Sharp local decay estimates for the Ricci flow on surfaces

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Sharp local decay estimates for the Ricci flow on surfaces
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There are many tools available when studying 2D Ricci flow, equivalently the logarithmic fast diffusion equation, but one has always been missing: how do you get uniform smoothing estimates in terms of local L^1 data, i.e. in terms of local bounds on the area. The problem is that the direct analogue of the geometrically less-useful L^p smoothing estimates for p bigger than 1 are simply false. In this talk I will explain this problem in more detail, and show how to get around it with a new local decay estimate. I also plan to sketch the proof and/or give some applications. No knowledge of Ricci flow will be assumed
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what kind of and what they will thank you very much thinks introduction thanks very much limitations which appreciated for a nice to be back in and out of the year here about 20 years ago and for much enjoyed sewing the tolling abounds Ricci flow plants suddenly Elizabeth big subject um there will be necessary to know anything about Ricci flow told parliament fact the content of what I want to talk about is you can think of is being focused on a very simple half differential equations the so-called a logarithmic past equation and this is simply the equation the lady is couldn't live plants along so this is for functions positive functions from let's say some domain autumn you see a little bit more later pocket so this is a very heavily studied equation at this very hour a very elegant properties of the solution of referring bring us equation and the way I see subject the reason for all these nice properties is the equation is very late this particularly natural equation the reach of the equation so let me just say a couple but the land so the set of his we 1 work now on a this set is not necessarily to hold part Martin but just take a and just some Smith said the government then Ricci flow right on sir Ricci Fla is along parameters family of reunion metrics but the appeal of the lessening of of the act the team programming some time interval so each point on surfaces and in a practical way of measuring distances and its cost souls a PD and the PD it is good the Jeev ITT gears well in high dimensions it would be minus 2 times the Ricci curvature G so because working on services that simplifies very nicely to minus 2 times the gals curvature times case against him so example maybe he would be if you tell them to this new started the Ricci flow with around metric then that would have constant gals curvature it would just be shrinking and which shrink it would shrink to nothing in volatile and that this is going to be relevant you know it's a bit of a phenomenon called it will be relevant in the 2nd so would actually shrink to nothing the Tomahawk so what's the connection here between these 2 equations was very simple so I could do 1 in Bukan pick special coordinator quite family wouldn't cost strategy can be written all locally as just some positive number times the the Euclidean met clusters writer accidents like this so locally it would just be Euclidean space where distances deformed y use the hall that would be working with so the area the service will be locally so the connection then is that achieving Ricci flow is equivalent to you at solving the logarithmic constitution so I'm not really signed the equations criminal uh grown and saying This is being sold in each "quotation mark child the right side so part of the motivation for the new estimates I want to use um explained today is to try and understand the topic of how do we run the Ricci flow starting with an initial data which is not necessarily just remains so will want to to stop the Ricci flow will a rough objects so you're familiar with this sort of idea from the other about the rough object we might want to start with are you might not be something of Soviet Sobolev regularity all something similar to that in my actually b The we wanna start with a certain metric space but in reality says was so just to give you some moral of the story what that will amount to in many situations is that we want to basically solve this equation with L 1 initial data so I'll come back to that in the 2nd you before we do
anything however let's have a look at the serious situation so although reachable in 2 dimensions is and by far the simplest situation for each of this point a fair amount of development over the last few years and actually shine the well-chosen history is dramatically better than Isfahan dimensions so there'll be lots of fun contributions over decades of the US trials may be settled about the earlier results after the theory the thoroughness form is due to gradual myself from in 2011 and the uniqueness part will be computer maker last year and so is the following so I suppose we have I that's because we haven't seen connected on Runnin Down justice assume that's not connected Romanian service endit g 0 well I Wanna Runner Ricci flow from the box and because this can be quite a general result let me actually .period held and how general before arriving else down so we're not assuming this is closed so we're not assuming that it has bounded curvature we're not even assuming there is complete so really just anything so I suppose we start off with His Romanian said then there was this 1988 the reach at the school In kitty the team Continental substance so the conditions are course that will satisfy the initial data on the 2nd condition is and GTE is complete obviously not the time 0 because we're not assuming that but for all tee oftentimes 0 so it's kind of important In an online clings so that seems like well existence and uniqueness but it's kind of a little bit we if you think about it because this is really this is dealing with an equation is too because a lot of nonunion and understanding was some situations that you consider us all divisions on settlement maybe I'll continue to safety this year the a minister of this book so without completeness you Senator that to cater to the needs of from yes so so so maybe it just know that some may be prompt discussion a little bit here said let's say I started off with a justified this concern nasa's smooth Romanian service so flat
