A LOCAL CURVATURE ESTIMATE FOR THE RICCI FLOW. 1. Introduction Let M be a smooth n-dimensional manifold and g(t) a solution to the Ricci flow

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1 A LOCAL CURVATURE ESTIATE FOR THE RICCI FLOW RETT OTSCHWAR, OVIDIU UNTEANU, AND JIAPING WANG Abstract. We show that the norm of the Riemann curvature tensor of any smooth solution to the Ricci flow can be explicitly estimate in terms of its initial values on a given ball, a local uniform boun on the Ricci tensor, an the elapse time. This provies a new, irect proof of a result of Šesǔm, which asserts that the curvature of a solution on a compact manifol cannot blow up while the Ricci curvature remains boune, an extens its conclusions to the noncompact setting. We also prove that the Ricci curvature must blow up at least linearly along a subsequence at a finite time singularity.. Introuction Let be a smooth n-imensional manifol an gt a solution to the Ricci flow. t g = 2Ricg on efine on a maximal interval [, T with < T. When is compact, Hamilton s long-time existence criterion [H] asserts that either T = or the the maximum of the norm of the Riemann curvature tensor blows up at T, that is, lim t T sup Rm gt =. In other wors, provie the sectional curvatures of a solution are uniformly boune on a finite interval [, T, the solution may be extene to a larger interval [, T + ɛ. The arrival of a finite-time singularity for a solution on a compact manifol is therefore characterize by the blow-up of Rm. It is of consierable interest to try to express this criterion in terms of a quantity simpler than the norm of the full curvature tensor. One of the first improvements in this irection was mae by Šesǔm [Se], who prove that, in the above situation, if T <, then lim sup t T sup Ric =. Her result has since been generalize in a number of irections. It has been conjecture, in fact, that the scalar curvature must also blow up in this case, that is, one must actually have lim sup t T sup R =. In imension three, this is a consequence of the Hamilton-Ivey estimate [H2, I], an it is true for all ähler solutions by a theorem of Zhang [Z]. Recent efforts have mae consierable progress towar resolving this conjecture an clarifying the relationship between scalar curvature an singularity formation. See, for example, [Z], [CT], [ET], [He], [n], [LS], [Si, Si2], [W, W2], an the references therein. On noncompact, the loss of a uniform curvature boun is no longer necessarily coincient with the arrival of a singularity. There are by now many constructions of smooth, complete solutions whose curvature is unboune on every time-slice, an Giesen an Topping [GT] an Cabezas-Rivas an Wilking [CW] have separately constructe examples of solutions gt which are efine smoothly for t [, T an The first author was partially supporte by Simons Founation grant # The secon author was partially supporte by NSF grant DS-5622.

2 2 RETT OTSCHWAR, OVIDIU UNTEANU, AND JIAPING WANG possess a uniform curvature boun on [, T ], but which have unboune curvature on [T, T for some T < T. It is possible, even, that there exist solutions for which sup Rm g x, < but sup [,ɛ Rm gt = for all ɛ >, although there are currently no complete examples known. A recent result of Topping [T] implies that they o not occur in two imensions. The preservation of a uniform curvature boun is closely relate to the uniqueness of Shi s solutions [Sh], which, in general imensions, is only presently known to hol within the class of complete solutions of boune curvature [CZ]; cf. [o], [T]. Hamilton s long-time existence criterion nevertheless has a partial analog for noncompact : it is still true that if gt is a complete solution on [, T with < T < an sup [,T Rm gt <, then gt can be extene smoothly to a solution on [, T + ɛ. Here, through the funamental estimates of ano [] an Shi [Sh], one can parlay the uniform curvature boun into instantaneous uniform bouns on all erivatives of the curvature. With these, one can show that gt converges smoothly to a complete limit metric gt of boune curvature, an, from there, use the short-time existence result of Shi [Sh] to restart the flow. As before, it is esirable to express this criterion in terms of a simpler object than the full curvature tensor. The first question to ask is whether an analog of Šesǔm s theorem is vali. As we have note, we must now conten with the new possibility that the Riemann curvature tensor may become unboune instantaneously. This is an obstacle to irectly aapting the argument-by-contraiction in [Se], which relies on the extraction of a limit of a sequence of solutions rescale by factors comparable to the spatial maximum of curvature. See, however, [C] for an approach along these lines. Recently, it was proven in [o] that, if the curvature of a smooth complete solution is initially boune, an if the Ricci curvature is uniformly boune along the flow, then the Riemann curvature tensor must also remain uniformly boune for a short time. However, this result is obtaine inirectly, as a consequence of a uniqueness theorem, an the length of the interval on which the curvature of the solution is guarantee to remain boune is nonexplicit. The statements of the short-time curvature boun in [o] an the long-time existence criterion in [Se] raise the question of whether it is possible to simply estimate the growth of the curvature tensor locally an explicitly in terms of the Ricci curvature, an thereby simultaneously obtain effective proofs of both of these results. Such estimates have some preceent for the Ricci flow: in [W], it is shown that, on a graient shrinking soliton, a boun on Ric implies a polynomial growth boun on Rm. In the present paper, we show that the integral estimates in [W] which are the basis of these bouns can be aapte to general smooth solutions to the Ricci flow. Our main result is the following local estimate. Theorem. Let n, g t be a smooth solution to the Ricci flow efine for t T. Suppose that there exist constants ρ, > an a point x such that the ball g x, ρ/ is compactly containe in an Ric on g x, ρ [, T ]. Then there are constants c, α, β >, epening only on the imension n, such that α β Rm x, T ce ct +ρ Λ + + T + ρ 2 + ρ 2 + Λ,

