid:3) ∗ u = δ -function at ( x , T ). Thenwe have µ ( M , g (0) , T ) ≤ W ( R ) (cid:16) M , g (0) , √ u , T (cid:17) ≤ W ( R ) (cid:16) M , g ( t ) , √ u , T − t (cid:17) ≤ lim t → T − W ( R ) (cid:16) M , g ( t ) , √ u , T − t (cid:17) = , (4.36)for every t ∈ (0 , T ). Therefore, we finish the proof of (4.35). Clearly, (4.35) becomes an equalityif the underlying flow is a static flow on R m . Now we assume (4.35) is an equality. Combining theequality form of (4.35) and (4.36), we obtain W ( R ) (cid:16) M , g ( t ) , √ u , T − t (cid:17) ≡ , ∀ t ∈ (0 , T ) . Note that the left hand side of the above equation is exactly R M v , where v ≤ v ≡
0. Then it follows from Proposition 4.7 that the flow isinduced by the Gaussian shrinking soliton on ( R m , g E ). (cid:3) roposition 4.9 ( Non-positivity of ν ) . Suppose ( M m , g ) is a complete Riemannian manifold withbounded curvature, T is a positive number. Then ν ( M , g , T ) ≤ . (4.37) Moreover, the equality in (4.37) holds if and only if ( M , g ) is isometric to the Euclidean space ( R m , g Euc ) .Proof. Since ( M , g ) has bounded curvature, starting from g , there exists a Ricci flow solution { ( M , g ( t )) , ≤ t ≤ ǫ } for some ǫ . Moreover, each time slice of this flow has bounded curvature.Without loss of generality, we may assume that 0 < ǫ < T . It follows from the definition of µ , ν and Proposition 4.8 that ν ( M , g , T ) ≤ ν ( M , g , ǫ ) ≤ µ ( M , g , ǫ ) = µ ( M , g (0) , ǫ ) ≤ , (4.38)which implies (4.37).If equality in (4.37) holds, then every inequality in (4.38) becomes equality. In particular, wehave µ ( M , g (0) , ǫ ) =
0. By Proposition 4.8 again, we know the flow { ( M , g ( t )) , ≤ t ≤ ǫ } is the static Ricci flow on Euclidean space. In particular, ( M , g ) = ( M , g (0)) is isometric to theEuclidean space ( R m , g Euc ). On the other hand, if ( M , g ) is isometric to ( R m , g Euc ), it is clear that ν ( M , g , T ) =
0. The proof of Proposition 4.9 is complete. (cid:3)
Remark 4.10.
Theorem 4.1 is implied by Theorem 4.2. Actually, fix an arbitrary point x ∈ M,we can choose small domains Ω i with smooth boundaries and diam g ( T ) ( Ω i ) → . Accordingly,we choose positive τ i << diam g ( T ) ( Ω i ) such that µ ( Ω i , g ( T ) , τ i ) → . Let ϕ i be the minimizer of µ ( Ω , g ( T ) , τ i ) , and u i be the conjugate heat solution (cid:3) ∗ u = , starting from ϕ i at time T . Then wedefine v i by Theorem 4.2. It is not hard to see that u i converges to a solution u such that (cid:3) ∗ u = on M × [0 , T ) and lim t → T − u is the δ -function based at ( x , T ) . In this way, it is clear that v is thesmooth limit of v i on M × [0 , T ) . Since each v i ≤ on M × [0 , T ) by Theorem 4.2, we obtain v ≤ and consequently prove Theorem 4.1. Theorem 4.11 ( Harnack inequality in terms of positive heat solution values ) . Same conditionsas in Theorem 4.2. Then the following di ff erential Harnack inequality holds:dd τ n √ τ f ( γ ( τ )) o ≤ √ τ (cid:16) R + | ˙ γ | (cid:17) + µ √ τ . (4.39) Suppose x , y are two points on M, γ ( τ ) = ( γ ( τ ) , τ T + T − t ) is a space-time curve parametrizedby τ = τ T + T − t ∈ [ τ T , τ T + T ] such that γ ( τ T ) = ( x , T ) and γ ( τ ) = ( y , . Then the followingHarnack inequality holds:u ( y , ≥ (4 πτ ) − m · e (cid:18) − + q τ T τ (cid:19) µ · e − L ( γ )2 √ τ · n (4 πτ T ) m · u ( x , T ) o q τ T τ (4.40) where L is the Lagrangian defined by L ( γ ) ≔ Z τ τ T √ τ (cid:16) R + | ˙ γ | (cid:17) d τ. (4.41)31 roof. Since (cid:3) ∗ u =
0, it is clear that f τ = ∆ f − |∇ f | + R − m τ . (4.42)The fact v ≤ ∆ f − |∇ f | + R ! + f − m − µ τ ≤ . (4.43)Combining (4.42) and (4.43) yields that f τ − R + |∇ f | + f − µ τ ≤ . Consequently, we have dd τ f ( γ ( τ )) = f τ + h∇ f , ˙ γ ( τ ) i ≤ f τ + |∇ f | + | ˙ γ ( τ ) | ≤ (cid:16) R + | ˙ γ | (cid:17) + µ − f τ , whence we obtain (4.39). Integrating (4.39), we can apply the definition of L in (4.41) to obtain √ τ f ( γ ( τ )) − √ τ T f ( γ ( τ T )) ≤ Z τ τ T √ τ (cid:16) R + | ˙ γ | (cid:17) d τ + (cid:16) √ τ − √ τ T (cid:17) µ ≤ L ( γ ) + (cid:16) √ τ − √ τ T (cid:17) µ . Thus, we have f ( γ ( τ )) ≤ L ( γ )2 √ τ + r τ T τ f ( γ ( τ T )) + − r τ T τ ! µ . Note that γ ( τ ) = y , γ ( τ T ) = x . Recall also that f = − log u − m log(4 πτ ). It follows thatlog u ( y , + m πτ ) ≥ − L ( γ )2 √ τ + r τ T τ (cid:18) log u ( x , T ) + m πτ T ) (cid:19) + − + r τ T τ ! µ , which is equivalent to (4.40). (cid:3) Remark 4.12.
The proof of Theorem 4.11 follows the route of the proof of Theorem 2.1 in thefundamental work of Li-Yau [30]. Similar argument was used by Perelman in Corollary 9.4 of [33].The precise definition of L under Ricci flow was first given by Perelman in Section 7.1 of hiscelebrated work [33], motivated by physics and infinite dimensional geometry(c.f. Section 5 and6 of [33]). In the setup of Theorem 4.11, the quest of the space-time curve γ minimizing L leadsnaturally to the concept of reduced distance and reduced geodesic of Perelman(c.f. Section 7of [33] or Section 7 of the current paper). Therefore, it seems that Theorem 4.11 provides a newperspective to understand the definition of reduced geodesic and reduced distance, which is similarto the ρ -functional of Li-Yau(c.f. equation (3.1) of [30]). Remark 4.13.
Suppose { ( M , g ( t )) , ≤ t < ∞} is the flat Ricci flow on Euclidean space R m . Let ~ a ∈ R m and let x = √ τ T ~ a and y = √ τ ~ a. Let u T ( · ) = (4 πτ T ) − m e − d · , , which is a minimizerfor µ = µ ( M , g ( T ) , τ T ) = . Let γ ( τ ) = ( √ τ~ a , τ T + T − τ ) , which connects ( x , T ) and ( y , .Then direct calculation shows that both (4.39) and (4.40) become equality. This means that allconstants in (4.39) and (4.40) are sharp. E ff ective monotonicity formulas for local functionals The formula of vu in Theorem 4.2 contains an extra term − µ = − µ ( Ω , g ( T ) , τ ), which carries theinformation of the Riemannian manifold(with boundary) ( Ω , g ( T )). We can use this informationto relate the geometry of ( Ω , g ( T )) with other domains at some time t < T . The study alongthis direction leads to e ff ective monotonicity formulas along the Ricci flow, which generalize theglobal monotonicity formulas of Perelman(c.f. Section 3 of [33]). Theorem 5.1 ( Harnack inequality in terms of local µ -functional values ) . Same conditions asin Theorem 4.2. Let Ω ′ ⊂ Ω ⊂ M and h : M → [0 , be a cuto ff function supported on Ω andh ≡ on Ω ′ . Set C h ≔ sup Ω (cid:12)(cid:12)(cid:12)(cid:12) ∇ √ h (cid:12)(cid:12)(cid:12)(cid:12) g (0) . (5.1) For each τ T > , set µ ≔ µ ( Ω , g (0) , τ ) , µ T ≔ µ ( Ω , g ( T ) , τ T ) . (5.2) Then we have µ − µ T ≤ R Ω \ Ω ′ n τ |∇ h | h − − h log h o u R Ω uh ≤ (cid:16) τ C h + e − (cid:17) · R Ω \ Ω ′ u R Ω ′ u . (5.3) Proof.
