HEAT KERNEL AND CURVATURE BOUNDS IN RICCI FLOWS WITH BOUNDED SCALAR CURVATURE 5 RICHARDH.BAMLER AND QI S.ZHANG 1 0 2 Abstract. InthispaperweanalyzeRicciflowsonwhichthescalarcurvatureisglobally orlocally boundedfrom abovebyauniform ortime-dependentconstant. OnsuchRicci v flowsweestablishanewtime-derivativeboundforsolutionstotheheatequation. Based o N on this bound, we solve several open problems: 1. distance distortion estimates, 2. the existence of a cutoff function, 3. Gaussian bounds for heat kernels, and, 4. a backward 8 pseudolocality theorem, which states that a curvature bound at a later time implies a 1 curvaturebound at a slightly earlier time. Usingthebackwardpseudolocalitytheorem,wenextestablishauniformL2curvature ] boundindimension4andweshowthattheflowindimension4convergestoanorbifold G at a singularity. We also obtain a stronger ε-regularity theorem for Ricci flows. This D result is particularly useful in the studyof K¨ahler Ricci flows on Fano manifolds, where . it can beused toderive certain convergence results. h t a m [ Contents 3 v 1. Introduction 1 1 2. Preliminaries 8 9 2 3. Heat kernel bounds, distance distortion estimates and the construction of a 1 cutoff function 10 0 4. Mean value inequalities for the heat and conjugate heat equation 18 . 1 5. Bounds on the heat kernel and its gradient 26 0 6. Backward pseudolocality 28 5 7. L2 curvature bound in dimension 4 34 1 : Acknowledgements 43 v i References 43 X r a 1. Introduction In this paper we analyze Ricci flows on which the scalar curvature is locally or globally bounded by a time-dependent or time-independent constant. We will derive new heat kernel and curvature estimates for such flows and point out the interplay between these two types of estimates. We brieflysummarizethemain results of this paper. More details can befoundtowards the end of the introduction. Consider a Ricci flow (M,(g ) ), ∂ g = 2Ric and assume that the scalar curvature R is locally bounded fromtatb∈o[0v,eT)by atgtiven−constagntt R . 0 Date: November20, 2015. 1 2 RICHARDH.BAMLERANDQIS.ZHANG We refer to thestatements of the theorems for a precise characterization whatwe mean by “locally”. Assuming the bound on the scalar curvature, we obtain the following results: Distance distortion estimate: The distance between two points x,y M changes by at ∈ mostauniformfactoronatime-intervalwhosesizedependsonR andthedistance 0 between x,y at the central time (see Theorem 1.1). This estimate addresses a question that was first raised by R. Hamilton. Construction of a cutoff function: We construct a cutoff function in space-time, whose supportis contained ingiven aparabolicneighborhoodandwhosegradient, Lapla- cian and time-derivative are bounded by a universal constant (see Theorem 1.3). Gaussian bounds on the heat kernel: Assuming a global bound on the scalar curvature, we deduce that the heat kernel K(x,t;y,s) on M [0,T) is bounded from above × and below by an expression of the form a (t s) n/2exp( a d2(x,y)/(t s)) for 1 − − − 2 s − certain positive constants a ,a (see Theorem 1.4). 