3 A LOCAL CURVATURE ESTIATE FOR THE RICCI FLOW 3 where Λ := sup ρ g x, Rm x,. This estimate gives a new, irect, proof of Šesǔm s theorem in the compact case, an extens its conclusion to solutions on noncompact manifols. Corollary. Suppose that n, gt is a smooth solution to the Ricci flow efine for t < T <, satisfying := sup Ric x, t <. [,T Then a There is a smooth metric ḡ on, uniformly equivalent to g, to which gt converges, locally smoothly, as t T. b If, in aition,, g is complete an Λ := sup Rm x, <, then sup Rm x, t <, [,T an gt extens smoothly to a complete solution on [, T + ɛ for some ɛ > epening on n,, T, an Λ. Part a may be proven by an argument analogous to that for compact solutions, by consiering the convergence on compact sets. See Section 4 of [H] an Section 6.7 of [C]. This emonstrates that, on an arbitrary smooth solution, the curvature tensor cannot blow up at a fixe point so long as the Ricci tensor remains boune in a surrouning neighborhoo. The uniform boun in b follows irectly from the application of Theorem to balls of fixe raius. It asserts, in particular, that the curvature tensor cannot instantaneously become unboune in space so long as the solution continues to move with boune spee. This boun is also Theorem.4 in the paper [C]. The proof there which is base on a blow-up argument, however, makes use of an implicit assumption that the curvature tensor remains boune for a short time [Ch]. There are a variety of other applications which one may obtain from the implication that an initial boun on Rm an a uniform boun on Ric guarantees a uniform boun on Rm. For example, it follows from Theorem 8.2 in [H2], that if a complete solution, gt has the property that Rm x, as x, then it continues to o so as long as Ric remains boune. Our estimates also immeiately imply an improvement of the epenencies of the constants in Shi s erivative estimates [Sh]. Corollary 2. Let, gt be a smooth solution to Ricci flow efine for t T. Suppose that g x, ρ is compactly containe in for some ρ > an x an let Λ := sup g x,ρ Rm x,, := sup g x,ρ [,T ] Ric x, t. Then, for all m =,, 2,..., sup t m/2 m Rm x, t Cm, n, ρ,, T, Λ. g x, ρ 2 [,T ]

4 4 RETT OTSCHWAR, OVIDIU UNTEANU, AND JIAPING WANG Our proof of Theorem is base on the following integral estimate, which may itself be of some inepenent interest. Proposition. Let n, g t be a smooth solution to the Ricci flow efine for t T. Assume that there exist constants ρ, > an a point x such that the ball g x, ρ/ is compactly containe in an that Ric on g x, ρ [, T ]. Then, for any p 3, there exists c = c n, p > so that for all t T g x, 2 Rm p x, t v t ce ct Rm ρ ρ g x p x, v, +c p + ρ 2p e ct ρ Vol gt g x,. The proof of this proposition is an aaptation of the metho in [W]. In that reference, the reuction from Rm to Ric is mae possible by the soliton ientities. Here, it is mae possible by the fact that the evolution of Rm can be expresse purely in terms of the secon covariant erivatives of Ric, see Lemma below. An analog of the above inequality is also vali for p = 2, see Remark below. We note that Xu [X] cf. [Ya] has also obtaine local curvature estimates for the Ricci flow by relate integral methos. We prove Proposition in the following section by the means of an energy estimate. In Section 3, we combine it with a stanar iteration argument to prove Theorem. In Section 4, we use a variation on the above methos to investigate the possible rates of blow-up of the Ricci curvature at a finite-time singularity. Note that Part b of Corollary implies that, if a complete solution gt to the Ricci flow is efine on a maximal interval [, T with T < an sup Rm x, t < for all t < T, then the Ricci curvature must blow-up as t T, i.e., there must exist a sequence of times t i T along which lim ti T sup Ric x, t i =. The theorem oes not say anything, however, about how rapily the blow-up must occur. y contrast, for the Riemann curvature tensor, it is known not only that one actually has lim t T sup Rm x, t = at T, but that, in fact, sup Rm x, t /8T t. This is a simple consequence of the parabolic maximum principle; see, e.g., Lemma 8.7 of [CLN]. It is natural to ask whether there is a corresponing minimal rate of blow-up for the Ricci curvature. When is compact, it was proven by. Wang [W2] that.2 lim sup t T T t sup Ric x, t x ε for some ε = εn, κ where κ is the noncollapsing constant along the flow. In subsequent work [ChW], X.-X. Chen an. Wang eliminate the epenency of ε on κ. The proof of this statement in [W2] relies on an estimate of Rm by the time-integral of the spatial supremum of Ric obtaine via a series of argumentsby-contraiction using sequences of solutions in an appropriate mouli space. y these methos,. Wang also obtains aitional results on the rate of blow-up of Ric which relate it to to that of Rm.

