ricci flows with bursts of unbounded curvature

Transcription

1 rii flows with bursts of unbounded urvature Gregor Giesen and Peter M. Topping st Otober 04 Abstrat Given a ompletely arbitrary surfae, whether or not it has bounded urvature, or even whether or not it is omplete, there exists an instantaneously omplete Rii flow evolution of that surfae that exists for a speifi amount of time [GT]. In the ase that the underlying Riemann surfae supports a hyperboli metri, this Rii flow always exists for all time and onverges after saling by a fator t to this hyperboli metri [GT], i.e. our Rii flow geometrises the surfae. In this paper we show that there exist omplete, bounded urvature initial metris, inluding those onformal to a hyperboli metri, whih have subsequent Rii flows developing unbounded urvature at ertain intermediate times. In partiular, when oupled with the uniqueness from [Top3], we find that any omplete Rii flow starting with suh initial metris must develop unbounded urvature over some intermediate time interval, but that nevertheless, the urvature must later beome bounded and the flow must ahieve geometrisation as t, even though there are otheonformal deformations to hyperboli metris that do not involve unbounded urvature. Anotheonsequene of ouonstrutions is that while our Rii flow from [GT] must agree initially with the lassial flow of Hamilton and Shi in the speial ase that the initial surfae is omplete and of bounded urvature, by uniqueness, it is now lear that our flow lasts for a longer time interval in general, with Shi s flow stopping when the urvature blows up, but our flow ontinuing stritly beyond in these situations. All ouonstrutions of unbounded urvature developing and then disappearing are in two dimensions. Generalisations to higher dimensions are then immediate. Introdution Hamilton [Ham8] and Shi [Shi89] proved that given a omplete Riemannian manifold M, g 0 with bounded urvature, there exists a omplete Rii flow gt on M for a short time, with g0 = g 0 see [Top06] for an introdution to this topi. The urvature of this Rii flow is initially bounded, and the flow an be extended until suh time that the urvature beomes unbounded. Rii flows with possibly unbounded urvature in their initial ondition and/or during the flow itself, were studied by the seond author in [Top0] in the speial ase of surfaes, and in [GT] we proved that one an always find an instantaneously omplete Rii flow starting at a ompletely arbitrary initial surfae, whether of unbounded urvature or not, or indeed whetheomplete or not, whih exists for a speifi amount of time, and in [Top3] this solution was shown to be unique. More preisely, we proved:

2 Theorem. Part of [GT, Theorem.3] and [Top3, Theorem.]. Let M, g 0 be a smooth Riemannian surfae whih ould have unbounded urvature or be inomplete. Depending on the onformal lass, we define T 0, ] by T := vol g0 M 4π χm if M, g 0 onf = S, g S or RP, g S or C, dz, otherwise.. Then there exists a smooth Rii flow gt t [0,T suh that i g0 = g 0 ; ii gt t [0,T iii gt t [0,T is instantaneously omplete i.e. gt is omplete for all t 0, T ; is maximally strethed see Remark.5, and this flow is unique in the sense that if g t is any other Rii flow on M t [0,T satisfying i and ii, then T T and g t = gt for all t [0, T. If T <, then we have vol gt M = 4π χm T t 0 as t T,. and in partiular, T is the maximal existene time. It has been understood sine the work of Hamilton and Chow [Ham88, Cho9] that the Rii flow on losed surfaes has exellent geometrisation properties in the sense that after appropriate resaling, the flow onverges to a metri of onstant urvature. It is then natural to ask to whih extent this geometrisation ours in the ase that the underlying surfae is nonompat. In [GT, Theorem.3] we also proved that geometrisation does indeed our in the hyperboli ase. Theorem. Speial ase of part of [GT, Theorem.3]. Let M, g 0 be any surfae, possibly inomplete or of unbounded urvature, that is onformal to a omplete hyperboli metri H. Then the Rii flow from Theorem., whih exists for all time t 0, must onverge in the sense that gt H smoothly loally as t. t Moreover, if there exists M < suh that g 0 MH, then we have global onvergene for any k N. t gt H in Ck M, H as t, In this paper we prove that although the Rii flow ahieves geometrisation in this partiularly simple form, the route it takes to get to the onstant urvature metri is neessarily more ompliated than one might initially expet. Indeed, the following theorem finds smooth, omplete initial metris with bounded urvature, whose subsequent Rii flows onverge uniformly in C k after resaling to hyperboli metris, but for whih the Rii flow develops a burst of unbounded urvature on some intermediate time interval. Coupled with the uniqueness of Rii flows proved in [Top3], this shows that Rii flow has no hoie but to develop this unbounded urvature, despite its nie initial and final behaviour, even though alternative deformations to onstant urvature without developing unbounded urvature will exist. Note that in the lattease, also T = if vol g0 C =.