1 2 Backward Pseudolocality Theorem: Named in homage to Perelman’s forward pseudolocal- ity theorem, this theorem states thatwhenever thenormof theRiemanniancurva- ture tensor on an r-ball at time t is less than r 2, then the Riemannian curvature − is less than Kr 2 on a smaller ball of size εr at earlier times [t (εr)2,t] (see − − Theorem 1.5). Strong ε-regularity theorem: If the local isoperimetric constant at a time-slice is close to theEuclideanconstant,thenwehaveacurvatureboundatthattime(seeCorollary 1.6). L2-curvature bound in dimension 4: If the scalar curvature is globally bounded on M × [0,T), then the L2-norm of the Riemannian curvaturetensor, Rm(,t) , is k · kL2(M,gt) bounded by a uniform constant, which is independent of time (see Theorem 1.8). Convergence to an orbifold in dimension 4: If the scalar curvature is globally boundedon M [0,T), then the metric g converges to an orbifold with cone singularities as t × t T (see Corollary 1.11). ր Let us motivate the study of Ricci flows with bounded scalar curvature. Consider a Ricci flow (M,(g ) ) that develops a singularity at time T < . It was shown by t t [0,T) ∈ ∞ Hamilton (cf [Ha1]), that the maximum of the norm of the Riemannian curvature tensor diverges as t T. Later, Sesum (cf [Se]) showed that also the norm of the Ricci tensor ր has to become unbounded as t T. It remained an open questions whether the scalar ր curvaturebecomesunboundedaswell. Indimensions2, 3andintheKa¨hlercase(cf[ZZh]) this is indeed the case. In order to understand the higher dimensional or non-K¨ahler case, it becomes natural to analyze Ricci flows with uniformly bounded scalar curvature and to try to rule out the formation of a singularity. Another motivation for the scalar curvature bound arises in the study of Ka¨hler-Ricci flows on Fano manifolds. Let M be a complex manifold with c (M) > 0 and g a Ka¨hler- 1 0 metric on M such that the corresponding Ka¨hler form satisfies ω c (M). Then g can 0 1 0 ∈ be evolved to a Ricci flow on the time-interval [0, 1) and the scalar curvature satisfies 2 the (time-dependent) bound C < R(,t) < C(1 t) 1 (cf [ST]). In other words, if we − · 2 − − consider the volume normalized Ricci flow (g ) , g = etg , t t [0, ) t (1 e−t)/2 ∈ ∞ − ∂ g = Ric+g , t t t −e e then the scalar curvature of g satisfies C/t < R(,t) < C for some uniform constant C. t e− e· e HEAT KERNEL AND CURVATURE BOUNDS IN RICCI FLOWS 3 Ricci flows with boundedscalar curvaturehave been previously studied in [CT], [CW1], [CW2],[CW3],[Wa],[Z11]. Wewillparticularlyusearesultobtainedbythesecondauthor in [Z11], which gives us upper volume bounds for small geodesic balls and certain upper an lower bounds on heat kernels for Ricci flows with bounded scalar curvature. Before presenting our main results in detail, we introduce some notation that we will frequentlyuseinthispaper. WeuseMtodenotea,mostlycompact,Riemannianmanifold and g to denote the metric at time t. For any two points x,y M we denote by d (x,y) t t ∈ the distance between x,y with respectto g and for any r 0 we let B(x,t,r) = y M : t ≥ { ∈ d (x,y) < r bethegeodesicballaroundxofradiusrattimet. Forany(x,t) M [0,T), t r 0 and ∆}t R we denote by ∈ × ≥ ∈ P(x,t,r,∆t) = B(x,t,r) [t,t+∆t] or B(x,t,r) [t+∆t,t], × × dependingon thesign of ∆t, theparabolic neighborhood around(x,t). For any measurable subset S M and any time t, we denote by S the measure of S with respect to the t ⊂ | | metric g . We use and ∆ to denote the gradient and the Laplace-Beltrami operators. t ∇ Sometimes we will employ the notation , ∆ or , ∆ to emphasize the dependence ∇t t ∇gt gt