5 A LOCAL CURVATURE ESTIATE FOR THE RICCI FLOW 5 Our methos here allow us to give an alternative proof of the estimate.2 with ε = εn which extens that result to complete solutions on noncompact. This requires a slightly more careful application of the iteration argument above, an is carrie out in Theorem 2 of Section 4 below. 2. The main estimate The following ifferential inequality is the primary ingreient in the proof of Proposition. Let t be a positive C -function on [, T ]. Proposition 2. Let gt be a smooth solution to the Ricci flow on n, efine for t T, satisfying the uniform Ricci boun sup Ric x, t t Ω on some open Ω, for all t T. Then, for any p 3, there are constants c, c 2 > epening only on n an p, such that Rm p φ 2p + Ric 2 Rm p φ 2p + c Rm p φ 2p t 2. c 2 Rm p φ 2p + c 2 Rm p φ 2 φ 2p 2 2 Ric 2 Rm p φ 2p + c Rm p φ 2p, for any Lipschitz function φx with support in Ω. 2.. Proof of Proposition 2. The proof requires a bit of computation. We first recall the evolution equations for the curvature see, e.g, Chapter 6 of [C]. Lemma. There exists a constant c > such that Ric 2 2 t Ric 2 2 Rm, Rm 2 2 t Rm 2 + c Rm 3, t Rijk l = g lq i q R jk + j i R kq + j k R iq g lq i j R kq + i k R jq + j q R ik, on Ω [, T ]. In the argument below, we will use c to represent a positive constant epening only on n an p, an will simply write for the various tensor norms gt inuce by gt. All integrals are taken relative to the evolving Riemannian measure v t = v gt, an we use the stanar convention that A represents some linear combination of contractions of the tensor prouct A relative to the metric gt. As above, we use R to enote the scalar curvature of gt. To begin, we use equation 2.4 to write t Rm 2 = t g ia g jb g kc g l RijkR l abc = 2 Ric Rm + Ric Rm Rm.

6 6 RETT OTSCHWAR, OVIDIU UNTEANU, AND JIAPING WANG Since the volume form v t evolves by t v t = R v t, we then have t Rm p φ 2p = c t Rm p φ 2p R Rm p φ 2p Rm p 2 2 Ric Rm φ 2p Rm p φ 2p for some c >. Note that φ is a Lipschitz function with support in Ω, which is inepenent of t. Integrating by parts, we fin that Rm p 2 2 Ric Rm φ 2p = from which we reaily obtain that 2.5 t Rm p φ 2p c + c + + Rm p 2 Ric Rm φ 2p Ric Rm p 2 Rm φ 2p Rm p 2 Ric φ 2p Rm, Ric Rm Rm p 2 φ 2p Ric Rm p φ φ 2p Rm p φ 2p. We may estimate the first term on the right of 2.5 by 2.6 c Ric Rm Rm p 2 φ 2p 2 an the secon term, similarly, by 2.7 c Ric Rm p φ φ 2p 2 Ric 2 Rm p φ 2p Using 2.6 an 2.7 in 2.5, it follows that 2.8 t Rm p φ 2p Ric 2 Rm p φ 2p Ric 2 Rm p φ 2p Rm 2 Rm p 3 φ 2p, Rm 2 Rm p 3 φ 2p Rm p φ 2 φ 2p 2 Rm p φ 2p. Rm p φ 2 φ 2p 2.