Since − x log x ≤ e − for each positive x , it is obvious that the second inequality of (5.3)follows from the first inequality of (5.3). Therefore, we only need to prove the first part of (5.3),which will be discussed in details in the next paragraph.Without loss of generality, we assume T =
1. For simplicity of notations, we define S ≔ Z Ω uh (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = ≤ Z Ω u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = ≤ Z M u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = = . (5.4)Let v = n τ (2 ∆ f − |∇ f | + R ) + f − m − µ o u as in Theorem 4.2. At time t =
0, let ˜ u be uhS . Then˜ u satisfies the normalization condition(c.f. (5.25)) R M ˜ u = Ω . Accordingly,we define˜ f ≔ − log ˜ u − m πτ ) = − log u − m πτ ) − log h + log S = f − log h + log S , where τ = τ +
1. Plugging √ ˜ u into the formula (2.2), we obtain µ + m ≤ W ( R ) (cid:16) Ω , g (0) , √ ˜ u , τ (cid:17) + m = Z Ω n τ ( R + ∆ ˜ f − |∇ ˜ f | ) + ˜ f o ˜ u = S Z Ω n τ (cid:16) R + ∆ f − ∆ log h − |∇ f | − |∇ log h | + h∇ f , ∇ log h i (cid:17) + f − log h + log S o uh . v into the above inequality, we arrive at µ ≤ log S + Z Ω (cid:18) vu + µ (cid:19) ˜ u + S Z Ω n τ (cid:16) − ∆ log h − |∇ log h | + h∇ f , ∇ log h i (cid:17) − log h o uh = log S + µ + S Z Ω vh + S Z Ω n τ |∇ log h | − log h o uh = µ + ( log S + S Z Ω vh ) + S Z Ω (cid:26) τ (cid:12)(cid:12)(cid:12)(cid:12) ∇ √ h (cid:12)(cid:12)(cid:12)(cid:12) − h log h (cid:27) u . (5.5)At time t =
0, note that both (cid:12)(cid:12)(cid:12) ∇ √ h (cid:12)(cid:12)(cid:12) and h log h are supported on Ω \ Ω ′ . Recall the fact that v ≤ < S ≤ µ − µ ≤ ( log S + S Z Ω vh ) + S Z Ω \ Ω ′ (cid:26) τ (cid:12)(cid:12)(cid:12)(cid:12) ∇ √ h (cid:12)(cid:12)(cid:12)(cid:12) − h log h (cid:27) u ≤ S Z Ω \ Ω ′ (cid:26) τ (cid:12)(cid:12)(cid:12)(cid:12) ∇ √ h (cid:12)(cid:12)(cid:12)(cid:12) − h log h (cid:27) u , (5.6)whence we obtain (5.3) by the choice of µ , µ , τ and S (c.f. (5.2) and (5.4)). (cid:3) Theorem 5.2 ( E ff ective monotonicity of local functionals ) . Same conditions and notations asin Theorem 4.2. For each λ ∈ [0 , τ T ] , let ϕ ( λ ) T be the minimizer function of µ ( Ω , g ( T ) , λ ) . Let u ( λ ) be the conjugate heat solution starting from (cid:16) ϕ ( λ ) T (cid:17) . Setc u ≔ inf λ ∈ (0 ,τ T ] Z Ω ′ u ( λ ) . (5.7) Define ν ≔ ν ( Ω , g (0) , τ ) , ν T ≔ ν ( Ω , g ( T ) , τ T ) . (5.8) Then we have ν − ν T ≤ inf τ ∈ [ T ,τ T + T ] µ ( Ω , g (0) , τ ) − ν ( Ω , g ( T ) , τ T ) ≤ (cid:16) τ C h + e − (cid:17) · n c − u − o . (5.9) Consequently, we have max (cid:8) µ − µ T , ν − ν T (cid:9) ≤ (cid:16) τ C h + e − (cid:17) · n c − u − o . (5.10) Proof.
It is clear that the first inequality of (5.9) follows directly from the definition of ν (c.f.equation (2.4) and (2.7)). According to inequality (5.3), we have µ (cid:16) Ω , g (0) , τ ′ T + T (cid:17) ≤ (cid:16) τ ′ T + T ) C h + e − (cid:17) · R Ω \ Ω ′ u R Ω ′ u + µ (cid:16) Ω , g ( T ) , τ ′ T (cid:17) , u = u ( τ ′ T ) is clearly a positive function on M × [0 , T ). Consequently, we have R Ω \ Ω ′ u R Ω ′ u ≤ R M \ Ω ′ u R Ω ′ u = R M u − R Ω ′ u R Ω ′ u = R Ω ′ u − ≤ c − u − µ (cid:16) Ω , g (0) , τ ′ T + T (cid:17) ≤ (cid:16) τ ′ T + T ) C h + e − (cid:17) · n c − u − o + µ (cid:16) Ω , g ( T ) , τ ′ T (cid:17) (5.11)for each τ ′ T ∈ (0 , τ T ]. Letting τ ′ T run over [0 , τ T ] and taking infimum of the right hand side of theabove inequality, we obtain the second inequality of (5.9). So we finish the proof of (5.9).The inequality (5.10) follows from the combination of (5.9) and (5.11) by setting τ ′ T = τ T . (cid:3) A particular case of Theorem 5.2 is to choose
Ω = Ω = Ω ′ = M . Then we obtain themonotonicity of the global functionals µ and ν of Perelman(c.f. Section 3 of [33]). Theorem 5.3 ( Monotonicity of µ and ν -functionals, Perelman ) . Let { ( M m , g ( t )) , ≤ t ≤ T } bea Ricci flow solution on a closed manifold. Then we have µ ( M , g ( T ) , τ T ) − µ ( M , g (0) , τ T + T ) ≥ , (5.12) ν ( M , g ( T ) , τ T ) − ν ( M , g (0) , τ T + T ) ≥ , (5.13) for every τ T > . Moreover, if equality in (5.12) or (5.13) holds, then the flow is induced by agradient shrinking soliton metric.Proof. Without loss of generality, we assume T = Ω ′ = M , it is clear that R Ω ′ u ( λ ) ≡ λ ∈ (0 , τ ]. It follows from (5.7) that c u = − ( log S + S Z Ω vh ) ≤ ( µ − µ ) − S Z Ω (cid:26) τ (cid:12)(cid:12)(cid:12)(cid:12) ∇ √ h (cid:12)(cid:12)(cid:12)(cid:12) − h log h (cid:27) u . In the current situation, we have h ≡ Ω = M and µ − µ =
0. So the right hand side ofthe above inequality vanishes. Recall that the left hand side of the above term is nonnegative(c.f.(5.4) and (4.7) in Theorem 4.2). Therefore, it is forced to be zero and we obtain S = v ≡ M at t =
0. In light of the second part of Theorem 4.2, we obtain that the flow is induced by agradient shrinking Ricci soliton metric.If (5.13) becomes an equality, we shall show the flow is also induced by a soliton metric.Actually, it follows from Proposition 4.9 that ν ( M , g (1) , τ ) < M is closed. On the other hand,from the proof of Proposition 2.11, it is clear thatlim s → + ν ( M , g (1) , s ) = . τ ′ ∈ (0 , τ ] such that ν ( M , g (1) , τ ) = inf τ ∈ (0 ,τ ] µ ( M , g (1) , τ ) = µ ( M , g (1) , τ ′ ) . Using (5.12), we then have ν ( M , g (1) , τ ) = µ ( M , g (1) , τ ′ ) ≥ µ ( M , g (0) , + τ ′ ) ≥ ν ( M , g (0) , + τ ) . Since now (5.13) is an equality, we know all the inequalities in the above line become equalities.In particular, we obtain µ ( M , g (1) , τ ′ ) = µ ( M , g (0) , + τ ′ ) . Therefore we return to the equality case of (5.12) and obtain that the flow is induced by a gradientshrinking Ricci soliton metric. (cid:3)
In light of Theorem 5.2, the estimate of the di ff erence of the local functionals is reduced to theestimate of two numbers: the upper bound of C h (c.f. (5.1)) and the lower bound of c u (c.f. (5.7)).If we assume Ω and Ω ′ to be concentric geodesic balls, say Ω = B ( x , r ) and Ω ′ = B ( x , r ),then there is a natural way to estimate C h . Actually, we can choose √ h as a cuto ff function suchthat (cid:12)(cid:12)(cid:12) ∇ √ h (cid:12)(cid:12)(cid:12) < r − . Then it follows from (5.1) that C h < r − . (5.14)The di ffi cult step is to estimate the lower bound of c u . In the remaining part of this section, weshall estimate c u in the case that Ω ′ is very large compared to Ω . However, the really hard case isthat Ω ′ is very small compared to Ω , which situation will be discussed in Section 7. Theorem 5.4 ( Almost monotonicity of local- µ -functional ) . Let A ≥ m be a large constant.Let { ( M m , g ( t )) , ≤ t ≤ T } be a Ricci flow solution satisfyingt · Rc ( x , t ) ≤ ( m − A , ∀ x ∈ B g ( t ) (cid:16) x , √ t (cid:17) , t ∈ (0 , T ] . (5.15) Then we have µ ( Ω ′ T , g ( T ) , τ T ) − µ ( Ω , g (0) , τ T + T ) ≥ − A − , ∀ τ T ∈ (0 , A T ) , (5.16) where Ω ′ T = B g ( T ) (cid:16) x , A √ T (cid:17) and Ω = B g (0) (cid:16) x , A √ T (cid:17) . In particular, we have ν ( Ω ′ T , g ( T ) , τ T ) ≥ − A − + inf τ ∈ [ T ,τ T + T ] µ ( Ω , g (0) , τ ) ≥ − A − + ν ( Ω , g (0) , τ T + T ) . (5.17)Note that the almost monotonicity inequality (5.16) in Theorem 5.4 can also be regarded as ageneralization of the monotonicity inequality (5.12) in Theorem 5.3. Actually, for a given Ricciflow M × [0 , T ], a given point x ∈ M and a given scale τ T >
0, we can always choose a very large A such that (5.15) holds. Then applying (5.16) and letting A → ∞ , in light of (2.15), we obtain(5.12). Similarly, one can obtain (5.13) by (5.17).A key step of the proof of Theorem 5.4 is to apply the condition (5.15) to obtain the c u lowerbound in Theorem 5.2. For the convenience of the readers, we recall the following distance esti-mates which will be used repeatedly in this paper. Their detailed proofs can be found in Section26 of Kleiner-Lott [26]. 36 emma 5.5 ( Distance estimate , c.f. Lemma 8.3 of Perelman [33] and Section 17 of Hamil-ton [24]) . Suppose { ( M , g ( t )) , ≤ t ≤ T } is a Ricci flow solution on a complete manifold M andt ∈ [0 , T ] .(a). If Rc ( x , t ) ≤ ( m − K in the ball B g ( t ) ( x , r ) , then we have (cid:3) d = ( ∂ t − ∆ ) d ≥ − ( m − Kr + r − ! (5.18) whenever t = t and d = d g ( t ) ( · , x ) > r .(b). If Rc ( x , t ) ≤ ( m − K in the union of balls B g ( t ) ( x , r ) ∪ B g ( t ) ( y , r ) , then we haveddt d g ( t ) ( x , y ) ≥ − m − Kr + r − ! (5.19) at time t = t . Combining Lemma 5.5 with the condition (5.15), we have the following Lemma.