desk so you could keep that as applied this that's a reach flow that's not complete you could also write down the equation well over 40 saying that this is the surmise parabolic equation I can specified boundaries and look for as many solutions I want this matter to the police this very extreme loneliness and the situation there is even a complete solution starting so of course it's not complete initially even get to the boundary in get infinity in the distance once in particular find this so something has to change dramatically and is also the region rules of exactly so that the solution would block and of the boundary so his UBS skin and initially your you there would just be 0 are just be 1 a man for a little time he would have had the block so if you accept integrate squarish review Elton Infiniti Q like Elton boundary venues have get so to be a certain rate block immediately to make a complete impact this will have a geometric picture this would be In then boundary layer around the desk that would be the hyperbolic metric of coach minus 1 1 2 2 the idea of using the your choice of our is unique getting this solution was evaluated here it's "quotation mark to erase the this uniqueness that's the whole point of the 2015 always always unique so we're feeling data in With the feeding heat and from vanity I think so taken off in a style you're getting something isn't justified as well that's not totally unreasonable but there's only 1 or if you didn't put as much heat and preventing you would fail to get completeness the plot rate would not be you put more attractive more and that would fail because it is more you put in the high blows up then as a damn thing you getting in equation because of this nominee so it might actually be instructive to look at the equation for the which is half of the you and that satisfies the equation DTV the 2 miners to the the past the so as the increases you really slowing down the diffusion so that is giving you then you because
if there is not just saying if you start with a flat this is solution saying he stuck with anything could be the worst fretful whatever boundary grows I don't know what you always get a solution and always anyone the question is not so good at 1st but you don't know on still on the unit as unless you willing to fail the test of completeness send insensitivity well a crater is always a bit of a conundrum a lot lot of papers on this all combatants In a few moments but uniqueness was a bit of regret the intention is to use
His you could in this situation right In this situation it would have happened but you could for instance take expanded a set equal to some constant increase at
constant but look that is the situation we she makes sense of boundary I don't care if you take Frankel said no boundary to you and I wanna talk about banning the bandits other won't about boundary and I was not even further the you get asymptotic Yukon formatted to vanity so it's not a question of social setting Everything to infinity on the mountain has just not true sometimes you have yeah on Euclidean space it would just stated space component factories they 1 for instance but there was no longer most of the in this situation that's what I was saying as a 2nd ago that the EU would look like in the boundary led these thickness you could say depending on time it would look like .period parametric scaled hot that increases the coach was once that's the I'm plotting his you know sold lot messages what's your yes so so I just mean the EU corresponding to the hyperbolic of the hyperbolic match it would be cars pointing to a one-liners I'm not exclude squared and then just scalable solutions opposition there is a the Muriel can probably give a different told here which is that the reach of flow does uniform ization for you in this case that's kind of amazing To me that you start off with a knee metric on on on a service like this it supports online metric and it will fluctuate In the sense that it will converge as and often if you divide them Judy but to take To re-normalised consider is expanding huddle except setting re-normalised by dividing my duty it will congressionally locally To the unique you authorization matching the makeup allotment but only smoothly lately because in general may be all right down to general the on the curvature supreme isn't convinced political freedom for the National Weather Service not you try voluntary so uniformly that's both the couple was about via the teeth the T is explicit existence time so basically normally teak wasn't in the clear there are exceptions regretted sellout and too obviously because as to Jewish around this to shrinks to nothing and you know what as to will shrink to ran in eventually come around Washington to nothingness of their of Hamilton and channels and which is sort of part of the don't don't don't and also if if you will welcome formerly the plane see if you can formerly the plane mean universal coverage is formed .period sort of versus way but if you conformity the playing so if you're working all too then you can have also flows where everything disappears the former on sofa is given another example take the CIA but remove 1 .period so this is no longer composers fears conform with plane Buster rejection of matter to the plane and it's no longer complete because you can get outside the service and employment you go to was punctured so this is is a really unique flora and what will happen is we'll develop a hyperbolic straight away and on the other hand the ball but he would assure shrink to nothing In a fight on and you can analyze yes investment drunk people that know did this sort of ulcer surgery were interesting stuff it's not going that way right now I so maybe I mention some more names involved so you in a closed case than the existence of the flow and uniqueness of is due to help from 1982 In the case where you if you have like completeness about curvature there is a flood to she existence but it