7 A LOCAL CURVATURE ESTIATE FOR THE RICCI FLOW 7 We now set about to estimate the first two terms on the right of 2.8. Using equation 2.2 from Lemma, we get Ric 2 Rm p φ 2p 2 Hence, integrating by parts, it follows that 2.9 Ric 2 Rm p φ 2p t + 2 t Ric 2 Rm p φ 2p Rm p φ 2p. Ric 2, Rm p φ 2p Ric 2, φ 2p Rm p Ric 2 Rm p φ 2p Ric 2 t Rm p φ 2p Rm p φ 2p. Since Ric t t on the support of φ, for the first term in 2.9, we have 2. 2 Furthermore, as in 2.7, we have 2. 2 Ric 2, Rm p φ 2p c Ric 2, φ 2p Rm p c Ric Rm p φ φ 2p Ric 2 Rm p φ 2p Ric Rm Rm p 2 φ 2p Ric 2 Rm p φ 2p Rm 2 Rm p 3 φ 2p. Rm p φ 2 φ 2p 2 for the secon term in 2.9. Hence, using 2. an 2., equation 2.9 implies Ric 2 Rm p φ 2p c + 2 Rm 2 Rm p 3 φ 2p 2 t Ric 2 t Rm p φ 2p Rm p φ 2 φ 2p 2. Ric 2 Rm p φ 2p Rm p φ 2p

8 8 RETT OTSCHWAR, OVIDIU UNTEANU, AND JIAPING WANG Applying again equation 2.4 of Lemma, we may estimate the thir term on the right of 2.2 by Ric 2 t Rm p φ 2p c Ric 2 2 Ric Rm Rm p 3 φ 2p Rm p φ 2p where we have use that Ric c Rm. Integrating by parts on the first term of the right han sie of 2.3, it follows that Ric 2 2 Ric Rm Rm p 3 φ 2p = Ric Ric 2 Rm Rm p 3 φ 2p Ric Rm Ric 2 Rm p 3 φ 2p Ric Rm p 3 Rm Ric 2 φ 2p Ric φ 2p Rm Ric 2 Rm p 3. Hence, using again that Ric c Rm an also that Ric, we can estimate the above by c Ric 2 2 Ric Rm Rm p 3 φ 2p c Using this estimate in 2.3 it follows that Using the inequality c Ric 2 t Rm p φ 2p c + c + c + c c + c Ric Rm Rm p 2 φ 2p Ric 2 Rm p 2 φ 2p Ric Rm Rm p 2 φ 2p Ric φ Rm p φ 2p Ric Rm Rm p 2 φ 2p Ric φ Rm p φ 2p. Ric Rm Rm p 2 φ 2p Ric φ Rm p φ 2p Rm p φ 2p. Ric 2 Rm p φ 2p Rm 2 Rm p 3 φ 2p,

9 A LOCAL CURVATURE ESTIATE FOR THE RICCI FLOW 9 together with inequality 2., we then obtain Ric 2 t Rm p φ 2p 5 Substituting 2.5 into 2.2 then gives Ric 2 Rm p φ 2p c Rm 2 Rm p 3 φ 2p 2 Rm p φ 2p Ric 2 Rm p φ 2p Rm 2 Rm p 3 φ 2p Rm p φ 2p Rm p φ 2 φ 2p 2. Ric 2 Rm p φ 2p t Rm p φ 2 φ 2p 2. This completes our estimate of the first term on the right of 2.8. Upate, inequality 2.8 now reas 2.7 t Rm p φ 2p t Ric 2 Rm p φ 2p Rm p φ 2 φ 2p 2 Rm 2 Rm p 3 φ 2p Rm p φ 2p. It remains only to estimate the secon term on the right of 2.7. This may be one more simply. Using 2.3 of Lemma, we fin first that 2 Rm 2 Rm p 3 φ 2p t Rm 2 Rm p 3 φ 2p c Rm p φ 2p. Integrating by parts, an using that p 3, we get that Rm 2 Rm p 3 φ 2p = Rm 2, Rm p 3 φ 2p c + c Rm 2, φ 2p Rm p 3 Rm φ Rm p 2 φ 2p Rm 2 Rm p 3 φ 2p φ 2 Rm p φ 2p 2,

10 RETT OTSCHWAR, OVIDIU UNTEANU, AND JIAPING WANG an, furthermore, that t Rm 2 Rm p 3 φ 2p 2 p t Therefore, it follows from 2.8 that 2.9 Rm 2 Rm p 3 φ 2p 2 p t + c + c Rm p φ 2p + c Rm p φ 2p. Rm p φ 2p Rm p φ 2p Rm p φ 2 φ 2p 2. Inserting 2.9 into 2.7, we conclue that, for any p 3, there exist constants c > an c 2 >, epening only on n an p, such that Since t Rm p φ 2p + t c 2 t Ric 2 Rm p φ 2p + c t Rm p φ 2p + c 2 Ric 2 Rm p φ 2p = t Rm p φ 2 φ 2p 2. Rm p φ 2p Ric 2 Rm p φ 2p + 2 Ric 2 Rm p φ 2p an Rm p φ 2p = Rm p φ 2p Rm p φ 2p, t t this completes the proof of Proposition Proof of Proposition. Proposition now follows easily. Our assumptions are that gt is a smooth solution efine on [, T ], an that there exist constants ρ, > an there exists x so that the ball g x, ρ/ is compactly containe in an we have Ric on g x, ρ [, T ]. We wish to prove that, for any p 3, there exists c = c n, p > so that Rm p x, τ ce ct Rm ρ ρ g x p x,, +c p + ρ 2p e ct ρ Vol gτ g x,, g x, 2 for all τ T. Due to the space-time invariance of the Ricci flow, it suffices to prove the proposition for =. We will assume below, then, that 2.2 Ric x, t on g x, ρ [, T ]. It follows that 2.2 e 2t g ij x, g ij x, t e 2t g ij x,,