Lemma 5.6.
Let { ( M m , g ( t )) , ≤ t ≤ T } be a Ricci flow solution satisfyingt · Rc ( x , t ) ≤ ( m − A , ∀ x ∈ B g ( t ) (cid:16) x , √ t (cid:17) , t ∈ (0 , T ] . Let d ( x , t ) = d g ( t ) ( x , x ) . Then we have (cid:3) (cid:16) d + A √ t (cid:17) ≥ . (5.20) Proof.
At time t ∈ (0 , T ], let r = √ t and K = A ( m − t . Then we have Rc ( · , t ) ≤ ( m − K in B g ( t ) ( x , r ). Consequently, we can apply (5.18) to obtain (cid:3) d ≥ − ( m − · A m − + ! · t − = − A + ( m − ! t − ≥ − A √ t , (5.21)which is equivalent to (5.20). (cid:3) Now we are ready for the proof of Theorem 5.4.
Proof of Theorem 5.4.
Without loss of generality, we assume T =
1. Same as Lemma 5.6, we set d be the function d g ( t ) ( · , x ). Let ψ be a cuto ff function such that ψ ≡ −∞ , ψ ≡ , ∞ )and − ≤ ψ ′ ≤ ψ satisfies ψ ′′ ≥ − ψ, ( ψ ′ ) ≤ ψ. (5.22)The choice of ψ is inspired by the proof of pseudo-locality theorem of Perelman [33]. The proofof the existence of such ψ can be found at the beginning of Section 34 of Kleiner-Lott [26]. SeeFigure 5 for a graph of ψ . 37 = y ss = y = ψ ( s )0Figure 5: The choice of bounded cuto ff function ψ By abusing of notations and setting ψ = ψ (cid:18) d + A √ t A (cid:19) , we can regard ψ as a function on space-time.For each t ∈ [0 , Ω t ≔ B g ( t ) (cid:16) x , A − A √ t (cid:17) , Ω ′ t ≔ B g ( t ) (cid:16) x , A − A √ t (cid:17) . (5.23)It follows from the definition that ψ ( x , t ) = , ∀ x ∈ Ω ′ t ;0 , ∀ x ∈ M \ Ω t . (5.24)We shall study the behavior of the conjugate heat solutions on the space-time support set of ψ . Thedi ff erent domains are illustrated in Figure 6. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) M t = t = t ∂ B g ( t ) (cid:16) x , √ t (cid:17) ∂ Ω ′ t ( x , ∂ Ω t Figure 6: Di ff erent domains for almost monotonicityApplying (5.21) and (5.22), we have (cid:3) ψ = (cid:3) ψ d + A √ t A ! = A (cid:3) d + A √ t ! ψ ′ − A ) ψ ′′ ≤ ψ A . Let h = e − t A ψ . Then we have (cid:3) h ≤
0. At time t =
1, the support set of h is Ω = B ( x , A )and h ≡ Ω ′ = B ( x , A ). Let ϕ ( τ )1 be a minimizer for µ = µ ( Ω ′ , g (1) , τ ) for some positivenumber τ . Starting from u = (cid:16) ϕ ( τ )1 (cid:17) , we solve the equation (cid:3) ∗ u =
0. Recall that u > × [0 , ddt Z M uh = Z M (cid:8) u ( (cid:3) h ) − h (cid:3) ∗ u (cid:9) ≤ . Integrating the above inequality yields that Z M uh (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = ≥ Z M uh (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = = Z Ω uh (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = = e − A Z Ω u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = = e − A . At time t = h = ψ by definition. Therefore, h ≡ Ω ′ and h ≡ Ω ′ . It follows that1 ≥ Z Ω u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = ≥ Z M uh (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = ≥ e − A . (5.25)Note that Ω ′ = B g (0) ( x , A ) and 10 A = · A . Similar to (5.25), we have1 ≥ Z Ω ′ u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = ≥ e − A . (5.26)Also, the gradient estimate of h at t = (cid:12)(cid:12)(cid:12)(cid:12) ∇ √ h (cid:12)(cid:12)(cid:12)(cid:12) = |∇ h | h = ( ψ ′ ) A ψ ≤ A . Applying (5.10) in Theorem 5.2 and setting C h = A , we obtain µ − µ ≤ (cid:26) τ A + e − (cid:27) · R Ω ′ u − . (5.27)Plugging (5.26) into the above inequality, we obtain µ − µ ≤ (cid:26) τ A + e − (cid:27) · (cid:18) e A − (cid:19) . (5.28)Since τ = + τ ∈ (1 , + A ) and A ≥ m , the right hand side of the above inequality can bebounded by A − . Therefore, we finish the proof of (5.16). (cid:3) The almost monotonicity inequality (5.17) implies the following no-local-collapsing Theorem.
Theorem 5.7 ( A local version of no-local-collapsing ) . Suppose { ( M m , g ( t )) , ≤ t ≤ T } is a Ricciflow solution satisfying (5.15) for some A ≥ m. Then for each r ∈ (cid:16) , √ T i and geodesic ballB g ( T ) ( x , r ) ⊂ B g ( T ) (cid:16) x , A √ T (cid:17) where R ( · , T ) ≤ r − , we have | B g ( T ) ( x , r ) | ω m r m ≥ κ, (5.29) where κ is a positive constant and can be chosen as e − m + − + inf τ ∈ [ T , T ] µ ( Ω , g (0) ,τ ) . roof. It follows from (5.17) and Proposition 2.1 that ν (cid:16) B g ( T ) ( x , r ) , g ( T ) , r (cid:17) ≥ ν (cid:16) B g ( T ) (cid:16) x , A √ T (cid:17) , g ( T ) , r (cid:17) ≥ inf τ ∈ [ T , T + r ] µ (cid:16) B g (0) (cid:16) x , A √ T (cid:17) , g (0) , τ + T (cid:17) − A − ≥ inf τ ∈ [ T , T ] µ (cid:16) B g (0) (cid:16) x , A √ T (cid:17) , g (0) , τ (cid:17) − . Note that R ( · , T ) ≤ r − in the ball B g ( T ) ( x , r ). Therefore, (5.29) and the choice of κ follows from(3.11) in Theorem 3.3. (cid:3) Remark 5.8.
Note that the constant κ in (5.29) does not depend on A whenever A ≥ m.Therefore, Perelman’s no-local-collapsing theorem, i.e., Theorem 4.1 of [33], is implied by The-orem 5.7 by choosing A large enough. In the paper [33], to obtain the volume lower bound, oneneed the Riemannian curvature bound in the ball B g ( T ) ( x , r ) . This condition of Riemannian cur-vature bound is then replaced by scalar curvature bound by Perelman later, using a contradictionargument(c.f. Remark 13.13 of Kleiner-Lott [26]). Theorem 5.7 has the advantage that the lowerbound of volume ratio is calculated explicitly, due to estimate (3.11) in Theorem 3.3. In Theorem 5.7, κ depends only on the initial condition inf τ ∈ [ T , T ] µ ( Ω , g (0) , τ ), which is totallydetermined by the local geometry of Ω = B g (0) (cid:16) x , A √ T (cid:17) . As A is a big constant, the geodesicball Ω seems to be large. Therefore, the requirement of inf τ ∈ [ T , T ] µ ( Ω , g (0) , τ ) at the initial timeseems to be strong. Such condition can be replaced by a weaker one(c.f. second part of (1.3)) inTheorem 1.1, with the help of stronger conditions in space-time(c.f. first part of (1.3)). In this section, we shall study the reduced distance and reduced volume density function, startingfrom a probability measure at some fixed time slice T . Such concepts are natural generalization ofthe ones(c.f. Definition 6.1 and Remark 6.2) defined by Perelman in [33].We first recall the concept and basic properties of the reduced distance and reduced volume ofPerelman [33]. Suppose { ( M m , g ( t )) , ≤ t ≤ T } is a Ricci flow solution. Fix y as a base point. Let γ be a spacetime curve parametrized by τ = T − t satisfying γ ( τ ) = ( γ ( τ ) , T − τ ) ∈ M × [0 , T ] , τ ∈ [0 , ¯ τ ] . For each such γ , we can calculate the Lagrangian L by L ( γ ) = Z ¯ τ √ τ (cid:16) R + | ˙ γ | (cid:17) g ( T − τ ) d τ. Note that the above definition coincides with (4.41) by letting τ T →
0. Fix ( x , T − ¯ τ ). Among allsuch γ connecting ( y , T ) and ( x , T − ¯ τ ), there exists at least one curve α such that L is minimized.40uch α is called a shortest reduced geodesic. The reduced distance between ( y , T ) and ( x , T − ¯ τ )is defined as l (( y , T ) , ( x , T − ¯ τ )) = √ ¯ τ L ( α ) . (6.1)Fix y ∈ M , then we denote l ( x , t ) = l (( y , T ) , ( x , t )) . By this notation, l is a function defined on the space-time M × [0 , T ). It was proved by Perelman(c.f.Corollary 9.5 of [33] and Theorem 2.23 of [53]) that (cid:3) ∗ n { π [ T − t ] } − m e − l o ≤ (cid:3) (cid:26) T − t ) (cid:18) l − m (cid:19)(cid:27) ≥ . (6.3)As pointed out by Perelman in section 7 of [33], the maximum principle implies that R ≥ − m T − t ) on M × [0 , T ). Then direct calculation yields that l + m is a nonnegative function on M × [0 , T ). Inother words, we have l (( y , T ) , ( x , t )) + m ≥ , ∀ x , y ∈ M . (6.4)Recall that lim t → T − T − t ) l (( y , T ) , ( · , t )) = d g ( T ) ( · , y ) , which is a large deviation formula dates back to Varadhan [48]. Therefore, the minimum valueof 4( T − t ) (cid:16) l − m (cid:17) is nearly zero as t → T − . Consequently, applying the maximum principle on(6.3) implies that the minimum value of 4( T − t ) (cid:16) l − m (cid:17) is always non-positive for each t ∈ [0 , T ).Therefore, we have 0 ≤ min x ∈ M (cid:26) l (( y , T ) , ( x , t )) + m (cid:27) ≤ m . (6.5)We now generalize the concept of reduced distance and reduced volume density. Definition 6.1.