3 Theorem.3. There exist a omplete, onformal, immortal Rii flow gt t [0, on the dis D arising from Theorem., and a time t [, 3 suh that < for all t [0, sup Kgt = for all t t, t + M /00.3 < for all t [4,, and so that t gt H in Ck D, H as t, where H is the Poinaré metri on D and k is any natural number, and in partiular so that K t gt uniformly as t. Moreover, there exist a omplete, onformal, immortal Rii flow gt t [0, on C arising from Theorem., and a possibly different time t [, 3 suh that.3 holds again. Note that by work of Chen [Che09], any omplete Rii flow on a surfae has K gt t, so when the urvature is unbounded for later times, it is unbounded from above. We pik our underlying Riemann surfaes to be D and C here to have examples in both the hyperboli and paraboli ases. It is not hard to generalise to other underlying Riemann surfaes, at the ost of inreased tehniality. Our next theorem also has bearing on how the Rii flow ahieves its geometrisation, as we will explain after stating the result. Theorem.4. On any nonompat underlying Riemann surfae, there exists a omplete, onformal, immortal Rii flow gt arising from Theorem. suh that t [0, { sup Kgt < for all t [0, M = for all t [3,. In partiular, we ould onsider suh an example gt on the dis D, on whih the Poinaré metri H lives. Sine H is hyperboli, we know from Theorem. that tgt onverges to H smoothly loally, and in partiular that the urvature of gt onverges to zero smoothly loally as t. On the other hand our Theorem.4 laims that the urvature remains unbounded for all t 3. We dedue that although the unbounded urvature neessarily forms, the Rii flow organises itself in order to push the regions of large urvature out to spatial infinity as t. In higher dimensions, a theorem of the generality of Theorem. annot be true. However, one an hope to prove the existene of Rii flows starting with unbounded-urvature manifolds with ertain positivity-of-urvature onditions, and Cabezas-Rivas and Wilking [CRW] have done this for positive omplex setional urvature. The Rii flows in our seond result, Theorem.4 have similar properties to four-dimensional Rii flows onstruted by the same authors [CRW]. Of ourse, by taking the produt of our examples with Eulidean spae, our work immediately yields examples also in all dimensions larger than two. One of the onsequenes of the examples above is that they reveal a striking differene between the Rii flows from Theorem. and the lassial Rii flows of Shi. In partiular, in the speial ase that M, g 0 is omplete and of bounded urvature, then in 3

4 addition to the Rii flow gt from Theorem., one also has the Hamilton-Shi Rii flow g t with g 0 = g 0, for t [0, T, for some T > 0, and by Theorem. or earlier theory in [GT] we must have T T and g t = gt for all t [0, T, i.e. the lassial flow agrees with our flow for as long as the lassial flow exists. We see that Theorems.3 and.4 resolve the natural question of whether it an atually our in pratie that our flow exists for a longer time interval than Shi s flow. In all the examples above, Shi s flow stops when the urvature blows up, while ours arries on beyond the time that the urvature beomes unbounded, a property shared with the four-dimensional examples of Cabezas-Rivas and Wilking [CRW] mentioned earlier. In partiular, the traditional use of the term maximal solution needs revision. An additional onsequene of our results is then that Shi s flow would not be able to geometrise an arbitrary omplete, bounded urvature initial surfae that is onformal to a hyperboli metri H, whereas our flow will geometrise even a ompletely general metri onformal to H. Remark.5. Reall from [GT] that the maximally strethed ondition from part iii of Theorem. means that if gt is any Rii flow on M for t [0, T ] with g0 g0 with gt not neessarily omplete or of bounded urvature then gt gt for every t [0, min{t, T }]. There are several alternative, but ultimately equivalent, ways of writing this ondition. For example, one ould allow ompetitors gt only with g0 = g0. Alternatively, one ould ask that whenever 0 a < b T and gt is any Rii flow on M for t [a, b with ga ga with gt not neessarily omplete or of bounded urvature then gt gt for every t [a, b. An inspetion of the part of the proof of Theorem. that an be found in [GT] shows that this apparently stronger property also holds for our flow. Remark.6. Given any Rii flow gt, t [0, T arising from Theorem., and given any t 0 [0, T, the new Rii flow ĝt := gt + t 0 defined for t [0, T t 0 must, by uniqueness [Top3], as in Theorem. above, be preisely the unique Rii flow that Theorem. would produe with initial metri ĝ0 = gt t 0. Applying this priniple to the flow of Theorem.4, with t 0 = 3, i.e. translating the onstruted Rii flow to start at time t = 3, we see that an initial surfae of unbounded urvature need not immediately put itself in the lassial situation by beoming of bounded urvature under our flow, but instead an have unbounded urvature for all time. This was originally proved in [GT3] on general surfaes, with a somewhat simpleonstrution. Speifi higher-dimensional examples of Rii flows with this type of behaviour were onstruted by Cabezas-Rivas and Wilking [CRW]. One of the underlying tehniques of this paper was introdued in [Top4] where a sequene of omplete Rii flows with loally-ontrolled urvature was onstruted that onverged in the Cheeger-Gromov sense to an inomplete Rii flow. With a great deal of extra tehnial effort, the same ideas should allow one to onstrut a Rii flow with unbounded urvature preisely on an extremely general subset of time [0,. For example, one one has a single Rii flow that has urvature unbounded preisely on a time interval [, a] with a > arbitrarily lose to and with appropriate spatial asymptotis then multiple opies of this one flow, with appropriate saling and translation in time, ould be ombined together into one onneted flow with unbounded urvature on a given union of losed time intervals. Finding Rii flows developing unbounded urvature and maintaining this unbounded urvature as in Theorem.4 is tehnially less involved than requiring the urvature to beome bounded again as in Theorem.3, and so we will only sketh the proof of Theorem.4. Aknowledgements: This work was partially supported by The Leverhulme Trust, EPSRC Programme grant EP/K00865X/ and the SFB 878: Groups, Geometry & Ations. 4

5 Strategy of the proof We will prove Theorem.3 by making a very preise onstrution of a suitable Rii flow, with preise onstants, whih is somewhat tehnial in parts. However it is important to digest the piture behind ouonstrution first, and this may be enough to understand fully what is going on, without further tehniality. l r b r 0 U i U U b Figure : b-surfae lollipop : Cylinder with bulb ap The basi building blok of ouonstrution is a lollipop surfae, pitured in Figure, that onsists of a plane that has had a loal region drawn out into a long thin ylinder U in the figure with a bulb U b on the end. This an easily be onstruted Setion 3 to have urvature bounded uniformly above and below whilst allowing the ylinder to be as thin as we like. The length of the ylinder is hosen so that its area is of order one i.e. the ylinder is long and thin and the bulb on the end will be of a uniform size and area. Consider now what happens to suh a surfae under Rii flow when we flow it for all time using Theorem.. Beause the urvature is initially bounded, and our flow initially agrees with the flow of Shi, the urvature of the flow will remain bounded for a uniform time independent of how thin the ylinder is. The ylinder, being flat, will essentially try to remain a ylinder. This an be made preise quikly by using Perelman s Pseudoloality Theorem [Per0], but here we will use barrier arguments. Meanwhile, the bulb part of the surfae will shrink, something like a shrinking round sphere, losing area at a onstant rate. After a time proportional to the initial area of the bulb whih is ontrolled uniformly, independently of how small the radius of the ylinder might be the Rii flow will now look similar to how it looked initially, up to and inluding the long thin ylinder, but now instead of having a big bulb attahed to the end, we will show that it will now have a small ap attahed Figure. r 0 Figure : b-surfae at some later time: Cylinder with small ap 5