delivers losses long results in general only to last until the capture blows worse for us the coaches does blow up in general and you just keep going so the uniqueness and In the
bounded curvature complete cases due to Chen's Georgia isn't that much shorter proof due to Koch from which I would recommend existence also overlaps of existence very full this equation which is accused lecture but I think the main results all the existence Aaron onto and off we go because of a desktop Wilson dull Pena and Benedetto Della and investors Esteban Rodriguez shortfall in what was so it's this big lecture but I think everything is subsumed into In this result which we will use some of the at the earliest .period get this looking at the sun do want talk about flying with rough and should says as alluded to a little bit of what you really need to do is trying this well 1 data so we have to talk about it
trying to prove what's normally called L 1 and smoothing estimates so what that really means is if you start off with a L 1 initial data then do you get an bounds at a later time depending on the L-1 data at the time but this year for the ordinary heat equation for the award of the so the ordinary linear heat equation of course this is trivial so if you don't have a nominee energy then you know you could write user represent in terms of representation followed in the 1st year undergrad so you get a compilation of the he and an initial data so immediately Yukon Ballmer why the bill but the size of and the media the size of this year's using held so what so you'd get anywhere abound like 4 .period T 7 in them To somehow out of work and a 2nd user and L. P so this would be the peas 1 B minus and on to so in general I think it's fair lettuce and on 2 feet reducing home so that they keep point here is that peak was 1 works saw but that's the key .period so the course extreme cases when you start with adult function in its new cells and that the reason I'm telling you this sort of baby stuff is that that's exactly what files the logarithmic Oster fusion cuisine and the moral of the story is that if you start with delta function then you float staying as adults and because the moment so the weight the this the serious I don't think geometrically again so what are we gonna do we have to go back to this shrinking sphere for example he said state so so just to clarify here here it is around the unit sphere this normalization to work OK so wanna do is another take these nice localized the corner just using stereo graphic projection and I have a right down my use of .period get a solution the going so some when do that again so the shrinking sphere which reaching the you can write down as considering them so I know this is Gibson notation or right to you 0 but also can put an extra brand in here and but if 1 2nd because son-in-law .period isn't just a 2nd salute this would be the the magic of the shrinking sphere but of course and so just taking stir graphic audience like it takes to records and then just pull back the by dilation so that amounts to changing it's a little bit more than landing lends clarity about loss talk to him in a way that this is the magic of the sphere course so far the largest some positive numbers so because it's totally open that this is a solution to the reach by inspection I immediately get a solution here without actually having to compete the Boston a lot of it and it would use scale this gives solution here Laverty said that would just be 2 1 wonders to take you 0 land so why 1 I tell this will cost signal limit Islamic ghost and if you do that then you get this massive for climax of area which is concentrated in the art so uses planned we're converged adults and then at the end landed the worldly converting to the not as nice spreading out downstairs but just still a scale telephone so as nonsense diffusion Ricci flow a lover and past diffusion equation is having relative to itself should think about and so in particular it's not true that's if you give me the year but if you give me the L 1 but and the time you can't get them 20 estimates of unit costs you 0 lambda don't want us to support the area of the city but the he landed the decisive that the origin it's right away now is just 1 minus 2 tee times 4 times that this is going to infinity for has landed at the which fixed the cost when the site's abated the moral is normally interpreted as this is at the moment repeated as the best is no L 1 L entities who so although the point of the legislators proved 1 and and he's moving so you'll see with where the catches some right so let me let me say 1 reason why this is what is essential for the go back to the motivation was not the Ricci flow further and quite a few reasons with rough days so what you would expect that they would be to take your data is only a time-honored tradition approximated mice metadata Ramos made data but prove a prior estimates on the smooth flows from this new data and then passed the limit the players that's the refreshing so in some sense the estimate the euro Kwan To make To gets the the rights a prior estimates on on the on the solutions is exactly at L 1 and then the so the instant apparently you know once you have an affinity control on solutions of this equation then you can you do your enough Maria shouted Traitors anything system and your way I want you to maximize solutions would have to face the same in movies procedure calls this condition getting this this loophole and are not signed Indiana kind what I said is what doesn't work so all right out there in a minute and to see what we can but I let me let me just say another thing that is is which is the so the closest existing it is believed held here hello moving for peace strictly won so that is due 2 not so completely clear and the visit of a nice paper to Benedetto Diller where these 2 Georgie iteration to make this work and there is all the work of other factors which may have been earlier mutual where he he was a centralization techniques time to make this work you consumer tries member dues to one-dimensional which has Lin easier to handle OK so in some sense of the society you know if you're LP you're a little bit more spread out controlled how concentrated often the moral issues take away the business it once you spread out of bit then you've got all the diffusion is having relative to itself and they want to spread out but then it really gets going you read it that unfortunately this result is actually not very useful for application because you just Jim you just don't have LP control uniform data in just basic the basic situation would