Suppose pdv g ( T ) is a continuous probability measure of the Riemannian manifold ( M , g ( T )) , i.e., p is a nonnegative continuous function satisfying R M p ( y ) dv g ( T ) ( y ) = . For each ( x , t ) ∈ M × [0 , T ) , we definel ( x , t ) ≔ Z M p ( y ) l (( y , T ) , ( x , t )) dv g ( T ) ( y ) , w ( x , t ) ≔ Z M p ( y ) { π [ T − t ] } − m e − l (( y , T ) , ( x , t )) dv g ( T ) ( y ) . We call l as the reduced distance, w as the reduced volume density function, with respect to theprobability measure pdv g ( T ) at time T . emark 6.2. If we choose the base probability measure p as the Dirac measure at time t = T , ourreduced distance function and reduced volume density function coincide with the ones defined byPerelman in [33]. For simplicity of notations, we shall also use l and w for our reduced distanceand reduced volume density with respect to probability measures. However, it will be clear whatis the base measure p in the context.
Proposition 6.3 ( Subsolution and supersolution ) . Suppose l is the reduced distance function, wis the reduced volume density function, with respect to a probability measure u T dv g ( T ) at time T .Then we have (cid:3) (cid:26) T − t ) (cid:18) l − m (cid:19)(cid:27) ≥ , (6.6) (cid:3) ∗ w ≤ in the distribution sense. In particular, on the space-time M × [0 , T ) , we havel + m ≥ , (6.8) w − u ≤ , (6.9) where u is the conjugate heat solution (cid:3) ∗ u = satisfying the initial condition u = p at time t = T .Proof.
Actually, (6.6) and (6.7) follow from (6.3) and (6.2) respectively, due to the fact that (cid:3) and (cid:3) ∗ are linear operators. (6.8) follows from (6.4), by integration with respect to the probabilitymeasure p ( y ) dv g ( T ) ( y ). Note that w − u is a continuous function starting from 0 at time t = T ,and it satisfies (cid:3) ∗ ( w − u ) ≤
0. Therefore, (6.9) follows from the standard maximum principle forparabolic sub-solutions. (cid:3)
Lemma 6.4 ( Construction of “cuto ff ” function ) . For each A > m, there exists a non-decreasing positive C -function ψ : ( −∞ , . → [1 , ∞ ) such that ψ ≡ on ( −∞ , . and increase to ∞ on (0 . , . . Moreover, ψ satisfies ψ ′ ) ψ − ψ ′′ − A ψ ′ ≥ − F ( A ) ψ, (6.10) where F ( A ) is a constant depending only on A and can be chosen as A .Proof. Solving the second order ODE y ′′ − y ′ ) y + Ay ′ = y blows up at s = .
2, it is not hard to see the general solution is Ce A (0 . − s ) − .For simplicity, we choose C = e −
1) and denote e − e A (0 . − s ) − by y . At s = s = . − A , we have y ( s ) = > , y ′ ( s ) = A · e ( e − , y ′′ ( s ) = A · e ( e + e − . a ≔ e ( e − ∼ . , a ≔ e ( e + e − ∼ . . (6.11)We observe that the following interpolation holds. Claim 6.5.
For every k ∈ (2 , A ) , there is an increasing positive function f on [0 , kA ] such that f (0) = f ′ (0) = f ′′ (0) = , f (cid:16) kA (cid:17) = , f ′ (cid:16) kA (cid:17) = a A , f ′′ (cid:16) kA (cid:17) = a A , (6.12) where a and a are defined in (6.11). This is done by an elementary interpolation. On [0 , kA ], let h be defined as follows h ≔ A ( c (cid:18) Atk (cid:19) + c (cid:18) Atk (cid:19) + c (cid:18) Atk (cid:19) ) . (6.13)Let H be the anti-derivative of h with H (0) = H be the anti-derivative of H with H (0) = H = A ( c t (cid:18) Atk (cid:19) + c t (cid:18) Atk (cid:19) + c t (cid:18) Atk (cid:19) ) , H = A ( c t (cid:18) Atk (cid:19) + c t (cid:18) Atk (cid:19) + c t (cid:18) Atk (cid:19) ) . We want to figure out c , c , c such that h = f ′′ , H = f ′ and H = f . For this purpose, we need = f (cid:16) kA (cid:17) = H (cid:16) kA (cid:17) = k n c + c + c o , a A = f ′ (cid:16) kA (cid:17) = H (cid:16) kA (cid:17) = Ak n c + c + c o , a A = f ′′ (cid:16) kA (cid:17) = h (cid:16) kA (cid:17) = A { c + c + c } . Solving this equation, we obtain c = a − a k − + k − , c = − a + a k − − k − , c = a − a k − + k − . (6.14)Let θ = Atk . Then we have f ′ ( t ) = kA θ (cid:26) c + c θ + c θ (cid:27) . (6.15)Note that c > − c c = a − a k − + k − a − a k − + k − ) = − a a − k − + a − k − − a a − k − + a − k − . k >
2, it follows from a direct calculation that12 − a a − k − − a a − k − > . Consequently, we obtain thatmin θ ∈ [0 , (cid:26) c + c θ + c θ (cid:27) = min (cid:26) c , c + c + c (cid:27) = min (cid:26) c , a k (cid:27) > . It follows from (6.15) that f ′ is a positive function on (0 , kA ]. The condition (6.12) holds naturallyby the construction of f . Therefore, the proof of Claim 6.5 is complete.Recall that A is very large. Let k = √ A and define ψ ( s ) ≔ e A (0 . − s ) − , s ∈ [0 . − A , . + f ( s − . + k + A ) , s ∈ [0 . − k + A , . − A ];1 , s ∈ ( −∞ , . − k + A ] . Here f is the positive increasing function defined in Claim 6.5. Then ψ obviously satisfies all therequired conditions except the di ff erential inequality (6.10), which will be verified in the followingsteps. It is clear that ψ is an increasing C -function. On the interval ( −∞ , . − k + A ] ∪ [0 . − A , . ( ψ ′ ) ψ − ψ ′′ − A ψ ′ = . − k + A , . − A ], where we have 1 ≤ ψ ≤
2. Thus, we have2 ( ψ ′ ) ψ − ψ ′′ − A ψ ′ ≥ ( ψ ′ ) ψ − ψ ′′ − A ψ ′ = (cid:0) f ′ (cid:1) − f ′′ − A f ′ ≥ − A − f ′′ . However, it follows from the construction(c.f. (6.13)) of f thatmax s ∈ [0 , kA ] f ′′ = max s ∈ [0 , kA ] h ( s ) = max θ ∈ [0 , n c θ + c θ + c θ o ≤ | c | + | c | + | c | < , where we used the explicit value of c , c and c in the last step(c.f. (6.14) and (6.11)). Combiningthe previous two inequalities, we have2 ( ψ ′ ) ψ − ψ ′′ − A ψ ′ ≥ − A ≥ − A ψ on the interval [0 . − k + A , . − A ]. Therefore, (6.10) holds on all ( −∞ , . (cid:3) The existence of ψ in Lemma 6.4 was pointed out by Perelman [33]. Its construction wasdescribed in Theorem 28.2 of Kleiner-Lott [26]. In Lemma 6.4, we provide the full details ofthe construction and calculate the explicit value of F ( A ), which will be used in our forthcomingdiscussions(c.f. Remark 7.4). For intuition, the graph of ψ is shown in Figure 7.44 = y = ψ ( s ) y ss = ff function ψ The following property is inspired by the proof of Theorem 8.2 of Perelman [34].