6 This point in time marks a transition for the flow. From now on, for a uniformly ontrolled time, the apped ylinder will evolve somewhat like a igar soliton [Top06,..]. It will keep on looking like a apped ylinder, but the length of that ylinder will shrink at a onstant rate. During this phase, if the radius of the ylinder is, the urvature of the flow should be of the order of r, i.e. large. In a uniformly ontrolled amount of time, the ylinder will disappear entirely, and the flow will look like a plane with a trunated hyperboli usp attahed Figure 3 in some loal region. This moment marks a seond phase transition for the flow. Following [Top] we show that this hyperboli usp ontrats in a uniformly ontrolled way independently of how thin the ylinder was, and thus how long the usp is and the urvature returns to being ontrolled independently of the radius. 0 g hyp Figure 3: b-surfae at some further later time: Cusp with ap What this onstrution yields is almost what is required in Theorem.3, exept instead of the urvature being infinite on the intermediate time interval, it is of order r. However, by gluing together infinitely many opies of this basi onstrution, with smaller and smaller radii, our job is done. Note that it turns out not to be too diffiult, using pseudoloality tehnology, to show that the different opies of the basi building blok desribed above do not influene eah other too muh. Of ourse there are a number of things to hek to be sure that this onstrution an be arried through. Barrier arguments turn out to be very useful, and some of these an be reyled from [GT3, Top, Top4]. However, barrier arguments alone did not seem to be suffiient to fore the bulb part of the onstrution above to shrink to nothing in a ontrolled time. Instead, in Setion 4 we introdue a novel and very simple tehnique where we ombine the Rii flow with the double-speed mean urvature flow i.e. the urve shortening flow to derive the required width estimate. Roughly, we allow a small loop in the ylinder part U to flow over the bulb part U b, arguing that it must ahieve this in a ontrolled amount of time without its length inreasing signifiantly. The evolving urve ats as a noose, squashing thin the bulb. Finally, we remark that the Rii flow of the surfae we desribe here is also a useful example in the study of Harnak inequalities for Rii flow, as we will desribe elsewhere. 3 Constrution In order to define a metri like in Figure let us first desribe the required ingredients and then put them together expliitly in Definition 3.. Starting with a flat ylinder U of radius and length l, we ap it off on one side by a bulb U b of radius r b =, and on the other side we interpolate U i it with the flat omplex plane C, dz. 6

7 In order to do so, we are going to onnet the ylinder with hyperboli approximated usps on both sides. The one in U b an be joined to a round sphere and the other one in U i to the flat plane U e = C \ U i U U b. To see that eah transition is at least differentiable we are going to speify pieewisely the onformal fator e us,θ ds + dθ of eah building blok in logarithmi ylindrial oordinates s, θ R [0, π: R s s flat plane R s log ylinder of radius > 0 R s log oshs + log r b round sphere of radius r b = π π, s log r os s hyperboli approximated usp u b s U e U i U U b l r s 0 s s b log r log r b s l r Figure 4: Conformal fator of a ylinder with bulb ap in ylindrial oordinates Definition 3.. For a given length l > 0 and radius 0, the b-surfae is the rotationally symmetri plane C, g b where gb = e u bs ds + dθ has in logarithmi ylindrial oordinates z = e s+se+iθ the form u s = s + s e for s, s 0 ] U e u s = log r os s + l for s s 0, l / U i u b s = u 3 s = log r for s [ l /, 0] U u 4 s = log r os s for s ] } 0, s U b \ {0} u 5 s = log oshs s b + log for s s,, 3. with parameters s 0 l+ π, l, s e s 0, l π, s 0, and s b > s hosen uniquely suh that u b C R. For these parameters and l, we define the b-rii flow to be the omplete Rii flow starting from the b-surfae, that is given by Theorem.. Taking a geometri viewpoint, it is easy to verify the existene of the b-metri without omputation essentially one takes the three middle parts u, u 3, u 4 strething from l+π / all the way to π whih automatially ombine to give a C funtion and then moves in the graph of s s from the left until it touhes, and moves the graph s log oshs + log in from the right until it touhes. Lemma 3. Properties of b-surfae. For given length l > 0 and radius 0, /0 the b-surfae C, g b has the following properties: i K gb [, /] a.e. 7

8 u b s U b L s s s b s b log r b s u 4 s log 8 +r u 5 s ũ 5 s Figure 5: Estimate of s b in Lemma 3. ii s 0 = r l + artan r and se = s 0 + log + r ; in partiular D +r = U i U U b iii s b 7 4 r iv vol gb U b < 0π Proof. i follows from the fat that the urvatures of the segements in 3. take the onstant values 0,, 0, and respetively. Sine u b C R, we ompute s 0 = r l + artan r by solving the equation u s 0 = u for s 0, l. l+ π Consequently, we also have s e = s 0 + u s 0 = s 0 + u s 0 = s 0 + log + r, i.e. ii. Instead of an expliit formula for s b and s, it suffies to get upper bounds, whih we estimate as follows: As shown in Figure 5 we shift u 5 slightly to the right suh that a straight line L parallel to s s fits exatly inbetween u 4 and the shifted u 5. Let us define s b by denoting the shifted u 5 as ũ 5 s = log oshs s b + log. The line L touhes u 4 at s := r artan r i.e. s is a solution of u 4 s =, hene the expliit formula for L is s s s +u 4 s = s s + log + r. This determines s b := s log + r + 3 log = s + log 8 +r. Note that the graph of u 4 lying above L means that u 4 s s s + log + r while L lying above ũ 5 means that for all s [ π 0, s s + log + r log oshs sb + log for all s [0,. 8