11 A LOCAL CURVATURE ESTIATE FOR THE RICCI FLOW for all t T. Consier the cut-off function ρ g x, x 2.22 φ x := ρ which is Lipschitz with support g x, ρ. Thus, from 2.2 we see that Ric x, t on the support of φ. Furthermore, from 2.2 an 2.22, we see that 2.23 φ gt e T φ g ρ e T. Hence, we may apply Proposition 2 to obtain that Rm p φ 2p + Ric 2 Rm p φ 2p + c Rm p φ 2p t 2.24 c 2 Rm p φ 2p + c 2 Rm p φ 2 φ 2p 2. We now estimate the rightmost term in 2.24 using 2.23 an Young s inequality, obtaining Rm p φ 2 φ 2p 2 ρ 2 e 2T Rm p φ 2p Consier the function U t := Rm p φ 2p + Rm p φ 2p + + cρ 2p e 2T p Vol gt g x, ρ. Ric 2 Rm p φ 2p + c Rm p φ 2p. From 2.2, we have the simple estimate 2.26 Vol gt g x, ρ e ct Vol gτ g x, ρ for all t τ. inequality Together with 2.24 an 2.25, we obtain the ifferential U t cu + cρ 2p e ct Vol gτ g x, ρ for U on [, τ]. This implies e ct Ut cρ 2p e ct t Vol gτ g x, ρ, t which yiels, upon integration over [, τ], that 2.27 U τ e ct U + cρ 2p Vol gτ g x, ρ. However, again using Young s inequality an 2.26, we have that U c Therefore, there exists a constant c > so that Rm p x, φ 2p + e ct Vol gτ g x, ρ. Rm p x, τφ 2p ce ct Rm p x, φ 2p + + ρ 2p Vol gτ g x, ρ,

12 2 RETT OTSCHWAR, OVIDIU UNTEANU, AND JIAPING WANG for all τ T. Since φ 2p 2 2p on g x, ρ/2, we have gx, ρ2 Rm p x, τ 2 2p Rm p x, τφ 2p, an this completes the proof of Proposition. Remark. Our application to Theorem only requires the valiity of the above estimate for sufficiently large p, an the proof above requires p 3. However, an analogous estimate can be proven for p = 2 by a slightly more etaile analysis of the terms in the evolution equation for Rm. The iea is to write t R ijkl = j D kli i D klj R ijpl R pk R ijkp R pl. where D ijk ivrm ijk = l R ijkl = i R jk j R ik, an use the Lichnerowicztype ientity to estimate t j D kli i D klj, R ijkl φ 2 2 Rm 2 φ 2 in place of 2.5. D 2 φ Proof of Theorem D Rm φ φ It is now stanar to obtain a pointwise estimate from Proposition using an iteration argument. Proof of Theorem. As before, we may assume =. Since Ric on g x, ρ [, T ], we have from Proposition an 2.2 that 3. gx, ρ 2 Rm p x, t v ce ct Rm p x, v g x,ρ + c + ρ 2p e ct Vol g g x, ρ for any t [, T ], where v is the volume form with respect to g. y the ishop-gromov volume comparison theorem we get Vol g g x, ρ Vol g g x, ρ ce ct +ρ. 2 Consequently, it follows from 3. that for any p 3, 3.2 where gx, ρ 2 Rm p x, tv Λ := p ce ct +ρ Λ + + ρ 2, sup Rm x,. g x,ρ We procee to apply De Giorgi-Nash-oser iteration to the equation for Rm using 3.2 for p > n. This is an essentially routine application since the Ricci curvature boun an 2.2 imply a uniform boun on the Sobolev constant of, gt on g x, ρ/2. We will reuce it below to the iteration argument

13 A LOCAL CURVATURE ESTIATE FOR THE RICCI FLOW 3 presente in Lemma 9. of [L] for a fixe Riemannian metric. See also, e.g., [X], [Ye] for relate implementations. In orer to conform to the notation of Chapter 9 of [L], whose argument we will follow closely, we will write f = u = Rm. The evolution equation 2.3 an the simple inequality Rm Rm imply that 3.3 t u cfu weakly on [, T ]. Fixing a, this says 3.4 ϕ 2 u 2a u v t + 2a ϕ 2 u 2a t v t c ϕ 2 u 2a fv t, for any t [, T ] an nonnegative Lipschitz function ϕx with compact support in g x, ρ 2 an inepenent of t. As in [L], we integrate by parts to fin that ϕ 2 u 2a u v t = 2 a ϕu 2a u, ϕ v t + 2a ϕu a 2 v t a ϕ 2 u 2a v t. ϕ 2 u 2a 2 u 2 v t Here we have use a. Furthermore, note that as t v t = Rv t an R cf, ϕ 2 u 2a v t t t Combining with 3.4, we obtain 3.5 ϕu a 2 v t + 2 t ϕ 2 u 2a v t c ϕ 2 u 2a v t ca + ϕ 2 u 2a fv t. ϕ 2 u 2a fv t ϕ 2 u 2a v t. For < s < s + v < T, multiply 3.5 with the Lipschitz function for t s t s 3.6 ψ t := v for s < t s + v for s + v < t T an obtain ψ 2 ϕ 2 u 2a v t + ψ 2 ϕu a 2 v t 2 t 3.7 caψ 2 ϕ 2 u 2a f v t + ψ 2 ϕ 2 u 2a v t + ψψ ϕ 2 u 2a v t. 3.8 For τ [, T ], integrating 3.7 from t = to t = τ implies 2 ψ2 τ ca + τ τ ϕ 2 u 2a τ v τ + ψ 2 t ψtψ t τ ψ 2 t ϕ 2 u 2a tft v t + ϕ 2 u 2a t v t. τ ϕu a t 2 v t ψ 2 t ϕ 2 u 2a t v t

14 4 RETT OTSCHWAR, OVIDIU UNTEANU, AND JIAPING WANG We can now rewrite 3.8 on a single time slice g, using 2.2. We get 2 ψ2 τ ϕ 2 u 2a τ v + τ cae ct ψ 2 t τ 2 + ce ct ψ 2 t where = g x, ρ 2 an τ ψ 2 t ϕ 2 u 2a tft v ϕu a t 2 g v τ ϕ 2 g u2a t v + ce ct ψtψ t hv := v hv ϕ 2 u 2a tv, is the average on of a function h, with respect to g. Now, in view of our boun Ric on, we have a Sobolev inequality for g of the form for some µ = µ n we obtain ψ2 τ e cρ c ρ 2 φ 2µ µ v φ 2 g v, n n 2 an for any φ supporte in. Applying this to φ = ϕua, ϕ 2 u 2a τ v + c τ cae ct ψ 2 t τ + ce ct ψ 2 t e cρ ρ 2 τ ψ 2 t ϕ 2 u 2a tftv ϕu a t 2µ µ v τ ϕ 2 g u2a t v + ce ct ψtψ t Now fix q = q n > n once an for all, an efine A := sup t [,T ] f q q t v. We may apply the argument on pp of [L] irectly to estimate cae ct ϕ 2 u 2a tftv e cρ 2c ρ 2 ϕu a t 2µ µ v ρ cect ρ 2 α aaα ϕ 2 u 2a t v. ϕ 2 u 2a tv, where α := qµ µq q >. Using 3. to boun the first term on the right sie of 3.9 yiels 3. ψ 2 τ ϕ 2 u 2a τ v + c cect +ρ ρ 2 α aaα τ + ce ct ψ 2 t τ e cρ ρ 2 ψ 2 t τ ψ 2 t ϕ 2 u 2a tv ϕu a t 2µ µ v τ ϕ 2 g u2a tv + ce ct ψtψ t ϕ 2 u 2a tv.

15 A LOCAL CURVATURE ESTIATE FOR THE RICCI FLOW 5 The inequality 3. correspons to 9. in [L]. We can follow exactly the argument on pp of [L] to conclue cf. inequality 9.6 that sup u ce ct +ρ A α + ρ 2 + T 2µ qµ T q u q t v. gx, ρ 4 [ T,T] T 2 2 Recall that = g x, ρ 2. Hence, inequality 3.2 implies an upper boun of the form T q T u q q tv A T 2 2 In conclusion, we have sup u ce ct +ρ gx, ρ 4 [ T,T] 2 for appropriate α, β, an c epening on n. general follows after rescaling. ce ct +cρ Λ + + ρ 2. + Λ α + T + ρ 2 β Λ + + ρ 2 The statement of the theorem for 4. On the rate of blow-up of Ricci curvature Theorem implies that if g is complete with boune curvature, an the Ricci flow exists on [, T, but cannot be extene past time T <, then the Ricci curvature Ric cannot be uniformly boune on [, T. The following theorem strengthens this conclusion. Theorem 2. Let, g t be a smooth solution of the Ricci flow efine on [, T, such that each, gt is complete an has boune curvature. Assume that the Ricci flow cannot be extene past time T. Then there exists a constant ɛ >, epening only on n = im, such that lim sup t T T t sup Ric x, t x ɛ. Proof of Theorem 2. We argue by contraiction. Fix some t [, T an assume that ε 4. sup Ric x, t < 2T t, for all t [t, T. We will show that this leas to a contraiction if ε = εn is chosen sufficiently small. y hypothesis we know that there exists a constant C so that 4.2 sup Rm x, t C. Throughout the proof, we will enote by c a constant epening only on the imension n, an by C a constant epening only on T, t an sup Rm x, t. We will also take p = pn to be a suitably large constant to be specifie below in terms of the imension. Ultimately, we will first specify the value of pn an then set the value of εn = εn, p accoringly. Fix some x an choose the cut-off function 4.3 φ x := gt x, x +,