Proposition 6.6 ( Harnack type Estimate of positive super solution ) . Suppose { ( M m , g ( t )) , ≤ t ≤ T } is a Ricci flow solution satisfyingt · Rc ( x , t ) ≤ ( m − A , ∀ x ∈ B g ( t ) (cid:16) x , √ t (cid:17) , t ∈ (0 , T ] . (6.16) Suppose (cid:3) H ≥ and H > on M × [ t , T ] for some t ∈ [0 . T , T ] . Then we have min z ∈ ¯ B g ( t (cid:16) x , . √ T (cid:17) H ( z , t ) ≤ e F (1 − t T ) min z ∈ ¯ B g (1) (cid:16) x , A √ T (cid:17) H ( z , , (6.17) where F = F ( A ) is a large positive number satisfying (6.10).Proof. Let ψ be the cuto ff function defined in Lemma 6.4 and define auxiliary functions˜ d ( x , t ) ≔ d ( x , t ) − A ( t − t )1 − t , (6.18) h ≔ H ψ (cid:16) ˜ d ( x , t ) (cid:17) . (6.19)By the positivity assumption of H , it is clear that h ( · , t ) is positive and finite in B g ( t ) (cid:16) x , A ( t − t )1 − t + . (cid:17) for each t ∈ [ t , h ( · , t ) = H ( · , t ) in B g ( t ) (cid:16) x , A ( t − t )1 − t + . (cid:17) . Also, h ( x , t ) → ∞ whenever x → ∂ B g ( t ) (cid:16) x , A ( t − t )1 − t + . (cid:17) . In particular, at time t =
1, we have h ( · , = H ( · ,
1) in B g (1) ( x , A ). The di ff erent domains are illustrated in Figure 8.45 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) M t = t = t t = t ∂ B g ( t ) (cid:16) x , A ( t − t )1 − t + . (cid:17) ∂ B g ( t ) (cid:16) x , √ t (cid:17) x ∂ B g ( t ) (cid:16) x , A ( t − t )1 − t + . (cid:17) Figure 8: Di ff erent domains for applying conjugate heat equationsNote that (cid:3) h = H (cid:3) ψ + ψ (cid:3) H − h∇ ψ, ∇ H i , ∇ h = H ∇ ψ + ψ ∇ H . At space minimum points of h , we have ∇ h =
0, which implies that ∇ H = − ψ − H ∇ ψ . Conse-quently, by applying the condition (cid:3) H ≥
0, on such minimum points, we have (cid:3) h = H (cid:3) ψ + ψ |∇ ψ | ! + ψ (cid:3) H ≥ H (cid:3) ψ + ψ |∇ ψ | ! . (6.20)However, it is clear that (cid:3) ψ = ψ ′ (cid:3) d − A − t ! − ψ ′′ . (6.21)By our assumption (6.16), it follows from Lemma 5.6 and the fact t ∈ [0 . ,
1] that (cid:3) d − A − t ≥ − A √ t − A − t ≥ − A , ∀ t ∈ [ t , . (6.22)Combining (6.20), (6.21) and (6.22), we obtain (cid:3) h ≥ H ψ ′ (cid:3) d − A − t ! + ψ ′ ) ψ − ψ ′′ ! ≥ H − A ψ ′ + ψ ′ ) ψ − ψ ′′ ! . Plugging (6.10) into the above inequality, at each space minimum point of h ( · , t ), we have (cid:3) h ≥ − F H ψ ≥ − F h , ⇒ (cid:3) (cid:16) e F t h (cid:17) ≥ . Hence, it follows from the maximum principle that h min ( t ) ≤ e F (1 − t ) h min (1) , ∀ t ∈ [ t , . (6.23)Suppose h min ( t ) is achieved at point z t . In other words, h min ( t ) = h ( z t , t ). Clearly, h ( z t , t ) > z t ∈ B g ( t ) (cid:16) x , A ( t − t )1 − t + . (cid:17) . Using the fact that ψ ≥ H ( z t , t ) = h ( z t , t ) ψ ( d ( z , t )) = h min ( t ) ψ ( d ( z , t )) ≤ h min ( t ) ≤ h min (1) e F (1 − t ) . (6.24)46ote that min z ∈ ¯ B g ( t ( x , . H ( z , t ) ≤ H ( z t , t ) , h min (1) ≤ min z ∈ ¯ B g (1) ( x , A ) h ( z , = min z ∈ ¯ B g (1) ( x , A ) H ( z , . Combining the previous inequalities, we obtainmin z ∈ ¯ B g ( t ( x , . H ( z , t ) ≤ e F (1 − t ) min z ∈ ¯ B g (1) ( x , A ) H ( z , , which is equivalent to (6.17) after rescaling 1 to T . (cid:3) Proposition 6.7 ( Estimate of reduced distance ) . Let { ( M m , g ( t )) , ≤ t ≤ T } be a Ricci flowsolution satisfying t · | Rc | ( x , t ) ≤ ( m − A , ∀ x ∈ B g ( t ) (cid:16) x , √ t (cid:17) , t ∈ (0 , T ] . (6.25) Then for each point y ∈ B g ( T ) (cid:16) x , A √ T (cid:17) and x ∈ B g (0 . T ) ( x , . , we havel (( y , T ) , ( x , . T )) ≤ me . F (6.26) where F = F ( A ) is a large positive number satisfying (6.10).Proof. In light of the scaling invariance of the reduced distance, we may assume that T = y ∈ B g (1) ( x , A ), s ∈ [0 . , .
6] and let l ( x , t ) = l (( y , , ( x , t )). Wedefine an auxiliary function ˜ L ≔ − t ) (cid:18) l − m (cid:19) + m + . (6.27)It follows directly from (6.3) that (cid:3) ˜ L ≥ . (6.28)On the other hand, one can apply (6.4) to obtain˜ L = − t ) (cid:18) l + m (cid:19) + m + − m (1 − t ) ≥ mt − m + ≥ t ∈ [0 . , L . Then(6.17) reads as min z ∈ ¯ B g ( s ) ( x , . ˜ L ( z , t ) ≤ e F (1 − s ) min z ∈ ¯ B g (1) ( x , A ) ˜ L ( z , ≤ e . F min z ∈ ¯ B g (1) ( x , A ) ˜ L ( z , . (6.30)However, for each z ∈ M , we have˜ L ( z , = lim t → − ˜ L ( z , t ) = m + + d g (1) ( z , y ) . y ∈ B g (1) ( x , A ), it is clear that min z ∈ B g (1) ( x , A ) ˜ L ( z , = ˜ L ( y , = m +
1. Threfore, it followsfrom (6.30) that min z ∈ ¯ B g ( s ) ( x , . ˜ L ( z , t ) ≤ (2 m + e . F . Plugging the above inequality into the defining equation (6.27), we arrive atmin z ∈ ¯ B g ( s ) ( x , . l ( z , s ) ≤ m + m + − s ) (cid:16) e . F − (cid:17) < me . F , (6.31)as s ∈ [0 . , . z be a point such that l ( z , s ) = min z ∈ ¯ B g ( s ) ( x , . l ( z , s ) . (6.32) Claim 6.8.
For each t ∈ [ s , s + ǫ A ] , we haveB g ( s ) ( x , . ⊂ B g ( s + ǫ A ) ( x , . ⊂ B g ( t ) (cid:16) x , √ t (cid:17) . (6.33) where ǫ A is a small constant defined by ǫ A ≔ mA . (6.34)The proof of Claim 6.8 follows from a standard application of the maximum principle of ballcontainment. For simplicity, we shall only prove the second inequality of (6.33). The proofof the first inequality is almost the same and will be left to interested readers. Actually, since0 . < √ s + ǫ A , it is clear that the second inequality of (6.33) holds strictly whenever t = s + ǫ A .Suppose it fails for some t ∈ [ s , s + ǫ A ]. Then we can assume t ′ to be the smallest t such that (6.33)starts to fail. In other words, the second inequality of (6.33) holds on [ t ′ , s + ǫ A ] and ∂ B g ( s + ǫ A ) ( x , . ∩ ∂ B g ( t ′ ) (cid:16) x , √ t ′ (cid:17) , ∅ . (6.35)Therefore, we can find a point q in the above intersection set and a shortest geodesic σ connecting q to x , with respect to metric g ( s + ǫ A ). According to the choice of σ , it is clear that | σ | g ( s + ǫ A ) = . . (6.36)Also, as σ is contained in the ball B g ( t ) (cid:16) x , √ t (cid:17) for each t ∈ [ t ′ , s + ǫ A ], we can apply our assumption(6.16) to obtain | σ | g ( s + ǫ A ) | σ | g ( t ′ ) ≥ e − R s + ǫ At ′ ( m − At dt ≥ ( t ′ s + ǫ A ) ( m − A ≥ ( − mA ) ( m − A ∼ e − m − m > . , where we used the explicit value of ǫ A in (6.34). Combining (6.36) with the above inequality, weobtain that | σ | g ( t ′ ) < | σ | g ( s + ǫ A ) < . . (6.37)48owever, in view of the facts that q ∈ ∂ B g ( t ′ ) (cid:16) x , √ t ′ (cid:17) and that σ connects q and x , we have | σ | g ( t ′ ) ≥ d g ( t ′ ) ( x , q ) = √ t ′ > √ . > . , which contradicts (6.37). This contradiction establishes the proof of the second inequality of(6.33). The first inequality of (6.33) can be proved similarly and is left to the interested readers.Combining (6.31) and (6.33), we have min z ∈ ¯ B g (0 . + ǫ A ) ( x , . l ( z , s ) < me . F , B g (0 . ( x , . ⊂ B g (0 . + ǫ A ) ( x , . ⊂ B g ( t ) (cid:16) x , √ t (cid:17) , ∀ t ∈ [0 . , . + ǫ A ] . (6.38)Fix an arbitrary point x ∈ B g (0 . ( x , . β connecting ( z , . + ǫ A ) and ( x , . g (0 . + ǫ A ), we can find a piecewise smooth geodesic β connecting z and x . For example, β can be chosen as the concatenation of the shortest geodesic connecting z to x and x to x . The length of β is less than 0 .
4. Furthermore, we can parametrize β by τ = − t ∈ [0 . − ǫ A , .
5] such that β (0 . − ǫ A ) = z , β (0 . = x . Clearly, with respect to g (0 . + ǫ A ), we have | β ′ | g (0 . + ǫ A ) ≡ d g (0 . + ǫ A ) ( z , x ) ǫ A = mAd g (0 . + ǫ A ) ( z , x ) < mA , (6.39)where we used the fact that the length of β is bounded by 0 . β ⊂ B g (0 . + ǫ A ) ( x , . β ( τ ) whenever τ ∈ [0 . − ǫ A , . A is large, it follows from (6.39) that | β ′ | g (1 − τ ) ≤ e R . + ǫ A − τ ( m − At dt | β ′ | g (0 . + ǫ A ) < | β ′ | g (0 . + ǫ A ) < mA . (6.40)Hence, by (6.40) and the Ricci upper bound guaranteed by Claim 6.8, we obtain Z . . − ǫ A √ τ (cid:16) R + | β ′ | (cid:17) d τ < Z . . − ǫ A (cid:16) m ( m − A + | β ′ | (cid:17) d τ < mA . (6.41)In conclusion, we have a space-time curve γ ( τ ) = ( γ ( τ ) , − τ ) connecting ( y ,
1) to ( x , . τ = − t such that γ ( τ ) = α ( τ ) , if τ ∈ [0 , . − ǫ A ]; β ( τ ) , if τ ∈ [0 . − ǫ A , . . Here α is the space projection of a reduced geodesic α connecting ( y ,
1) and ( z , . + ǫ A ). Itfollows from (6.31) and (6.32) that L ( α ) = · (0 . − ǫ A ) · l (( y , , ( z , . + ǫ A )) ≤ { l (( y , , ( z , . + ǫ A )) , } ≤ me . F . (6.42)49he inequality (6.41) can be understood as L ( β ) < mA < A < F . (6.43)Combining (6.42) and (6.43), we obtain L ( γ ) = L ( α ) + L ( β ) < me . F + F . As ¯ τ = .