9 Beause s b < s b we shifted u 5 to the right by s b s b we an estimate s b < s b = s + log 8 + r s + 3 log π + 3 log 7 4 r, 3. beause < 0, thus proving iii. Sine u 5s < for all s R and u 4s for all s s, we know that s < s, and also u 5 s u 4 s for all s [s, s ]. Therefore, vol gb U b = π π = π onluding iv. ˆ s 0 ˆ s 0 ˆ s 0 e u4s ds + π ˆ s e u5s ds ˆ ˆ s e u4s ds + π e u4s ds + π s s r ˆ osr s ds + π s oshs sb ds = π + 4π + tanhs b s 3. < π e u5s ds tanh log = 8 9 π < 0π. Note that the essential point in iv is that we have a uniform upper bound for the volume that is independent of, and this fat is essentially obvious from the onstrution if one thinks geometrially. 4 Width estimate using urve shortening flow Under Rii flow, the bulb part of the b-surfae will shrink, and intuitively the area will shrink at a onstant rate of about 4π, independently of. The preise result we need is that after a definite amount of time, the width of the bulb is similar to the width of the ylinder, and to ahieve this we use a ombination of the urve shortening flow and the Rii flow. Proposition 4.. On a surfae M let U 0 M be a simply onneted domain with smooth boundary U 0. For T = 4π vol g0 U 0 let gt be a smooth Rii flow on t [0,T ] M and γt be a smooth embedded solution to the double speed urve shortening t [0,T flow within M, gt starting from the losed urve along U 0, i.e. t γt = κ gt,γtν gt,γt 4. im γ0 = U 0, where ν is the outward unit normal to γ and κ is the orresponding geodesi urvature. Then vol gt U t = 4πT t for all t [0, T ], where U t is the simply onneted domain with boundary U t = im γt on the same side as U 0. Proof. We alulate using the generalised Leibniz integral rule, the Gauss-Bonnet Theorem and the fat that d dt dµ gt = K gt dµ gt see [Top06,.5.7] ˆ ˆ ˆ d dµ gt = K gt dµ gt + dt U t U t im γt t γt, ν gt,γt ds gt ˆ ˆ = 4π χu t + κ gt ds κ gt,γt ds = 4π. U t im γt Therefore, vol gt U t = vol g0 U 0 4πt = 4πT t for all t [0, T ]. 9

10 This simple priniple will be behind the following width estimate. It will ontrol the length of any irle at some time, and then our loweurvature bounds will give ontrol on its length for all later times, and in partiular at the time T that we expet the bulb to be extinguished. Lemma 4.. For some > 0 and l > let gt be the b-rii flow i.e. the t [0, omplete Rii flow on C from Theorem. starting from the orresponding b-metri. Then we have the width estimate w gt U U b := max L gt B g0 0; r π T +, 4. r [0,l b +l ] where T = 4π vol g0 U b, l b = dist g0 0, Ub and LgT represents the length. In ylindrial oordinates using the onvention from 3 and writing gt = e ut,s ds + dθ, we an state the width estimate 4. in terms of the onformal fator: ut, s log + log T + for all s l,. 4. Proof. Let γt be the rotationally symmetri solution to 4. starting from t [0,T im γ0 = U b. Define a time-dependent radius ρ : [0, T 0, suh that B g0 0; ρt = im γt for all t [0, T. Now estimate using Chen s Theorem A. and the fat that K g0 ˆ d dt L gtγt = K gt + κ gt ds im γt t + L gtγt, whih we integrate to L gtγt t + L t+ gt γt for all 0 t < t < T. Similarly, for eah fixed t 0 [0, T, ˆ d dt L gtγt 0 = im γt 0 K gt ds t + L gtγt 0, 4.3 whih integrates L gtγt 0 t + t+ L gt γt 0 for all 0 t < t < T. Fix any radius r 0, l b at whih we would like to estimate the width. By Proposition 4. and symmetry we see that ρt 0 as t T, so by ontinuity we find a time t r := max { t [0, T ] : im γt = B g0 0; r } when the urve oinides with the boundary of the r-ball. Hene, we estimate L gt B g0 0; r T + t r + L gt r B g0 0; r = T + L g0 γ0 = T + π. T + t r + L gt rγt r Finally, the priniple behind 4.3 shows that L gt B g0 0; r π T + for all r [l b, l b + l ]. 5 Barriers First we show that the bulb does not shrink too fast. 0

11 Lemma 5. Coarse lower sphere barrier. For any parameters l > and 0, let e ut ds + dθ be the orresponding b-rii flow. Then we have the lower t [0, barrier ut, s log oshs s b + log t 5. for all t, s [0, R. Proof. Comparing the solution to an inner sphere in the bulb part, i.e. u 5 s in Definition 3. for the initial ondition, the result follows from the fat that the b-rii flow is maximally strethed see Theorem.. 5. Cigar barriers Hamilton s igar will serve as a useful barrier. Therefore, like in [Top4,..] we introdue the notation of the standard igar, in logarithmi ylindrial oordinates with the orresponding Rii soliton flow Cs := log e s + 5. t, s C s + t. 5.3 In pratie, we will need resaled and translated forms of the igar and its assoiated Rii flow, so for λ > 0, we define C λ s := Cs log λ and C λt, s = C λ λt + s. 5.4 We have the rough estimates log for s 0 log s for s > 0 } Cs { 0 for s 0 s for s > and a unit sphere is dominated by the igar: log oshs C s for all s R Thinking of the igar solution as a apped ylinder whih translates in time, we expet its area to behave like the area of a ylinder of length s t, i.e. π s t for s < t; more preisely, we have the lower estimate for all s < λt vol Cλ t, [s, [0, π = πλ log e λt+s + πλ λt + s. 5.7 Note, that we are abusing notation by writing geometrial quantities with respet to the onformal fator instead of the assoiated metri. Lemma 5. Coarse uppeigar barrier. For some parameters l > and 0, let e ut ds + dθ be the orresponding b-rii flow. Then we have for all t [0, t, s [ ] [ 0, r l, ut, s C t, s s b

12 Proof. Choosing ylindrial oordinates as usual, we an ompare the initial metri to a igar dominating the bulb part { [ ] u0, s log if s l, s b log log oshs s b if s s b, [ log + C s s b = C s s b for all s l 4 8, using 5.5 and 5.6 in the seond line. On the other hand, using Corollary A. we have for all t [ ] 0, r at the boundary s = l u t, l u 0, l + logt + = log rt + r r : log 8 + C t 4 l s b = C 8 t, l s b Hene, we may apply the omparison priniple and obtain the result. log 4 using 5.5. That uppeigar barrier is oarse in the sense that the irumferene of the igar is of order one so that it an dominate the entire bulb. We now make a refinement than says the bulb part is dominated by a igar of tiny irumferene of order one the bulb has been given time to ollapse. We also get lower bounds whih show that the ylinder itself has not ollapsed at that time. Lemma 5.3 Refined igar barriers. For parameters 0, 0 and l 8 r there exists a time t 3/4, / 5, suh that the orresponding b-rii flow e ut ds + dθ has the upper and lower barriers t [0, ut, s C 4r t t, s r ut, s C r for all t, s [ t, t + ] [ l,, 5.9 t t, s for all t, s [t, R. 5.0 Proof. Fix and l aording to the lemma s statement. First we find the time that the bulb has first ollapsed to a speified degree. Claim. There exist a time t 3 4, 5 and a point s > 0 suh that and ut, s = max ut, s = log + log 8 5. s l r { log for all s, s ] ut, s 5. s s + log for all s s,. Proof of Claim. Note first that given any t 5, we annot have s [ l, 0 ] satisfying 5. beause Corollary A. would onstrain ut, s u0, s + logt + log + log 6 i.e. the ylindeannot fatten up too quikly. By Lemma 3.iv, we know 4π vol g b U b < 5, thus the existene of t 0, 4π vol g b U b 0, 5/ and s > 0 satisfying 5. is a onsequene of Lemma 4. and in partiular estimate 4.. From Lemma 5. we see that t 4r 3 4. In order to show 5. we are going to ompare u with the flat, stati ylinder solution t, s log r [0,t] on [0, t ], s ] and with the flat plane t, s s s + log on [0, t ] [s, :

13 First observe, that ut, s > log for all t [0, t ] beause if at some time t 0 [0, t we had ut 0, s log, then using Corollary A. along with 5. we would obtain the ontradition log + log 8 = ut, s ut 0, s + log t + t 0 + log + log 6. By onstrution we have initially { log for all s, s ] u0, s = u b s s s + log for all s s,. Therefore, we an apply a omparison priniple on both sides of s and onlude 5.. // In order to show 5.9, we start omparing both solutions initially at time t. Combining the oarse upper barrier from Lemma 5. valid until time r 0 > 5 / > t with 5. and using estimate 5.5 we have for all s l { ut, s min log + } log 8, C t, s s 8 b { [ l, s log 8r for s s + s b t 4 + log 8 for s [ s, { = log + log [ 6r for s l, s log6r s + s for s s, [ C 4r s r for all s l,, where s = s b t 4 log r using Lemma 3.iii is roughly the intersetion of log 8r with C t, s s 8 b. At the boundary s = l we have for all t [0, t + ] using Corollary A. and 5.5 u t, l log + logt + log 8r log6r + C 6r l r = C 4r, l r C 4r t t, l r with the above hoie of l desired. 8 r. Thus the omparison priniple implies 5.9 as Finally, 5.0 follows from the fat that ut is maximally strethed and we t [0, have at the time t for all s R from 5. { log for all s, s ] ut, s log s s for all s s, C r s s = Cr 0, s s using 5.5; hene ut, s C r t t, s s Cr t t, s for all t t. 3

14 5. Bounding the metri at later times The igar barriers we have just onstruted give good ontrol on the b-rii flow after the bulb has deflated and while the ylinder part is shrinking. To regain ontrol on the urvature afterwards, we will need the following: Lemma 5.4 Undeusp. For some parameters 0, 0 and l = 8 r onsider the b-rii flow gt on C, then t [0, if we where g hyp is the omplete hyperboli metri on D \ {0}. g 7 / 56 g hyp on D \ {0}, 5.3 Proof. As usual we write gt = e ut,s ds + dθ, and fix t 3/4, / 5 from Lemma 5.3. Reall from Lemma 3.ii, we have s e = s 0 + log + r l+π /, l and D \ {0} s e, [0, π, in partiular g hyp = ds +dθ s s e. At the time t = t + we have on the one hand side for s A. and the onstrution of the metri g b ] s e, l by Corollary ut +, s log t u b s log 8 + log r os s + l for s s e, l [ ] log for s l, l s log 8 log + l + π log π for s s e, l r + log [ ] + π for s l, l log s s e + log 5 for all s s e, l ]. On the other hand, by virtue of Lemma 5.3 we know that ut +, s C 4r, s r for all s l, so it suffies to ompare the usp of g hyp with this igar in this part. Note that for all s l = 8 r + r we have d C ds 4r, s r = ds d C 8 r + s r C 0 = s + l s s e = d ds Combined with the following inequality at s = l C 4r, l we may onlude that log s s e. log 4 using 5.5 = log 4 log l + l + π + log + π log l s e + log 5, C 4r, s r logs se + log 5 for s l 4