16 6 RETT OTSCHWAR, OVIDIU UNTEANU, AND JIAPING WANG with support gt x,. Now fix τ [t, T. Applying Proposition 2 on [t, τ], with Ω = gtx, 2, it follows that there exist c = c n, p an c 2 = c 2 n, p, so that Rm p φ 2p + Ric 2 Rm p φ 2p + c Rm p φ 2p t 4.4 c 2 Rm p φ 2p + c 2 φ 2p 2 Ric 2 Rm p φ 2p + c for all t [t, τ]. We choose Rm p φ 2p, 4.5 t := ε T t, an we assume that ε = ε n, p is small enough so that c 2 ε <. It follows from 4.4 an 4.5 that the function satisfies 4.6 U t := U t c 2ε T t T t Rm p φ 2p + Rm p φ 2p + c 2 T t U + c 2 Rm p φ 2p + c Ric 2 Rm p φ 2p + c φ 2p. Let us note that 4. an 4.2 imply 4.7 Rm p φ 2p φ 2p + c ε T t 2 Rm p φ 2p Rm p φ 2p + c 2 C T C tε g ij t g ij t T t ε g ij t. y 4.7 an the efinition of φ in 4.3, we then have that φ 2 C T t ε an Vol gt gt x, C Vol T t nε gt gt x,. 2 Consequently, we euce from 4.6 that U t T t U + C Vol T t p+ n gt gt x, 2 ε+ for all t [t, τ]. This implies 4.8 C T t Ut t Vol T t p+ n gt gt x,. 2 ε φ 2p Integrating 4.8 from t = t to t = τ, an choosing ε = ε n, p small enough, it follows that U τ C T τ Vol gt gt x,,

17 A LOCAL CURVATURE ESTIATE FOR THE RICCI FLOW 7 for all τ [t, T. Consequently, it follows from 4.7 that C gt x, 2 Rm p x, t v t T t n 2 ε Ut C T t n 2 ε+ Vol gt gt x,. Using the ishop-gromov volume comparison theorem for gt, this proves that p C 4.9 gt x, 2 Rm p x, t v t T t p+ n ε, 2 for all t [t, T. We now claim that there exists c 3 >, epening only on n, so that 4. Rm x, t C T t c3ε+ p, for all t [T + t /2, T. Inee, we may argue as in the proof of Theorem, using DeGiorgi-Nash-oser iteration. As in the preceing section, we will closely follow the proof of of Lemma 9. of [L], an write f = u = Rm. Starting from 3.3 we obtain 3.8, which reas 4. 2 ψ2 τ ca + τ t τ t ϕ 2 u 2a τ v τ + ψ 2 t ψtψ t τ t ψ 2 t ϕ 2 u 2a tft v t + ϕ 2 u 2a t v t, τ t ϕu a t 2 v t ψ 2 t ϕ 2 u 2a t v t for any Lipschitz function ϕx with support in := gt x, 2 an for ψ given in 3.6. Now, using 4.7, we will rewrite the inequality 4. in terms of the measure v t an the metric gt. Transforme, the inequality reas ψ2 τ ϕ 2 u 2a τ v t + τ ac T τ cε ψ 2 t t τ + C T τ cε ψ 2 t t τ τ + C T τ cε ψtψ t t t ψ 2 t ϕu a t 2 gt v t ϕ 2 u 2a tft v t ϕ 2 gt u2a tv t ϕ 2 u 2a t v t, for some constant c epening only on n. The Sobolev inequality for the metric gt on the ball = gt x, 2 asserts that φ 2µ µ v t CS φ 2 gt v t, for some µ = µn n/n 2 an any φ supporte in. Since C S only epens on the lower boun of the Ricci curvature of g t on, we have C S C accoring to our convention. Obviously, we may assume C S.