on the time/metric. We also reserve R = R(x,t) for the scalar curvature of g , Ric = R t ij for the Ricci curvature and Rm = Rm for the Riemannian curvature tensor. Lastly, ijkl we define ν[g,τ] := inf µ[g,τ ], where µ[g,τ] denotes Perelman’s µ-functional. For 0<τ′<τ ′ more details on these functionals see section 2. The first main result of this paper is a local bound on the distortion of the distance function between two points, given a local bound on the scalar curvature. This result solves a basic question in Ricci flow, which was first raised by R. Hamilton (cf [Ha2, section 17]), under a minimal curvature assumption. We remark that previously, distance distortion bounds have been obtained under certain growth conditions on the curvature tensor or boundedness of parts of the Ricci curvature, see [Ha2, section 17], [P1, Lemma 8.3], [Si1] and [TW]. Observe that the following distance distortion bound becomes false if we drop the scalar curvature bound, as one can observe near a 3-dimensional horn (for more details see the comment after the proof of Theorem 1.1). Theorem 1.1 (short-time distance distortion estimate). Let (Mn,(g ) ), T < be t t [0,T) a Ricci flow on a compact n-manifold. Then there is a constant 0 < α∈< 1, which∞only depends on ν[g ,2T],n, such that the following holds: 0 Suppose that t [0,T), 0 < r √t , let x ,y M be points with d (x ,y ) r 0 ∈ 0 ≤ 0 0 0 ∈ t0 0 0 ≥ 0 and let t [t αr2,min t +αr2,T ). Assume that R r 2 on U [t,t ] or U [t ,t], ∈ 0− 0 { 0 0 } ≤ 0− × 0 × 0 depending on whether t t or t t , where U M is a subset with the property that U 0 0 ≤ ≥ ⊂ contains a time-t minimizing geodesic between x and y for all times t between t and t. ′ 0 0 ′ 0 Then αd (x ,y )< d (x ,y ) < α 1d (x ,y ). t0 0 0 t 0 0 − t0 0 0 NotethatU couldforexamplebetheballB(x ,t ,α 1d (x ,y )). Theupperboundof 0 0 − t0 0 0 Theorem 1.1 can be generalized for longer time-intervals. For simplicity, we are phrasing this result using a global scalar curvature bound, but localizations are possible. Corollary 1.2 (long-time distance distortion estimate). Let (Mn,(g ) ), T < be t t [0,T) a Ricci flow on a compact n-manifold that satisfies R R < everyw∈here. Then∞there 0 ≤ ∞ is a constant A < , which only depends on ν[g ,2T],n, such that the following holds: 0 ∞ 4 RICHARDH.BAMLERANDQIS.ZHANG Suppose that t [0,T), let x ,y M be points and set r = d (x ,y ). For any 0 ∈ 0 0 ∈ 0 t0 0 0 t [r2,T) with A2(r2+ t t ) min R 1,t we have ∈ 0 0 | − 0| ≤ { 0− 0} d (x ,y ) < A r2+ t t . t 0 0 0 | − 0| Similartechniques alsoimply theexistenceqofawell behaved space-time cutoff function. Theorem 1.3 (cutofffunction). Let (Mn,(g ) ), T < be a Ricci flow on a compact t t [0,T) ∈ ∞ n-manifold. Then there is a constant ρ > 0, which only depends on ν[g ,2T] and n, such 0 that the following holds: Let (x ,t ) M [0,T) and 0 < r √t and let 0 < τ ρ2r2. Assume that R r 2 0 0 ∈ × 0 ≤ 0 ≤ 0 ≤ 0− on P(x ,t ,r , τ). Then there is a function φ C (M [t τ,t ]) with the following 0 0 0 ∞ 0 0 − ∈ × − properties: (a) 0 φ< 1 everywhere. ≤ (b) φ> ρ on P(x ,t ,ρr , τ). 0 0 0 − (c) φ= 0 on (M B(x ,t ,r )) [t τ,t ]. 