15 and thus ut +, s log s s 0 + log 5 for all s s e, or equivalently gt + 5 g hyp on D \ {0}. Consequently, we have gt t t + 5 g hyp on D \ {0} for all t [t, beause the right-hand side is a maximally strethed Rii flow on D \ {0}; in partiular, we have 5.3. Following [Top4, Lemma.9] we use the following result from [Top] in order to bound uniformly the onformal fator. Lemma 5.5 Speial ase of [Top, Lemma 3.3]. If e vt dz is any smooth t [0,] Rii flow on D with e v0 dz g hyp on D \ {0}, then there exists a universal onstant β 0, suh that sup vt β D t for all t 0, ]. 5.4 Lemma 5.6 Variant of [Top4, Lemma.9]. There exists a universal onstant C, suh that if we onsider the b-rii flow gt t [0, on C with 0, 0 and l = 8 r, then gt C dz for all t [ 5 /4,. 5.5 Proof. Writing gt = e ut dz, we define the parabolially resaled and translated Rii flow e vt dz := 56 g7 / + 56t. By Lemma 5.4 we have e v0 dz = 56 g7 / g hyp on D\{0}, satisfying the hypothesis of Lemma 5.5, and we may onlude g 5/4 = 56 e v /04 dz 56 e 048β dz on D. For the upper bound outside of D, first note that in logarithmi ylindrial oordinates we have D \{0} s e+log, [0, π U U b, hene u b s e +log = u s e +log using the notation from Definition 3.. Sine u b s for all s, s e + log ], it suffies to estimate it at s = s e + log u b s e + log = log r os s e + log + l = log r log r os log + log + r artan r os artan r = log + r s e + log + s e + 3 log, i.e. u b s s + s e + 3 log for all s s e + log. Using Corollary A., we get u 5/4, s u b s+ log 5 + s+s e + log 68 for all s, s e + log ], or g 5/4 68 dz on C\D. Choosing C := 56e048β > 68, the stati and maximally strethed Rii flow C dz t [ 7 /, is an upper barrier for gt from time t = 5 /4 onwards. 5

16 6 Burst of large urvature At this point, we have derived the full set of estimates on the b-rii flow, in partiular to show that it will have a burst of large urvature on some time interval during the flow. However, the Rii flow required for Theorem.3 will evolve from infinitely many opies of the b-surfae, and eah of these will evolve as slight perturbations of the b-rii flow. Therefore we need: Proposition 6. Loweurvature bound. For parameters 0, 56 and l 8 r let g b t be the b-rii flow on C. t [0, Suppose gt is a Rii flow on t [0,3] D C onformally equivalent to g b 0 D suh that for α ], 0 /00 α g b α t D gt α g b α t D for all t [0, 3]. 6. Then there exists a time t 3 4, 3 8 suh that we have the urvature estimate [ t, t + 00 sup D K gt for all t ]. 6. Proof. Let t 3 4, 5 be the time when the barriers from Lemma 5.3 begin to hold for gb t. Define t t [0, := α t 3 4, 3 8 and } ρt := max {r > 0 : vol gt B gt 0; r π. To see that B gt 0; ρt Ub U for suffiiently long after t = t, we an use the lower barrier 5.0 and 5.7 to estimate vol gt U U b vol α g b α tu U b [ α vol Cr r α l t t, r l π α + t α 4 t π [ for t α 4 t, α 4 t + [0, π α l ]. Note that α 4 t + α l = t + α 4 t + α l t + 00, so [ [ t, t + 00] α 4 t, α 4 t + r α l ]. The upper barrier 5.9 and 5.5 gives us an estimate of the length of the boundary at s = l L gt B gt 0; ρt Lα g b α t B gt 0; ρt α 8π r for all t [ α t, α t + ] = [t, t + α ] [ t, t + 00]. Applying Bol s isoperimetri inequality Theorem A.4 to B gt 0; ρt yields for all t [ t, t + 00]. sup K gt sup K gt D B gt 0;ρt 4π vol gt B gt 0; ρt r 6α r 6 Lgt B gt 0; ρt vol gt B gt 0; ρt 7 r

17 To apply this proposition, we need a way to hek that the Rii flows that we onstrut do satisfy the hypothesis 6.. Lemma 6.. For any 0, let g b t be a b-rii flow on C. For all t [0, α > and T > 0, there exists R > suh that if gt is a omplete Rii flow on t [0,T ] any surfae M suh that M, g0 ontains an isometri opy of D R, g b 0 then we have α g b α t D gt α g b α t D for all t [0, T ]. Proof. First note that from Lemma 3.ii we have g b 0 C\D B > 0 be the onstant from Theorem A.3. With { } Bα 4α r 0 = max T, T log α = dz. For v 0 = π let 6.3 we hoose R = r 0 + and have the isometry ψ : D R, g b 0 U R, g0 M, g0. We are going to apply Theorem A.3 to g b t t [0,α T ] and gt at points p t [0,T ] D r0+ and q ψ D r0+ UR. With the above hoies of p, q, r 0, g b 0 and g0 the Rii flows g b t and gt satisfy onditions i,ii and iii of Theorem t [0,α T ] t [0,T ] A.3, suh that we may onlude K gb tp r0 for all p D r0+ and t [ 0, α T ] and K gt q r0 for all q ψ D r0+ and t [0, T ]. Integrating the Rii flow equation t gt = K gtgt along with this urvature estimate, we obtain using 6.3, i.e. e 4r 0 t α for all t [ 0, α T ] α g b 0 e 4r 0 t g b 0 g b t e 4r 0 t g b 0 α g b on D r0+ for all t [ 0, α T ] and α g0 e 4r 0 t g0 gt e 4r 0 t g0 α g0 6.5 on ψ D r0+ for all t [0, T ]. Hene we may estimate on Dr0+ α g b α t α g b 0 = α ψ g0 ψ gt αψ g0 = αg b 0 α g b α t for all t [0, T ]. Therefore, we may apply a omparison priniple and onlude the lemma s statement. 7 Burst of unbounded urvature Proof of Theorem.3. We begin by proving the last part of the theorem, i.e. assuming that the underlying Riemann surfae is C, and proving.3. For T = 3 and α = 0 /00 we obtain from Lemma 6. some radius R >. Pik any sequene of disjoint diss Bp j ; R j N in C, of radius R. We obtain the metri g 0 on 7