5, we have 2 √ ¯ τ = √
2. It follows from the definition(c.f. (6.1)) of reduced distance l andthe above inequality that l < me . F + F √ < me . F , whence the inequality (6.26) holds. (cid:3) Theorem 6.9 ( Lower bound of reduced volume density and conjugate heat solution ) . Let { ( M m , g ( t )) , ≤ t ≤ T } be a Ricci flow solution satisfying (6.25). Suppose ϕ T is a nonnegativefunction supported on B g ( T ) (cid:16) x , A √ T (cid:17) satisfying R M ϕ T dv g ( T ) = . Let w be the reduced volumedensity function with respect to the probability measure ϕ T dv g ( T ) . Let u be the conjugate heatsolution starting from ϕ T . In other words, (cid:3) ∗ u = ( ∂ τ − ∆ + R ) u = on M × [0 , T ] and u ( · , T ) = ϕ T .Then for every x ∈ B g (0 . ( x , . , we haveu ( x , . T ) ≥ w ( x , . T ) ≥ (2 π ) − m e − me . F T − m (6.44) where F = F ( A ) is a large positive number satisfying (6.10).Proof. The first inequality of (6.44) follows directly from (6.9). We focus on the proof of thesecond inequality of (6.44). By (6.26), direct calculation implies that w ( x , . T ) = Z M (2 π T ) − m e − l (( y , T ) , ( x , . T )) ϕ T ( y ) dv g ( T ) ( y ) ≥ (2 π T ) − m e − me . F Z M ϕ T ( y ) dv g ( T ) ( y )which yields (6.44) since R M ϕ T ( y ) dv g ( T ) ( y ) = (cid:3) As discussed in the paragraph before Theorem 5.4, to relate the local functionals of di ff erenttime slices, it is a key step to estimate the uniform lower bound of c u . Now Theorem 6.9 providessuch a lower bound, by comparing u with the reduced volume density function w starting froma probability measure(c.f. Definition 6.1). Consequently, we are now ready to compare the localfunctionals of Ω with the local functionals of Ω , even when Ω is very small compared to Ω , interms the notation of e ff ective monotonicity theorem, i.e., Theorem 5.2. The purpose of this section is to generalize Section 8 of Perelman’s paper [33]. Let us first recallTheorem 8.2 of Perelman([33]). 50 heorem 7.1 (Theorem 8.2 of Perelman [33]) . For any A > there exists κ = κ ( m , A ) > withthe following property. If g i j ( t ) is a smooth solution to the Ricci flow ∂∂ t g i j = − R i j , ≤ t ≤ r , which has | Rm | ( x , t ) ≤ m − r − for all ( x , t ) satisfying d g (0) ( x , x ) < r , and the volume of the metricball B g (0) ( x , r ) is at least A − r m , then g i j ( t ) cannot be κ -collapsed on the scales less than r at apoint ( x , r ) with d g ( r ) ( x , x ) ≤ Ar . We have modified the statement of Theorem 8.2 of Perelman([33]) slightly following Kleiner-Lott [26]. One can check Theorem 27.2 and Remark 27.3 of Kleiner-Lott [26] for more details.Clearly, such modifications do not a ff ect its application at all. Note that the “ κ -collapsed” inTheorem 7.1 means that | B g ( t ) ( x , r ) | g ( t ) r − m < κ whenever | Rm | ( x , t ) ≤ r − for all ( x , t ) satisfying d g ( t ) ( x , x ) < r and t − r ≤ t ≤ t . This seemsto be a strong condition and sometimes hard to obtain, especially in the high dimensional case.In the following theorem, we shall show that the Riemannian curvature bound in a space-timeneighborhood of ( x , t ) is not needed, a scalar curvature bound in a space neighborhood of ( x , t )is su ffi cient to draw similar non-collapsing conclusion.Our starting point is the relationships between volume ratios and the local ν -functionals, ex-pressed explicitly in Theorem 3.3 and Theorem 3.6. Based on the e ff ective monotonicity(c.f.Theorem 5.2) and the uniform lower bound of c u (c.f. Theorem 6.9), we can first set up an estimateof the propagation of the local ν -functionals. Then we use the equivalent relationships to transformit to an estimate of the propagation of the volume ratios. Theorem 7.2 ( Propagation estimate of local ν -functional ) . Suppose { ( M m , g ( t )) , ≤ t ≤ T } isa Ricci flow solution. Suppose < r < √ T and B g ( T ) ( y , r ) is a geodesic ball where the scalarcurvature R ( · , T ) < r − . Suppose there is a big constant A > m such that t | Rc | ( x , t ) < ( m − A , for each x ∈ B g ( t ) (cid:16) x , √ t (cid:17) , < t < T ; B g ( T ) ( y , r ) ⊂ B g ( T ) (cid:16) x , A √ T (cid:17) . (7.1) Define ν a ≔ ν (cid:16) B g (0 . T ) (cid:16) x , . √ T (cid:17) , g (0 . T ) , . T (cid:17) , ν b ≔ ν (cid:16) B g ( T ) (cid:16) x , A √ T (cid:17) , g ( T ) , T (cid:17) , ν c ≔ ν (cid:16) B g ( T ) ( y , r ) , g ( T ) , r (cid:17) . (7.2) Then the following inequality holds: ν c ≥ ν b ≥ ν a − e − ν a + G , (7.3) where G is a large constant chosen as G ≔ me A . (7.4)51 roof. The first part of (7.3) follows directly from the definition (2.4) and Proposition 2.1. Wefocus on the proof of the second part of (7.3).Without loss of generality, we assume T =
1. For simplicity of notations, we define(c.f.Figure 9): Ω ′ a ≔ B g (0 . ( x , . , Ω a ≔ B g (0 . ( x , . , Ω b ≔ B g (1) ( x , A ) , Ω c ≔ B g (1) ( y , r ) . (7.5) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) M t = t = . t = t Ω c ∂ B g ( t ) (cid:16) x , √ t (cid:17) Ω ′ a Ω a Ω b x y Figure 9: Di ff erent domains for comparing ν -functionalsWe shall apply Theorem 5.2, regarding Ω a as Ω , Ω ′ a = Ω ′ and Ω b = Ω respectively. Naturally,the default metric for the discussion with respect to Ω ′ a and Ω a is g (0 . Ω b and Ω c is g (1).Using the terminology in Theorem 5.2, it follows from (6.44) in Theorem 6.9, (5.7) and (5.14)that C h ≤ , c u ≥ (2 π ) − m e − me . F | Ω ′ a | . (7.6)As mentioned before, the lower bound of c u is a key. Applying (5.9) of Theorem 5.2, it followsfrom the above inequality and the fact that τ . ≤ . ν a − ν b ≤ τ . + c u < π ) m e me . F | Ω ′ a | . (7.7)Recall that(c.f. (7.1) and (7.5)) we have Rc ≥ − m − A = ¯ Λ in B g (0 . (cid:16) x , √ . (cid:17) , which contains5 Ω a and Ω ′ a . Using Gromov-Bishop volume comparison, similar to the discussion around (3.36),we can apply the fact 1 < sinh tt < e t to obtain | Ω ′ a || Ω a | ≥ R . (cid:18) sinh √ Ar √ A (cid:19) m − dr R . (cid:18) sinh √ Ar √ A (cid:19) m − dr ≥ m − (0 . m e . m − √ A · m − · (0 . m = − m e − . m − √ A . (7.8)Now we combine (7.7) and (7.8). Because F = A (c.f. the choice of F in Lemma 6.4) isvery large, we can absorb the extra constants and obtain that ν a − ν b < e . me . F | Ω a | . (7.9)52owever, in light of the scalar curvature upper bound R ≤ m ( m − A in Ω a , we can bound | Ω a | from below | Ω a | ω m (0 . m ≥ e ν − m + − m ( m − A · (0 . , (7.10)where ν = ν ( Ω a , . . < .