18 8 RETT OTSCHWAR, OVIDIU UNTEANU, AND JIAPING WANG Applying this to 4.2, we obtain 4.3 Defining 2 ψ2 τ ϕ 2 u 2a τ v t + τ ac T τ cε ψ 2 t t τ + C T τ cε ψ 2 t t τ τ + C T τ cε ψtψ t Qτ := t sup t [t,τ] t ψ 2 t ϕ 2 u 2a tft v t ϕu a t 2µ µ v t ϕ 2 gt u2a tv t ϕ 2 u 2a t v t. Rm p p x, t v t for some p = pn to be chosen later, we may estimate as on pp of [L] that ac T τ cε ϕ 2 u 2a tft v t ϕu a t 2µ µ v t C T τ cε a α Q α τ ϕ 2 u 2a t v t, pµ µp p. for all t t τ, where c = cn an α = Using 4.4 to estimate the first term on the right-han sie of 4.3, we obtain τ ψ 2 τ ϕ 2 u 2a τ v t + ψ 2 t ϕu a t 2µ µ v t 4.5 C T τ cε a α Q α τ t τ τ + C T τ cε ψ 2 t t t τ + C T τ cε ψtψ t t ψ 2 t ϕ 2 u 2a t v t ϕ 2 gt u2a tv t ϕ 2 u 2a tv t. This inequality, which correspons to 9. in [L], now hols relative to the fixe manifol, g t. We can therefore follow the remainer of the argument on pp of that reference to conclue that, if p > µ/µ, we have Rm x, t CQt T t cε + Q α t + t t 2µ pµ, for some constant c > epening only on n. Consequently, if p > µ/µ + an t T + t /2, it follows that 4.6 Rm x, t Hence, using 4.9, we have that CQt T t cε + Qp t 2µ pµ. C Rm x, t T t c3ε+ p

19 A LOCAL CURVATURE ESTIATE FOR THE RICCI FLOW 9 for some c 3 = c 3 n an all t [T + t /2, T as claime in 4.. Thus, if we now choose, say, p = pn > max{µ/µ +, 4c 3 } an then choose ε = εn, p so that ε < /4c 3 in aition to the restrictions we have alreay impose above, it follows that we have 4.7 sup Rm x, t C T t 2 for all t [T + t /2, T. On the other han, each, gt is complete an has boune curvature, so we may apply the parabolic maximum principle to the equation t Rm 2 Rm Rm 3 an use that T is a singular time for the solution to euce that sup Rm x, t 8 T t, as in Lemma 8.7 of [CLN]. This contraicts 4.7 an proves the theorem. References [Z] R. amler an Q. S. Zhang, Heat kernel an curvature bouns in Ricci flows with boune scalar curvature, Preprint, 25 arxiv:5.29 [math.dg]. [] S. ano, Real analyticity of solutions of Hamilton s equation, ath. Z , no., [CW] E. Cabezas-Rivas an. Wilking, How to prouce a Ricci flow via Cheeger-Gromoll exhaustion, J. Eur. ath. Soc. 7 25, no. 2, [CT] X. Cao an H. Tran, ean value inequalities an conitions to exten Ricci flow, ath. Res. Lett , no. 2, [CZ].-L. Chen an X.-P. Zhu, Uniqueness of the Ricci flow on complete noncompact manifols, J. Diff. Geom , no., [ChW] X.-X. Chen an. Wang, On the conitions to exten Ricci flow III, Int. ath. Res. Not. 23, no., [Ch] L. Cheng, Personal communication. [C]. Chow an D. nopf, The Ricci flow: an introuction, athematical Surveys an onographs,, American athematical Society, Provience, RI, 24, xii+325 pp. ISN: [CLN]. Chow, P. Lu, an L. Ni, Hamilton s Ricci flow, Grauate Stuies in athematics, 77, American athematical Society, Provience, RI; Science Press, New York, 26. xxxvi+68 pp. [He] F. He, Remarks on the extension of the Ricci flow J. Geom. Anal , no., 8 9. [GT] G. Giesen an P. Topping, Ricci flows with bursts of unboune curvature, Preprint 23, arxiv: [math.ap]. [ET] J. Eners, R. üller an P. Topping, On type-i singularities in Ricci flow, Comm. Anal. Geom. 9 2, no. 5, [H] R. Hamilton, Three-manifols with positive Ricci curvature, J. Diff. Geom , no. 2, [H2] R. Hamilton, The formation of singularities in the Ricci flow, Surveys in ifferential geometry, Vol. II Cambrige, A, 993, 7 36, Int. Press, Cambrige, A, 995. [I] T. Ivey, Ricci solitons on compact three-manifols, Differential Geom. Appl , no. 4, [n] D. nopf, Estimating the trace-free Ricci tensor in Ricci flow, Proc. Amer. ath. Soc , no. 9, [o]. otschwar, Short-time persistence of boune curvature uner the Ricci flow, Preprint, 25 arxiv: [math.dg]. [LS] N. Q. Le an N. Šesǔm, Remarks on the curvature behavior at the first singular time of the Ricci flow, Pacific J. ath , no.,

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