18 C by replaing those diss with b-metris g DR b with ylinder radius = 56j and length l = 8 r = 3j for eah j N. Now observe that by onstrution Lemma 3. we have both g 0 dz and K g0 [, ]. Then Theorem. provides a omplete and maximally strethed Rii flow gt t [0, starting with g0 = g 0. For a short time this Rii flow oinides with the Shi solution whih has bounded urvature, so we an apply a maximum priniple to the evolution equation of the Gaussian urvature t K gt = gt K gt + K gt see [Top06, Proposition.5.4] and onlude K gt t for all t [0,. 7. By virtue of Lemma 6. we may apply Proposition 6. to gt t [0,3] on eah dis Bp j; and obtain a sequene of times t,j j N 3 4, 8 3 suh that for eah j N sup K gt 56j Bp j; for all t [ t,j, t,j + ]. 00 Therefore, there exists a t [ 3 4, ] 8 3 suh that for every j0 N and t t, t + an find j j 0 with t [ t,j, t,j + 00]. Consequently, we have sup K gt = C for all t t, t By 7. we know that t, hene we may assume that t [, we Let C > be the universal onstant from Lemma 5.6. Using again Lemma 6. we obtain 5 g 4 α α C dz on Bp j ; for all j N. On the omplement we an simply use Corollary A. in order to onlude 5 5 g 4 α α + dz on C \ Bp j ;. j N Assuming without any restrition C > 5 α +, we have g 5 4 α α C dz on C, and onsequently gt α C dz for all t [ 5 4 α,, beause the right-hand side is a stati maximally strethed Rii flow on C. On the other hand we have gt dz for all t [0,, beause gt is maximally strethed too. With t [0, the solution sandwihed between dz and α C dz, paraboli regularity shows that we have uniform C k bounds for t [4, ; in partiular, the urvature is bounded for all t 4. This ompletes the proof in the ase of C. Consider now the ase that the underlying spae is D, and let H be the Poinaré metri thereon. As before, we set T = 3 and α = 0 /00, and obtain R > from Lemma 6.. We begin by flattening out H in a small neighbourhood of some point in D, to give a new metri g. More preisely, we make a smooth onformal perturbation of H in some domain Σ D so that the new surfae D, g ontains some say small flat dis, while only inreasing the metri and retaining the nonpositivity of the urvature, and all the resulting metri g. Think of flattening the onformal fator in a neighbourhood of the origin Figure 6. We now do the same perturbation near a sequene of points heading out to infinity in D, with the perturbed regions being disjoint, to give a new metri g. More preisely, pik 8

19 u h Σ flat dis D Figure 6: Conformal fators for metris H = e h dz and g = e u dz on the dis D any isometry Γ of D, H suh that the images Γ k Σ are pairwise disjoint for k N for example one an take a suffiiently large translation in the upper half-spae model of the hyperboli plane and define g at eah point to be the supremum of Γ k g as k varies within N. The resulting metri g is bounded below by H, and bounded above by some multiple of H. Now homothetially expand g by a large enough fator so that the infinitely many flat diss that lie isometrially within D, g end up of radius at least R, and so that the urvature lies within [, 0]. Still, the metri is sandwihed from below by H and from above by some large multiple of H. We all this expanded metri g 3, and one an think of it as the substitute for the metri dz in the C ase, into whih we inserted b-metris. We now replae eah of the flat diss of radius R by a sequene of more and more extreme b-metris as we did in the C ase, to get the metri g 0 that we are going to flow to obtain the Rii flow required in the theorem. By onstrution, we have g 0 g 3 H, although g 0 is not bounded above by any multiple of H. Note that g 0 has urvature lying within [, /], so as in the ase for C, the urvature will initially be bounded as ditated by 7., and just as before, we then have a time interval t, t + /00 when the urvature is unbounded. Moreover, at time t = 5 4 α we may onlude that the flow gt lies below the metri α C g 3, in a manner idential to how we dealt with the C ase. Thus, at this time, the flow is sandwihed between H and some large multiple of H. We may then appeal to Theorem., oming from [GT], applied to the Rii flow g t α, to dedue that t gt must onverge in Ck to H as t. 9