0 0 0 0 0 \ × − (d) φ < r 1 and ∂ φ + ∆φ < r 2 everywhere. |∇ | 0− | t | | | 0− The proofs of Theorem 1.1, Corollary 1.2 and Theorem 1.3 can be found in section 3. Next, we analyze the kernel K(x,t;y,s) for the heat equation coupled with Ricci flow and establish Gaussian bounds on K(x,t;y,s) and its gradient. This addresses a question of Hein and Naber (cf [HN, Remark 1.15]). Over the last few decades, similar questions have been subject to active research, especially after Li-Yau’s paper [LY]. Among numer- ous useful papers, let us mention [BCG], [ChH], [GH], [LT], [HN], the books [L] and [Gr] and the reference therein. Previous bounds usually relied on boundedness assumptions of the Ricci curvature and distance functions, see for instance [Z06]. Our proof makes use of a recent integral bound for the heat kernel obtained by Hein-Naber in [HN] as well as our distance distortion bounds. We mention that a lower bound on the heat kernel has been proven in [Z11], which matches the upper bound in this paper up to constants. Since many geometric quantities such as the scalar curvature obey equations of heat-type, we expect this result to have further applications. For example, our bounds enable us to convert integral curvature bounds into pointwise bounds in certain settings. Theorem 1.4. Let (Mn,(g ) ), T < be a Ricci flow on a compact n-manifold t t [0,T) that satisfies R R < eve∈rywhere. The∞n for any A < there are constants C = 0 1 ≤ ∞ ∞ C (A),C = C (A) < , which only depend on A,ν[g ,2T],n, such that the following 1 2 2 0 ∞ holds: Let K(x,t;y,s) be the fundamental solution of the heat equation coupled with the Ricci flow (see section 2 for more details), where 0 s < t < T. Suppose that t s AR 1 ≤ − ≤ 0− and s >A 1(t s). Then − − 1 C d2(x,y) K(x,t;y,s) > exp 2 s , C (t s)n/2 − t s 1 − (cid:16) − (cid:17) C d2(x,y) K(x,t;y,s) < 1 exp s , (t s)n/2 −C (t s) 2 − (cid:16) − (cid:17) C d2(x,y) K(x,t;y,s) < 1 exp s . |∇x |gt (t s)(n+1)/2 −C (t s) 2 − (cid:16) − (cid:17) In the last line the gradient is taken with respect to the metric g . t HEAT KERNEL AND CURVATURE BOUNDS IN RICCI FLOWS 5 Note that the lower bound for s is necessary since we do not assume curvature or injectivity radius bounds on the initial metric. On the other hand, if we assume such bounds on the initial metric, then one can derive upper and lower Gaussian bounds by standard methods, since (M,g ) will have bounded geometry at least for small times. t Using the reproducing formula and Theorem 1.4 we can derive Gaussian bounds up to any finite time. Observe also that by Theorem 1.1, one can replace the distance d (x,y) s by d (x,y) freely after adjusting the constant C in the above statements. This will be t 2 made clear during the proof of the theorem. The proof of this theorem and further useful results, such as mean value inequalities for heat equations on Ricci flows, can be found in sections 4 and 5. Next, we prove the following backward pseudolocality theorem assuming a local scalar curvature bound. Previously, similar backward pseudolocality properties have been estab- lished in two other settings: First, Perelman (cf [P2, Proposition 6.4]) obtained a similar result in the three dimensional case without assuming a scalar curvature bound. Second, X. X. Chen and B. Wang (cf [CW1]) proved a backward curvature bound under the addi- tional assumption that the curvature tensor has uniformly bounded Ln/2 norm. Recently, thesameauthorspresentedalong-time backward pseudolocality propertyforKa¨hlerRicci flows on Fano manifolds (cf [CW3]). Theorem 1.5 (backward pseudolocality). Let (Mn,(g ) ), T < be a Ricci flow t t [0,T) on a compact manifold. Then there are constants ε > 0, ∈K < , whic∞h only depend on ∞ ν[g ,2T],n, such that the following holds: 0 Let (x ,t ) M (0,T) and 0 < r √t and assume that 0 0 0 0 ∈ × ≤ (1.1) R r 2 on P(x ,t ,r , 2(εr )2) ≤ 0− 0 0 0 − 0 and Rm(,t ) r 2 on B(x ,t ,r ). | · 0 | ≤ 0− 0 0 0 Then Rm < Kr 2 on P(x ,t ,εr , (εr )2). | | 0− 0 0 0 − 0 The proof of this theorem can be found in section 6. Note that it can be observed from this proof that the factor 2 in (1.1) can be replaced by any number larger than 1 if the constants ε, K are adjusted suitably. Asanapplication, thebackwardpseudolocalitytheoremcanbecoupledwithPerelman’s forwardpseudolocalitytheorem(cf[P1])todeduceastrongerε-regularitytheoremforRicci flows. Corollary 1.6 (strong ε-regularity). Let (Mn,(g ) ) be a Ricci flow on a compact t t [0,T) n-manifold. Then there are constants δ,ε > 0 and K∈< , where δ only depends on n,ε ∞ and K only depends on ν[g ,2T],n, such that the following holds: 0 Let (x ,t ) M [0,T) and 0 < r min √t ,√T t and assume that 0 0 0 0 0 ∈ × ≤ { − } R r 2 on B(x ,t ,r ) [t 2(εr )2,t +(εr )2]. ≤ 0− 0 0 0 × 0− 0 0 0 and ∂Ω n (1 δ)c Ω n 1 for any Ω B(x ,t ,r ), | |t0 ≥ − n| |t0− ⊂ 0 0 0 where c is the Euclidean isoperimetric constant. Then n Rm < Kr 2 on B(x ,t ,εr ) [t (εr )2,t +(εr )2]. | | 0− 0 0 0 × 0− 0 0 0 6 RICHARDH.BAMLERANDQIS.ZHANG The proof of this corollary can be found in section 6. Note that in Perelman’s forward pseudolocality theorem, the bound on the curvature tensor is K/(s t) for s (t,t+εr2], which blows up at time t. In contrast, the curvature − ∈ bound in the corollary above is indepent of time and extends in both directions in time. As an application, Corollary 1.6 combined with Shi’s curvature derivative bound [Sh], seems to simplify and fill in some details in section 3.3 of the paper [TZz] by G. Tian and Z. L. Zhang. For example, the curvature bound (3.45) there now holds in a fixed open set relative to the time level 0 instead of the variable time t . Note that if the Cα harmonic j radius at a point x and time t is r, then the isoperimetric condition holds in B(x,t,θr) for a fixed θ (0,1). Thus the corollary implies that the curvature is bounded in B(x,t,εθr). ∈ As further application of the backward pseudolocality Theorem, we derive an L2-bound for the Riemannian curvature in dimension 4, assuming a uniform bound on the scalar curvature. The precise statement of the result makes use of the following notion: Definition 1.7. For a Riemannian manifold (M,g) and a point x M we define ∈ r (x) := sup r > 0 : Rm < r 2 on B(x,r) . Rm − | | | | If (M,g) is flat, then we set r (cid:8)(x) = . If (M,(g ) ) is a Ric(cid:9)ci flow and (x,t) Rm t t [0,T) M [0,T), then r (x,t) is d|efin|ed to be∞the radius r ∈(x) on the Riemannian manifol∈d Rm Rm (M×,g ). | | | | t The result is now the following. Theorem 1.8 (L2 curvature bound in dimension 4). Let (M4,(g ) ) be a Ricci flow t t [0,T) on a compact 4-manifold that satisfies R R < everywhere. The∈n there are constants 0 ≤ ∞ A,B < , which only depend on the product R T and on ν[g ,2T], such that the following 0 0 ∞ holds: Let χ(M) be the Euler characteristic of M. Then for all t [T/2,T) the following ∈ bounds hold: 1/2 Rm(,t) = Rm(,t)2dg Aχ(M)+Bvol