5, it is clear(c.f. the first inequality of (3.9) in Proposi-tion 3.2) that ν = ν ( Ω a , . ≥ ν ( Ω a , . = ν a . Thus, putting the above inequality into (7.10) yields that | Ω a | ≥ e ν a − m + − . m ( m − A − m log 10 ω m ≥ e ν a − m + − . m A ω m > e ν a − m + − m A − log Γ ( m + , (7.11)where we used the volume formula (3.15). The combination of (7.9) and (7.11) yields that ν a − ν b ≤ e − ν a · e . me . F + m + + m A + log Γ ( m + < e − ν a + me . F , where we used again the fact that F = A is very large. Plugging F = A into the aboveinequality, we arrive at the second inequality of (7.3). The proof of the Theorem is complete. (cid:3) From the Ricci flow point of view, the inequality (7.3) in Theorem 7.2 is natural. However,many geometers are not acquainted with the local functional ν . Therefore, for the convenience ofthe readers, we also provide a volume ratio version of inequality (7.3) in the following Theorem.The basic idea is first to transform the volume ratio information into the local functional informa-tion in the domain with bounded Ricci curvature, by Theorem 3.6. Then we apply inequality (7.3)in Theorem 7.2 to obtain an estimate of the local ν -functional in a destination domain far away.Finally, we transform the local functional information back into volume ratio information at thedestination domain, by Theorem 3.3. In this way, we obtain a propagation estimate (7.13) of thevolume ratio. Theorem 7.3 ( Propagation of non-collapsing constant ) . Same conditions as in Theorem 7.2.Set ρ a ≔ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) B g (cid:16) . √ T (cid:17) (cid:16) x , . √ T (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω m (cid:16) . √ T (cid:17) m , ρ c ≔ | B g ( T ) ( y , r ) | ω m r m . (7.12) Then we have ρ c ≥ ρ m + a e − G ρ a (7.13) where G = me A , a large constant as defined in (7.4).Proof. Without loss of generality, we assume T =
1. It follows from (7.9) that ν a − ν c ≤ ν a − ν b ≤ ρ − a · e . me . F . (7.14)53ecall that ν a = ν ( B g (0 . ( x , . , g (0 . , . | R | ≤ m ( m − A on Ω a = B g (0 . ( x , . ν ( Ω a , . ≤ ν a + m + . · m ( m − A ≤ ν a + m A . Plugging the above inequality into (7.14), we arrive at − m A + ν ( Ω a , . − e . me . F ρ a ≤ v ( Ω c , r ) = v c . (7.15)Now we need to transform all ν ’s into the volume ratios. Notice that (3.33) in Theorem 3.6 can beapplied by choosing K = √ A and Λ = m ( m − A . Using the fact that A is large, we have − m A + ( m +
1) log ρ a ≤ ν ( Ω a , . . (7.16)Similarly, as R ≤ r − in Ω c at time t =
1, we can apply (3.33) again by choosing ¯
Λ = r − . So wehave ν c = ν ( Ω c , r ) ≤ log ρ c + n m + + o . (7.17)Putting (7.15), (7.16) and (7.17) together, we obtain − m A + ( m +
1) log ρ a − e . me . F ρ a ≤ log ρ c + m + . Absorbing the extra terms into e F , we havelog ρ c ≥ ( m +
1) log ρ a − e . me . F ρ a − e F = ( m +
1) log ρ a − n e . me . F + e F ρ a o ρ a . (7.18)However, it follows from Gromov-Bishop volume comparison that ρ a ≤ m · m Z . sinh √ Ar √ A m − dr ≤ e . √ A ( m − . Thus we obtain e F ρ a ≤ e F + . √ A ( m − ≤ e F << e . me . F . (7.19)Combining (7.18) with (7.19), noting that 2 e . me . F << e me . F , we thus arrive atlog ρ c ≥ ( m +
1) log ρ a − e me . F ρ a , which is equivalent to (7.13), since F = A as chosen in Lemma 6.4. The proof of theTheorem is complete. (cid:3) emark 7.4. Suppose the Ricci flow in Theorem 7.3 is Ricci flat. Then one can use standard ballcontainment argument and Bishop-Gromov volume comparison to obtain that ρ c ≥ (cid:12)(cid:12)(cid:12)(cid:12) B g ( T ) (cid:16) y , A √ T (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ω m (cid:16) A √ T (cid:17) m ≥ (cid:12)(cid:12)(cid:12)(cid:12) B g ( T ) (cid:16) x , . √ T (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ω m (cid:16) A √ T (cid:17) m = ρ a · (30 A ) − m . (7.20) Comparing the above inequality with (7.13) in Theorem 7.3, it is clear that the above inequalityis much stronger. This suggests that the estimate (7.13) should not be sharp. It will be interestingto ask whether (7.13) can be improved to an inequality similar to (7.20). One key step for such animprovement will be the construction of cuto ff function with better F ( A ) in Lemma 6.4. This ispartly the reason why we explicitly estimate F ( A ) in details there. Now we are ready to finish the proof of Theorem 1.1, which is nothing but a corollary ofTheorem 7.3. For the convenience of the readers, we rewrite Theorem 1.1 in the following slightlymore general way.
Theorem 7.5 ( Improved version of no-local-collapsing ) . For every A > there exists κ = κ ( m , A ) > with the following property. Suppose { ( M m , g ( t )) , ≤ t ≤ r } is a Ricci flow solutionof the type ∂ t g = − k {− Rc + λ ( t ) g } (7.21) where ≤ k ≤ and | λ ( t ) | ≤ . Supposer | Rm | ( x , t ) ≤ m − , ∀ x ∈ B g (0) ( x , r ) , ≤ t ≤ r ; r − m (cid:12)(cid:12)(cid:12) B g (0) ( x , r ) (cid:12)(cid:12)(cid:12) dv g (0) ≥ A − . (7.22) Then we have r − m (cid:12)(cid:12)(cid:12) B g ( t ) ( x , r ) (cid:12)(cid:12)(cid:12) dv g ( t ≥ κ (7.23) whenever A − r ≤ t ≤ r , < r ≤ r , and B g ( t ) ( x , r ) ⊂ B g ( t ) ( x , Ar ) is a geodesic ball satisfyingr R ( · , t ) ≤ . Comparing Theorem 7.5 with Theorem 7.1, the conclusion of Theorem 7.5 is much stronger.We obtain uniform non-collapsing estimates only assuming the scalar curvature’s local upperbound in a given time-slice, rather than the Riemannian curvature bound in a space-time neigh-borhood. The statement of Theorem 7.5 seems to be slightly more general than Theorem 1.1.However, by a standard parabolic rescaling and the fact that both log k and λ ( t ) are uniformlybounded, it is clear that Theorem 7.5 and Theorem 1.1 are equivalent.We close this section by providing the proof of Theorem 1.1. The proof is basically to useTheorem 7.3 on a very short period of Ricci flow before the time t = t to estimate the lowerbound of r − m (cid:12)(cid:12)(cid:12) B g ( t ) ( x , r ) (cid:12)(cid:12)(cid:12) dv g ( t by the volume ratio of a small ball in the domain with boundedgeometry. By Bishop-Gromov volume comparison, a lower bound of the volume ratio of the smallball can be obtained by the non-collapsing condition of the relatively bigger ball B g (0) ( x , r ). Thefull details are given below. 55 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) M t − A − t a t t B g ( t ) ( x , r ) ∂ n B g (0) ( x , × [0 , o ∂ B g ( t ) (cid:16) x , p t − t + A − (cid:17) Ω a x x Figure 10: A short period of Ricci flow
Proof of Theorem 1.1.
Without loss of generality, we assume r =
1. Fix a time t = t ∈ [ A − , ffi ces to show that r − m | B g ( t ) ( x , r ) | dv g ( t ≥ κ. (7.24)We focus attention on the Ricci flow space-time { ( M m , g ( t )) , t − A − ≤ t ≤ t } . Let ¯ A = A and s = t − t + A − . By the first part of (7.22), we have s | Rc | ( x , t ) ≤ ( m − A ≤ ( m −
1) ¯ A , ∀ x ∈ B g ( t ) (cid:16) x , √ s (cid:17) , t − A − ≤ t ≤ t . Note that B g ( t ) ( x , r ) ⊂ B g ( t ) ( x , A ) ⊂ B g ( t ) (cid:18) x , ¯ A p A − (cid:19) . Therefore, up to a time shifting, we can apply Theorem 7.3 with constant ¯ A = A and T = A − .The upshot is ω − m r − m | B g ( t ) ( x , r ) | dv g ( t ≥ ρ m + a e − A ρ a , (7.25)where ρ a is the volume ratio(c.f. the first part of (7.26)) of the ball B (cid:16) x , . A − (cid:17) at the time t = t − . A − . For simplicity, we denote this ball as Ω a and denote the time t − . A − by t a . Note that | t a − t | = . A − , which is a very small number. However, from the first part of(7.22) and the Ricci flow equation, we can compare small geodesic balls at time t a and time t . Inparticular, the following relationship holds: B g (0) (cid:18) x , . A − (cid:19) ⊂ Ω a ⊂ B g (0) (cid:18) x , . A − (cid:19) . Recall that the volume element evolution is dominated by − R . In the very short time period, thevolume element is almost fixed. In other words, we can apply the first part of (7.22) again andobtain that | Ω a | dv g ( ta ) ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) B g (0) (cid:18) x , . A − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dv g ( ta ) ≥ . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) B g (0) (cid:18) x , . A − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dv g (0) . t =
0, as | Rc | ≤ ≤ ( m − · , we have (cid:12)(cid:12)(cid:12)(cid:12) B g (0) (cid:16) x , . A − (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) dv g (0) (cid:12)(cid:12)(cid:12) B g (0) ( x , (cid:12)(cid:12)(cid:12) dv g (0) ≥ R . A − sinh m − rdr R sinh m − rdr ≥ (cid:16) . A − (cid:17) m e m − · m ≥ − m A − m , where we again used the fact that 1 < sinh ss < e s for each positive s , as done in (3.36). Recall that (cid:12)(cid:12)(cid:12) B g (0) ( x , (cid:12)(cid:12)(cid:12) dv g (0) ≥ A − according to our assumption. Combining the previous steps, we obtain | Ω a | dv g ( ta ) ≥ . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) B g (0) (cid:18) x , . A − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dv g (0) > − m − A − m − . In other words, it means that ρ a = ω − m (cid:18) . A − (cid:19) − m | Ω a | dv g ( ta ) ≥ ω − m − m − A − . (7.26)Plugging this into (7.25), we obtain (7.24) for some positive κ = κ ( m , A ), which can be chosen as ω m · (cid:16) m + ω m A (cid:17) − ( m + · e − · m + · ω m A . The proof of the Theorem is complete. (cid:3)
In this section, we briefly discuss one application of our general theory developed in previoussections. Let X be a projective manifold of complex dimension n . X is called a minimal model ofgeneral type if the canonical bundle K X is big ( K nX ,
0) and nef(numerically e ff ective). Startingfrom an initial K¨ahler metric g , we run the Ricci flow ∂∂ t g = − Ric − g . (8.1)Let ω be the metric form corresponding to g , i.e., ω ( · , · ) = g ( J · , · ) for the complex structure J . Then the K¨ahler condition ∇ J ≡ ω ( t ) be the metric formcompatible to both g ( t ) and J . Then (8.1) can be rewritten as the following evolution equation: ∂ω∂ t = − Rc ( ω ) − ω. (8.2)With some e ff orts(c.f. Tsuji [47], Tian-Zhang [45]), the above equation (8.2) can be simplified asan evolution equation of scalar functions. Actually, we can choose a smooth volume form Ω on X and denote √− ∂ ¯ ∂ log Ω by χ . It is clear that [ χ ] = c ( K X ) = − c ( X ). Then we have[ ω ( t )] = e − t [ ω ] + (1 − e − t )[ χ ] . (8.3)57herefore, up to an additive constant, ω ( t ) can be uniquely determined as χ + e − t ( ω − χ ) + √− ∂ ¯ ∂ u for some smooth function u = u ( · , t ). In terms of u , the equation (8.2) can be translated as thefollowing complex Monge-Amp`ere equation ∂ u ∂ t = log (cid:16) χ + e − t ( ω − χ ) + √− ∂ ¯ ∂ u (cid:17) n Ω − u , (8.4)starting from u ( · , ≡
0. By the work of Tsuji [47] and Tian-Zhang [45], the equation (8.4)has long time existence. It was also shown there that (8.4) converges in the distribution sense to aK¨ahler-Einstein current. Moreover, the convergence can be improved to be in the smooth topologyoutside the exceptional locus B , which is canonically determined and will be explained in the nextparagraph. A natural question is whether one have the “global” convergence of (8.2) or (8.4),including the behavior of the exceptional locus. The following conjecture is well-known. Conjecture 8.1 (c.f. Conjecture 4.1 of [40] and Conjecture 6.2 of [41]) . Suppose X is a smoothminimal model of general type. The normalized K ¨ahler-Ricci flow (8.2) converges to the unique(possibly singular) K ¨ahler-Einstein metric ω KE on X can in the Gromov-Hausdor ff topology ast → ∞ . Let us briefly explain the meaning and history of the above conjecture. The condition that X isa minimal model of general type is equivalent to the fact that K X is big and nef. Hence one can usesections of the pluri-canonical line bundle K ν X to define a map ι from X to CP N , for ν su ffi cientlylarge. This map is an embedding map on X \B for some exceptional set B . However, ι fails to bean embedding on B , which is at least complex co-dimension 1 and does not depend on ν . If theexceptional set B = ∅ , i.e., K X is ample or c ( X ) <
0, then the above conjecture holds automaticallyby the classical result of H.D. Cao [3]. In general, B , ∅ , ι ( X ) is isomorphic to X can , the canonicalmodel of X . There is a unique K¨ahler Einstein current ω KE on X such that [ ω KE ] = [ c ( K X )] and itsrestriction on X \B is a genuine smooth K¨ahler Einstein metric(c.f. [14] and the references therein).The metric completion of ( X \B , ω KE ) can be regarded as a canonical metric on the singular variety X can = ι ( X ). By abusing of notations, we denote this metric completion by ( X can , ω KE ), which isclearly also unique . To answer Conjecture 8.1, it is important to understand the degeneration ofmetrics ω ( t ) along the exceptional set B . Global estimates along the flow need to be developedalong the flow. The last decade witnessed many important progresses along this direction. First, itwas confirmed by J. Song and B. Weinkove(c.f. [42] and [43]) that Conjecture 8.1 holds for many2-dimensional manifolds. Then, B. Guo [18] proved Conjecture 8.1 under the assumption thatRicci curvature is uniformly bounded from below. Recently, Conjecture 8.1 was solved completelyin low dimension, by Guo-Song-Weinkove [19] in dimension 2 and Tian-Zhang [46] in dimension3. We confirm this conjecture for general dimension by Theorem 1.2. In our solution, the uniformscalar curvature bound by Z. Zhang [56] and the diameter bound of ( X can , ω KE ) by J. Song [40]will play important roles. For the convenience of the readers, we rewrite Theorem 1.2 as follows. Theorem 8.2.
Let X be a projective manifold with K X big and nef. Then the K ¨ahler Ricci flow (8.2)has uniformly bounded diameter and converges to the unique singular K ¨ahler-Einstein metric ω KE on X can in the sense of Gromov-Hausdor ff as t → ∞ .Proof of Theorem 8.2. It is known (c.f. Tsuji [47] and Tian-Zhang [45]) that ω t converges smoothlyto ω KE on X \B as time t → ∞ . Fix x ∈ X \B and choose r small enough such that the 2 r -neighborhood (with respect to the metric ω KE ) of x locates in X \B . Recall that ( X can , ω KE ) is the58etric completion of ( X \B , ω KE ). In light of the results of J. Song(c.f. Theorem 4.1 of [40]), wecan assume L < ∞ to be the diameter of ( X can , ω KE ). We claim thatlim t →∞ diam( X , g ( t )) ≤ L . (8.5)For otherwise, we can find t i → ∞ and y i ∈ X such that lim t →∞ d g ( t i ) ( x , y i ) ≥ L + ǫ for some positivenumber ǫ ∈ (0 , y i if necessary, we can further assume thatlim t →∞ d g ( t i ) ( x , y i ) = L + ǫ < L + . (8.6)In light of (8.3), we have the global volume estimatelim t →∞ | X | ω nt = | X \B| ω nKE . (8.7)For each compact set K ⋐ X \B , it follows from the choice of L and (8.6) that B g ( t i ) ( y i , ǫ ) ∩ K = ∅ for large i . Therefore the “volume-squeezing” implieslim i →∞ (cid:12)(cid:12)(cid:12) B g ( t i ) ( y i , ǫ ) (cid:12)(cid:12)(cid:12) ω nti ≤ inf K ⋐ X \B lim sup i →∞ | X \ K | ω nti = inf K ⋐ X \B |{ X \B} \ K | ω nKE = . (8.8)Let g i ( t ) = g ( t + t i − r ). By the currents convergence, we know B g i (0) ( x , r ) × [ − r , r ] smoothlyconverges to B g KE ( x , r ) × [ − r , r ]. Namely, the regularity theory of complex Monge-Amp`ereequations(c.f. Tsuji [47], Kolodziej [27] and Tian-Zhang [45]) implies that B g i (0) ( x , r ) × [ − r , r ]has uniformly bounded geometry. Recall that g i is a solution of (8.1), y i has bounded distanceto x by (8.6), scalar curvature is uniformly bounded by Λ along (8.1) in view of the estimate ofZ. Zhang [56]. Consequently, up to a parabolic rescaling, we can apply Theorem 1.1(or applyTheorem 7.5 directly for k = λ ( t ) ≡
1) to the flow g i and obtain that ǫ − n (cid:12)(cid:12)(cid:12) B g ( t i ) ( y i , ǫ ) (cid:12)(cid:12)(cid:12) ω nti = ǫ − n (cid:12)(cid:12)(cid:12)(cid:12) B g i ( r ) ( y i , ǫ ) (cid:12)(cid:12)(cid:12)(cid:12) dv gi (0) ≥ κ (8.9)uniformly for some positive κ = κ ( ω KE , x , L , n , Λ ). However, the above inequality contradicts(8.8). This contradiction establishes the proof of (8.5).The equation (8.5) implies that ( X , ω ( t )) has uniformly bounded diameter along the flow (8.2).Similar to the proof of (8.9), we can apply Theorem 1.1 again to obtain uniform non-collapsing.In other words, there is a constant κ > r − m | B ( y , r ) | dv g ( t ) > κ (8.10)uniformly for every ( y , t ) in the flow space-time and r ∈ (0 , X \B is an ǫ -almostisometry from ( X , ω ( t )) to ( X can , ω KE ), for every ǫ and correspondingly large t . It follows thatlim t →∞ d GH (( X , ω ( t )) , ( X can , ω KE )) ≤ ǫ, whence we finish the proof of convergence by letting ǫ → (cid:3) Remark 8.3.
The uniform κ -non-collapsing (8.10) and uniform diameter bound (8.5) are the keydi ffi culties for proving Theorem 8.2. The correspondent estimates in the Fano K ¨ahler Ricci flowwere discovered by Perelman and refined by Sesum-Tian [38].
59 natural idea to show (8.10) is to apply Perelman’s ν -functional. Let Ω be B g ( t ) ( x , r ). Thenwe have the following formal inequalities: ν ( Ω , g ( t ) , r ) ≥ ν ( M , g ( t ) , r ) ≥ ν ( M , g (0) , t + r ) ≥ ν ( M , g (0)) . (8.11)Consequently, we can apply the scalar curvature bound and Theorem 3.3 to obtain the volumeratio lower bound. However, a pitfall is that ν ( M , g (0)) could be −∞ , which makes the finalinequality trivial and prevent us from extracting useful information from the inequalities. In theproof of Theorem 8.2, we use Theorem 1.1, where the inequalities (8.11) were localized and theaforementioned pitfall was avoided. Remark 8.4.
Under the conditions of Theorem 8.2, beyond (8.10) and (8.5), many other uniformestimates hold along the flow. For example, there exists uniform non-inflation bound, dual to theFano K ¨ahler Ricci flow case(c.f. Q. Zhang [55] and Chen-Wang [6]). The limit length space ( X can , ω KE ) has a regular-singular decomposition R ∪ S such that R is a geodesic convex Ein-stein manifold and S has Hausdor ff codimension at least (c.f. Tian-Wang [44] and Song [40]).Furthermore, the convergence topology of Theorem 8.2 could be better and one can discuss the“space-time” convergence, in the so called ˆ C ∞ -Cheeger-Gromov topology, of the K ¨ahler Ricciflow on general-type minimal projective manifolds, which mirrors the picture of the K ¨ahler Ricciflow on Fano manifolds(c.f. Chen-Wang [7], [8]). The full details will be provided in a separatepaper [51]. References [1] C. B ¨ohm, B. Wilking,
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