20 Sketh of proof of Theorem.4. Let us begin by onsidering the ase that the underlying Riemann surfae is C, sine in that ase it is easiest to diretly apply the preise estimates we have derived to prove Theorem.3, even though they give muh more than we now require. Indeed, in this ase, the proof is similar to the proof of Theorem.3 on C, but in the onstrution we hoose muh longeylinders, e.g. l = r. Combined with the tehniques in [GT3] in partiular using pseudoloality in the ylinder region, or alternatively barrier arguments, we ould modify Lemma 5.3 suh that the upper barrier 5.9 holds for longer times. An inspetion of the proof of Proposition 6. shows that we also have the loweurvature bound 6. for longer and longer times, whih is enough to onlude the result in this ase. l= r b r 0 U U b Figure 7: Long lollipop surfae: Infinite ylinder with bulb ap In the ase that we work on a more general underlying Riemann surfae M, we take a slightly different approah, whih is a lot simpler than that of Theorem.3 for not having to worry about making the urvature bounded for large times. Consider instead of the b-surfae we desribe in Setion, just the ylinder U and bulb U b parts, but with the ylinder infinitely long Figure 7. This lollipop surfae has a subsequent Rii flow whose behaviour should be quite apparent by now: The urvature, although initially bounded above and below, will gradually blow up as the bulb part shrinks, until it reahes a large value depending on, from whih moment the Rii flow will look more and more like a thin igar, with the large urvature persisting. This an be made preise using slight variants of the estimates we proved in Setions 4 and 5 without requiring Setion 5. and in Proposition 6.. As in the C ase of this sketh proof, the variant of Lemma 6. will have a slightly adjusted proof using the tehniques from [GT3] or barrier arguments. Equipped with this long lollipop surfae, we onstrut the initial metri g 0 required to prove Theorem.4 on M as follows. Pik a sequene of points p j j N in M without any aumulation point i.e. heading off to infinity and let H be the omplete onformal hyperboli metri on M \ j N {p j}. Near eah of the puntures p j, the metri H will have the struture of a hyperboli usp, whih we an trunate further and further out as j inreases, and add in a lollipop surfae with smaller and smaller, that has itself had its ylinder part trunated. Providing we trunate the ylinder to have length l, where l as j, our new surfae will generate a Rii flow that has the properties demanded by Theorem.4. 0

21 A A priori estimates Theorem A. Chen [Che09, Corollary.3i]. Let gt t [0,T ] be a omplete Rii flow on a surfae M. If K g0 K 0 for some K 0 [0, ], then K gt t + K0 for all t [0, T ]. A. Corollary A.. Let gt t [0,T ] be a omplete Rii flow on a surfae M, suh that K g0 K 0 for some K 0 [0,. Then writing in a loal omplex isothermal oordinate gt = e ut dz, we have ut ut + log t + K0 t + K0 for all 0 t < t T. A. A more elaborate argument of Chen leads to the following pseudoloality-type result giving two-sided estimates on the urvature. Theorem A.3 Chen [Che09, Proposition 3.9]. Let gt t [0,T ] surfae M. If we have for some p M, r 0 > 0 and v 0 > 0 be a Rii flow on a i B gt p; r 0 M for all t [0, T ]; ii K g0 r 0 on B g0 p; r 0 ; iii vol g0 B g0 p; r 0 v 0 r0, then there exists a onstant B = Bv 0 > 0 suh that for all t [ 0, min { }] T, B r 0 K gt r 0 on B gt p; r 0 The following isoperimetri inequality due to G. Bol allows us estimate the maximum of the urvature on a surfae s domain from below if we know its area and the length of its boundary. For an alternative proof using urvature flows, and further generalisations see [Top98] and [Top99]. Theorem A.4. [Bol4, eqn. 30 on p. 30] Let Ω be a simply-onneted domain on a surfae M, g, then. Lg Ω 4π volg Ω vol g Ω sup K g. Ω A.3 Referenes [Bol4] Bol, Gerrit: Isoperimetrishe Ungleihungen für Bereihe auf Flähen. Jahresberiht der Deutshen Mathematiker-Vereinigung, 5:9 57, 94. [Che09] [Cho9] Chen, Bing-Long: Strong uniqueness of the Rii flow. Journal of Differential Geometry, 8:363 38, 009. Chow, Bennet: The Rii flow on the -sphere. Journal of Differential Geometry, 33:35 334, 99.

22 [CRW] Cabezas-Rivas, Esther and Burkhard Wilking: How to produe a Rii Flow via Cheeger-Gromoll exhaustion. Journal of the European Mathematial Soiety, 0. To appear. arxiv: [GT] Giesen, Gregor and Peter M. Topping: Existene of Rii Flows of Inomplete Surfaes. Communiations in Partial Differential Equations, 360: , 0. [GT3] Giesen, Gregor and Peter M. Topping: Rii flows with unbounded urvature. Mathematishe Zeitshrift, 73: , 03. [Ham8] Hamilton, Rihard S.: Three-manifolds with positive Rii urvature. Journal of Differential Geometry, 7:55 306, 98. [Ham88] Hamilton, Rihard S.: The Rii flow on surfaes. In Isenberg, James A. editor: Mathematis and general relativity Santa Cruz, CA, 986, volume 7 of Contemporary Mathematis, pages 37 6, Providene, RI, 988. Amerian Mathematial Soiety. [Per0] [Shi89] [Top98] Perelman, Grisha: The entropy formula for the Rii flow and its geometri appliations. arxiv:math/059, November 00. Shi, Wan-Xiong: Deforming the metri on omplete Riemannian manifolds. Journal of Differential Geometry, 30:3 30, 989. Topping, Peter: Mean urvature flow and geometri inequalities. Journal für die reine und angewandte Mathematik, 503:47 6, 998. [Top99] Topping, Peter: The isoperimetri inequality on a surfae. manusripta mathematia, 00:3 3, 999. [Top06] [Top0] [Top] [Top3] [Top4] Topping, Peter: Letures on the Rii Flow. Number 35 in London Mathematial Soiety Leture Note Series. Cambridge University Press, Otober Topping, Peter: Rii flow ompatness via pseudoloality, and flows with inomplete initial metris. Journal of the European Mathematial Soiety, 6:49 45, 00. Topping, Peter M.: Uniqueness and Nonuniqueness for Rii Flow on Surfaes: Reverse Cusp Singularities. International Mathematis Researh Noties, 00: , 0. Topping, Peter M.: Uniqueness of Instantaneously Complete Rii flows. Geometry and Topology, 03. To appear. arxiv: Topping, Peter M.: Remarks on Hamilton s Compatness Theorem for Rii flow. Journal für die reine und angewandte Mathematik, 69:73 9, 04.