M. k · kL2(M,gt) M| · | t ≤ 0 (cid:18)Z (cid:19) For any p (0,4) ∈ B ZM|Ric|p(x,t)dgt +ZMr|−Rpm|(x,t)dgt ≤ Aχ(M)+ 4−pvol0M. Finally, for all 0 < s 1 ≤ |{|Ric|(·,st)4≥ s−1}|t + |{r|Rm|(·s,4t) ≤ s}|t ≤ Aχ(M)+Bvol0M. Here we use the short form f a := x M : f(x) a . { ≥ } { ∈ ≥ } The proof of this theorem can be found in section 7. Note that the first bound of Theorem 1.8 has been obtained independently in [Si2]. Since the L2-bound of Rm is scaling invariant and descends to geometric limits, we immediately obtain that singularity models of 4-dimensional Ricci flows with bounded scalar curvature have L2-bounded curvature as well: HEAT KERNEL AND CURVATURE BOUNDS IN RICCI FLOWS 7 Corollary 1.9. Let (M4,(g ) ), T < be a Ricci flow on a compact 4-manifold t t [0,T) that satisfies R R < ever∈ywhere. Let (∞x ,t ) M [0,T) be a sequence with Q = 0 k k k ≤ ∞ ∈ × Rm (x ,t ) and assume that the pointed sequence of blow-ups (M,Q g ,x ) | | k k → ∞ k (Q−k1t+tk) k converges to some ancient Ricci flow (M ,(g ) ,x ) in the smooth Cheeger- ,t t ( ,0] Gromov sense. Then g = g is cons∞tant ∞in ti∈m−e∞and (∞M ,g ) is Ricci-flat and ,t ∞ ∞ ∞ ∞ asymptotically locally Euclidean (ALE). Corollary 1.9 also follows from recent work of Cheeger and Naber (cf [CN]). A direct consequence of Corollary 1.9 is (see [And2, Corollary 5.8]): Corollary 1.10. Let (M4,(g ) ), T < be a Ricci flow on a compact 4-manifold t t [0,T) M that satisfies the following top∈ological cond∞ition: the second homology group over every field vanishes, i.e. H (M;F) = 0 for every field F (for example, the 4-sphere satisfies this 2 condition). Then the scalar curvature becomes unbounded as t T. ր The L2-bound on the Riemannian curvature can also be used to understand the forma- tion of the ALE-singularities on a global scale: Corollary 1.11. Let (M4,(g ) ), T < be a Ricci flow on a compact 4-manifold t t [0,T) that satisfies R R < eve∈rywhere. Th∞en (M,g ) converges to an orbifold in the 0 t ≤ ∞ smooth Cheeger-Gromov sense. More specifically, we can find a decomposition M = Mreg Msing with the following ∪· properties: (a) Mreg is open and connected. (b) Msing is a null set with respect to g for all t [0,T). t ∈ (c) g smoothly converges to a Riemannian metric g on Mreg. t T (d) (Mreg,g ) can be compactified to a metric space (Mreg,d) by adding finitely many T points and the differentiable structure on Mreg can be extended to a smooth orbifold reg structure on M , such that the orbifold singularities are of cone type. (e) Around every orbifold singularity of (Mreg,d) the metric g satisfies mRm < T |∇ | o(r 2 m) and mRic < O(r 1 m) as r 0, where r denotes the distance to the − − − − |∇ | → singularity. Furthermore, for every ε > 0 we can find a smooth orbifold metric reg g on M (meaning that g pulls back to a smooth Riemannian metric on local ε ε orbifold covers) such that the following holds: kgT −gεkC0(Mreg;gε)+kgT −gεkW2,2(Mreg;gε) < ε Here, the C0 and W2,2-norms are taken with respect to g . ε A slightly weaker characterization was obtained independently in [Si3] and in [CW1] under an additional curvature assumption. The proof of this corollary can be found in section 7. This paper is organized as follows: In section 2, we introduce the notions and tools that we will use throughout the paper. In section 3, we prove a time derivative