be the year L 1 day and many other metrics she would end up considering as be limits a smooth metrics with certain curvature conditions would end up having seen like a HUD Casper Institute approximated by metrics of Commission below for instance and hyperbolic costs when you write it In call wouldn't will be along but not in a mail OK so what we need to do is find a and L 1 and smoothing which why this could not exist in the traditional sense that he is the idea so normal L 1 and moving says you give me the L 1 data and you do the time I immediately infinity but it was his fault all we can do it is you give me the bounds you want and you give me L 1 data that site I'll give you the time you have to wait this is a little bit twisted round and just by making little twist around it actually makes everything so that's right down the 3rd many of the Americans so this is the 1st in which is the sort of a straight PTV and little variation that which is the more geometric Pacelle said this is with us the and "quotation mark approved a few months ago it's going to be submitted soon so I suppose we have a a solution to this over the prosecution equation within that we're going to do it on the wall of 3rd this year is the unit in Austria so suppose this solve this equation here 1st Pristina threatened analysts say saying with initial data users In L. Paul for not making any assumptions about what happens as you approach the boundary wall you could be smart about these could be totally crazy but it was the conclusion so then don't get distracted by this adult is going to be a bit of a red herring it's just to make it super shop so for all this is going to be remember feeling in the upper bound on the time if the flow think of Raphel the chaos case being the upper so for any of Obama's K the and tea 12 I have to specify the team so teas such that final say once you at a certain time we good so what is at times for the time we have to wait before he is given by take the the difference user miners came take a positive part take measure and and now is the time modular a factor for and then wait a little bit longer spending on Delta so for all this anti-social such that we have and then morally the estimate saying when now below the balance came although I Infoseek if 1 you can replace key case by case and I'm also going notice that if I make see really larger than a different phenomenon kicks in you know have a hyperbolic guide that sort of expanding and making this big so that added t here but that's so negligible moved and sees depending on the the banks of the year for the 1st time so for buys amendment this a shop Yukon any better than this the various ways of saying but you can give an example is actually not the shrinking sphere so the shrinking sphere saying you know you like adult contraband only until time half OK and then you go according Belanger she 0 says and that's not that that would give you a factor to loss here In the estimates that actually shop example is a so-called golf salt on the trail of which is below that of other include Michigan to the camp having on time the other right at the end of May maybe thing to point out is that there is no boundary assumption here at home Arkansas I hope so but make sure you have for that to be true I will make sure the region interiors the most of the solutions evidence this is the year just make so I of in the thermal everything take everything to be smart and then you in a nice situation which has all the content you commando approximations the she so even use their take to be smoother when assets and L wonders Maine it if you integrate Atsmon function you get something that would not look you not to be a regular on the Internet the what yes there's no boundaries there's no boundary assumptions which is again someday there would be a nonlinear outing of the if it is to do this nominee narrative this equation this sort of thing would be impossible to win local estimates purely local estimates which eventually can be applied to any Ricci flow in an arbitary chance you could possibly hope for lower bounds of the same people that we just work the wrong way so given any salute any initial data there would be a solution which just disappears nothing in a short time as as you like so this is a complete contrast the solids quickly give you a slightly more geometric form with the equation and what's a few words about fruit so to see them more geometric form of the equation we have to consider the metric I was talking about earlier the wandering metric that you have standard so I kind of put Mexican In quotes here because I'm reconsidering the conformal factor so as I wrote down before this is one-liners what clear square so that would hand there will be a metric of constant gals curvature minus 1 OK so it looks something like this on the wall and if you evolve under Ricci flow just lift out instead of being multiplied by small small in number and shrinking nothing eventually multiply the 1 plus 2 TVs expense I think so the theory lines and so to think streamlining the so let's do it a slight variant on
this very 1 there 2 still with him and says that under the same assumptions here but not this 0 1 assumption here and make it a little bit different so in the same year you 0 one-liner the hyperbolic metric positive Partizan 0 1 but any scaling the hyperbolic metric so let's take To that end to so Elfers just some so if it for an L 1 Morris the same conclusion certainty is big enough and now it's going to be the L-1 this "quotation mark status no 1 course the officers still divided by 4 times and then just on a little bit beyond then we get is a conclusion now on the whole world that where lying underneath the seat teapots Alpha H analysis of existing on the hold or not this not even interested anymore so what was the same for running a little bit of time let let me try and summarizes what it's saying is so you got this hyperbolic metric which evolves nicely by expanding I'm saying if you take any other metric even if schools will really sharp .period then eventually this expanding hyperbolic rhetoric will overtake the other 1 ones some of the cityscape that hopefully overtake this is not something you can get from the tried and tested maximum principal it's so it's more sort of its own and on the other end OK alright so I think we should probably say something about the way the proof so why is it true that for a definite time interval I definitely don't have up abound online yeah but after the specific times than I do so why not wise you what's the mechanism is making that work this well from withdrawing explained so after after a shorter than this time we won't have any element that controls from 1 even have any LP control but as soon as we ever knew that we had LP control then the LP smoothing instantly and give us element better immediatly so for some reason there's no LP control for indefinite mountainous suddenly you get this nice inventory control so let me try and so this summer some of the ideas in the approved so we end up doing is declining the potential which is derived "quotation mark 0 is derived From solving the equation that's just restricted the Alpha equals 1 case solving equations as follows so this is now going to be on the way and we're going to take 0 boundary data country is sold in the example you need the loss of a new life in the area the last the was Of the example you mean the shrinking sphere and not because that will be on attitude so if you want 1 that show that this is shot then you should take this a gospel :colon scalar properly so the scales it also means that slows slightly different metrics which geometrically as like a hostel under which is being cattle you you choose that correctly then it's circles steady salt on the equation of a later time it would be isometric the real solution but it's not static solution the sense that To be isometric wine it is not the identity but to the scale and the right way then exactly what you say is true sir yeah at that particular time it would that Delta masks the way and the way he would end up standing in the Delta last winter so I and disorders the paint symbol pressures last time but what you end up daisies considerable tension which is like a variant on His islam which is so funny it's a quantity which you would consider when selling carriage on convective them we have to worry about but told it would be basically this nice simple potential plus something that you can control that you would we have had she come up with a particular instance of so-called carry a potential however because we're working locally but we end up cooking up a potential which we call 5 minutes to go very wrong which is sort of like this plus something which blows up in Avery prescribed way infinity and then we let our ball under the PD which looks a little like the longer the conservation equation and a lot of loss in the Roma so you end up with this equation which is there's something that would crop up in case your and the consensus in this what you have to be very careful we should and we have to do something to make sure you have the right boundary conditions which urges boundary group condition so then you look at this evaluation and then there's a Heineken quality which we at least that borrowed from a a relatively recent paper 2 guys from Toulouse good and very hot trying nation's name and I'm 3 adapted that what they did to our situation local case and it ends up giving you the harmonic estimate on a sort of evolved potential of this so far perhaps is that I can tell you what we managed to extract from that Honecker inequality and that that will be enough to think so what we managed to extract is that of a later time you at any time in our time beyond some magic threshold at any later time we get an estimate which is like each other the solution of this original potential divided mighty so so this is going to be true on bail so you don't have to worry about all that derivation today we you have to think about is this these 2 and boxes so ,comma assembler PT than that I can't really get much simpler estimate in the coming not nice because the right hand side here now magically only involves a potential times ties actually the H is hyperbolic gun rights I the way you prove is that it is taken slightly more geometric right you can think of
it mean locally would pick up you know Ralph H. would be a little like right so users but it's just convenient to be able to work globally the late actually it's like a K which blows up the inventing so warmly and do this estimate so this have a look and sound the speedy Sony or components left but this quality here by some Chanel 1 so we get here in 2 dimensions which doesn't quite in W 2 1 and the if we element Kennedy then this will be an anti-missile rebounded so rewarding rebound so actually you know if you if you could get a W 2 1 estimate you of the borderline case for elliptic theory then you wouldn't have this shows shrinking sphere example it is completely concrete on the other hand there is an estimate of the which in this setting normally the attributed to Brazil's now the super super simple very pretty indeed take a few months proved that said basically if you have an equation like this we have a positive something is an L 1 function then this you get exponential while the various exponential and 1 of the these things as you all of outside 1 we need is the 1 remaining in the Prizzi's melting and he will say at the time Saudis says is that if you minus plus an eater His 701 and 0 on the boundary he then huh if you will expire it is less than 4 . 0 it would be the only 1 or 2 then you get each the in helping so that's what we can apply in Sudan apply so he said is not going to be size 0 let's just divine through my T may as well and then this will be Alameda and this will be an so if you unravel what that woman actually says but then says that Preston 1 1 the team is bigger than this magic threshold number man the exponential of sigh zero-gravity His NLP Popperian done forward the please website and so you have to get to the threshold time then this country's LP and that's exactly we have the right hand side of this system so that .period you're an LP & L Pelosi's moving kicks so immediately and infinite well but that's not what we do because the LP empathy similarly estimates that are in the lecture a little that we can have more we can deduce from Allah estimates so that we go we just double respect here we get this in LP and we just do the same argument again and get the network self so I could use a lot of ideas of proves not not not less tricky I'm was going to explain how you get them the myself Helen 30 smoothing the only takes a few months but the amount of time you have to read it in the paper of I think this was followed we the this yeah he going back to the beginning there was right you need a lot of energy from complying with the law exactly exactly exactly gives love right because without injecting you can on the so how do you get to you have to find the right quality which is a combination the scientists derivatives which then which we could borrow this is exactly the way we far from Monday evening will show the 1st paper with consider this but this is the paper legal and at once you got this quantity he committed the violation quite for that but they've got a bill that carefully but make sure you doing the geometric thing In another looking devolution equation in all too you go look at the village conclusion had walks so if you do that then you can apply the maximum principal you can show that on a quantity by the way that we could come hello potential infinity you can show that it actually vanishes and the all-you-can-eat in fact that will be bounded on it would grow to battle and inventing than then even argue that that particular quality that you cook up a 0 initially and that was parabolic Paul for different recalled so it ends up having signed and then when you consider the consequences of that yes you got you gotta think about constantly you good you gotta have that quantity and all the more so the villagers had occasion this whole so says well that there there's several application for the 1 on move was a alluding to earlier is prices starting Ricci flow with Raffarin intraday so what sort of rough data might you consider so in practice when you're actually using which you might end up having to consider the data which is limit of smooth God so let us move levels is just nothing in general but in practice it from approve their armor belt maybe manifolds with Ricci curvature band below for instance this situation we reach bagasse below that would be so if take a sequence of such memorable then again you don't have smoothness and lemon Bob there is a nice theory of convergence such objects to metric spaces with lots of structure so Alexander of spaces all variants of ,comma spaces that little lots of flexibility in the actual then we rightly so so these metric spaces that you you are not you can view as ultimately why would people at you can view them as Raemon services we come fall back to you that have very little regard to how we flow something that I want you to go along with parabolic equation chilis and love but of course we're just seen in the delta functions don't know that don't get smoothing general so you need to make sure that you can take this sort of L 1 time data and and flights of course the way you do it is to smooth approximated by some flows through guns which fire out there and will just exist typically brought on must pass the limit and the estimates unity L 1 Infiniti's meetings but it was working this is the key to the success of our well there is is I mean need depends which but you won't want to extrapolate so sunny the idea that you trying to consider metric spaces that all limits of smooth guys with various geometric conditions because you made the goal is to prove results due matches also Bell manifolds with in Rijeka Japan below and in volume growth was made whatever so yeah an argument In a contradiction on continue might be trying to prove something about such objects so you end up saying Well if it is not true that you unit the sequence with the Ricci coaches the that has a uniform Ballack such research and then you'd end up having to postulate that many consider the Ricci flow with that led to this something Is it does actually works at mall Saunders the pioneer in sort of a line of work I work since while Simon so yeah so on the other hand then still looking really lost a lot to be done in as thank you have 1 of the leaders of the 7th year of the losers in the history of the the the the the funding that would make it work yeah right so but this is a much more generous labor which was alluding to file messages asking his question before so the firm says the following you get me so I already gave said you can always flow any service if you give me a circus which well look is component it's either as universal Colorado workspace will play by the formalization of hyperbole which in almost every case it is a minimalist if there exists I would match then Eltham always says you have existed for all time and he said but that is another worse if you divide by the few divide duty by To take serves scaled down then that will always convinced me to look at the hospital right so yes In I think thank you thank you