bound for solutions of theheat equation on a Ricci flow, which is independentof the Ricci curvature. Then we use this bound to derive the distance distortion estimates, Theorem 1.1 and Corollary 1.2, and construct a well behaved cutoff function in space-time in Theorem 1.3. Next, in section 5, we use this cutoff function to establish mean value inequalities 8 RICHARDH.BAMLERANDQIS.ZHANG for solutions of the heat and conjugate heat equation in section 4 and eventually the Gaussian bounds for the heat kernel, Theorem 1.4. In section 6, we prove the backward pseudolocality Theorem 1.5 and the strong ε-regularity theorem, Corollary 1.6. Finally, we prove the L2 curvature bound in dimension 4, Theorem 1.8, and its consequences. 2. Preliminaries Consider a Ricci flow (M,(g ) ), T < on a compact n-manifold M. For any t t [0,T) (x ,t ) M [0,T), r > 0, we wi∈ll frequently ∞use the notation 0 0 ∈ × Q+(x ,t ,r) = (x,t) M [0,T) : d (x ,x) < r, t t t +r2 0 0 t 0 0 0 { ∈ × ≤ ≤ } Q (x ,t ,r) = (x,t) M [0,T) : d (x ,x) < r, t r2 t t − 0 0 t 0 0 0 { ∈ × − ≤ ≤ } to denote the “forward” and “backward” parabolic cubes. Next, note that by applying the maximum principle to the evolution equation ∂ R = t ∆R + 2Ric 2 ∆R + 2R2, we can deduce the following lower bound on the scalar | | ≥ n curvature: n R(,t) > . · −2t So, for instance, if we also have an upper bound of the form R R < and if we are 0 ≤ ∞ 1/2 working at a scale 0 < r0 ≤ min{R0− ,√t0}, then we have the two-sided bound R nr 2 on M [t 1r2,t ]. | |≤ 0− × 0− 2 0 0 In such a situation, we will often rescale parabolically and assume that r = 1, t 1 and 0 0 ≥ R n on M [t 1,t ]. | | ≤ × 0− 2 0 Assume for simplicity, that R R everywhere. By the evolution equation of the 0 | | ≤ volume form, ∂ dg = Rdg , the bound on the scalar curvature implies the following t t t − distortion estimate for the volume element: (2.1) e R0t sdg dg eR0 t sdg s,t [0,T). − | − | s t | − | s ≤ ≤ ∈ This implies that for any measurable subset S M, we have ⊂ (2.2) e R0t s S S eR0t s S s,t [0,T). − | − | s t | − | s | | ≤ | | ≤ | | ∈ Next, we recall Perelman’s -functional (cf [P1]): W [g,f,τ] = τ( f 2+R)+f n (4πτ)−n2e−fdg W M |∇ | − Z (cid:0) (cid:1) Here g is a Riemannian metric, f C1(M) and τ > 0. The fact that this functional ∈ involves no curvature term other than the scalar curvature, comes very handy in the study of bounded scalar curvature. Recall that the derived functionals µ[g,τ] = inf [g,f,τ] RM(4πτ)−n2e−fdg=1W and ν[g,τ] = inf µ[g,τ ], ν[g] = inf µ[g,τ] ′ 0<τ′<τ τ>0 are monotone in time and that ν[g,τ],ν[g] 0. By this, we mean that ∂ µ[g ,τ t t ≤ − t],∂ ν[g ,τ t],∂ ν[g ] 0. t t t t − ≥ HEAT KERNEL AND CURVATURE BOUNDS IN RICCI FLOWS 9 Wenow mention twoimportantconsequences, whichhave beenderived fromthismono- tonicity and which we will be using frequently in this paper without further mention. The first consequence is Perelman’s celebrated No Local Collapsing Theorem (see [P1]), which can be phrased as follows: Let (x ,t ) M [0,T), 0 < r < √t and assume that 0 0 0 0 ∈ × R < r 2 on B(x ,t ,r ). Then 0− 0 0 0 (2.3) B(x ,t ,r ) > κ rn, | 0 0 0 |t0 1 0 for some constant κ , which only depends on ν[g ,2T] and n. On the other hand, a 1 0 non-inflating property was shown in [Z11] (see also [CW2]). This property states that whenever (x ,t ) M [0,T), 0 < r < √t 0 0 0 0 ∈ × α R(,t) on Q (x ,t ,r ), − 0 0 0 · ≤ t t 0 − then (2.4) B(x ,t ,r ) < κ rn, | 0 0 0 |t0 2 0 where κ again only depends on ν[g ,2T] and n. 2 0 Next, wediscuss heatequations coupled with theRicci flow. Theforward heat equation ∂ u ∆u= 0, t (2.5) − (∂tgt = −2Ricgt has a conjugate, ∂ u ∆u+Ru= 0 t (2.6) − − (∂tgt = −2Ricgt, which evolves backwards in time. So for any two compactly supported functions u,v ∈ C (M (0,T)) we have ∞ × (∂ u ∆u) vdg dt = u ( ∂ v ∆v+Rv)dg dt. t t t t ZM×(0,T) − · ZM×(0,T) · − − We will denote by K(x,t;y,s), where x,y M, 0 s < t < T, the heat kernel associated ∈ ≤ with the heat equation (2.5), meaning that for any fixed (y,s) M [0,T) ∈ × (2.7) (∂ ∆ )K(, ;y,s) = 0 and limK(,t;y,s) = δ . t x y − · · t s · ց Then K(x,t; , ) is the kernel associated to the conjugate heat equation (2.6), meaning · · that for any fixed (x,t) M [0,T) ∈ × ( ∂ ∆ +R)K(x,t; , ) = 0 and limK(x,t; ,s) = δ . s y x − − · · s t · ր In [Z06, Theorem 3.2] (see also [CH, Theorem 5.1]), the second author has obtained the following derivative bound for positive solutions u C (M (0,T)) of the heat equation ∞ ∈ × (2.5): u(x,t) 1 J (2.8) |∇ | log , u(x,t) ≤ rts u(x,t) 10 RICHARDH.BAMLERANDQIS.ZHANG whenever 0 < u < J on M (0,t]. Note that this bound is sharp for the heat kernel × on Euclidean space with J =supM (0,t]u and does not assume boundedness of the scalar curvature. × Lastly, we mention two heat kernel estimates, which have been obtained by the second author in [Z11, equations (1.5), (1.7), etc.]. The first estimate is a global upper bound on the heat kernel. Assume that x,y M, 0 s < t < T and that we have the lower scalar ∈ ≤ curvature bound R R on M [s,t]. Then 0 ≥ − × C (2.9) K(x,t;y,s) < , (t s)n/2 − where C only depends on ν[g ,2T],n and (t s)R . 0 0 − Thesecondestimateisadistancedependentlowerboundontheheatkernel. Thisbound is a consequence of (2.8) and Perelman’s Harnack inequality. Let x,y M, 0 s < t < T ∈ ≤ and consider a smooth curve γ : [s,t] M between (y,s) and (x,t), i.e. γ(s) = y and → γ(t) = x. Its -length is defined as L t (γ) := √t t γ (t)2 +R(γ(t),t) dt. L − ′ | ′ ′ |t′ ′ ′ ′ Zs (cid:0) (cid:1) The reduced distance between (x,t) and (y,s) is defined as 1 ℓ (y,s) := inf (γ) : γ : [s,t] M between (y,s) and (x,t) . (x,t) 2√t s L → − (cid:8) (cid:9) Then by [P1, Corollary 9.5] we have 1 (2.10) K(x,t;y,s) e−ℓ(x,t)(y,s). ≥ (4π(t s))n/2 − Applying (2.10) for x = y and (2.8) at time t yields the following Gaussian lower bound on the heat kernel (see [Z11] for more details): Assume that we have the upper scalar curvature bound R(y,t) R for all t [s,t]. Then ′ 0 ′ ≤ ∈ C 1 d2(x,y) (2.11) K(x,t;y,s) > − exp t , (t s)n/2 −2(t s) − (cid:18) − (cid:19) where C only depends on (t s)R . 0 − 3. Heat kernel bounds, distance distortion estimates and the construction of a cutoff function In this section, we first derive a bound on the time derivative (or Laplacian) of positive solutions of the heat equation, which is independent of the Ricci curvature. This bound is then used to obtain distance distortion estimates and to construct a space-time cutoff function. Lemma 3.1. Let (Mn,(g ) ) be a Ricci flow on a compact n-manifold and let u t t [0,T) C (M [0,T)) be a positive∈solution to the heat equation ∂ u = ∆u, u(,0) = u that i∈s ∞ t 0 × · coupled to the Ricci flow. Then the following is true: