 Okay, well, thank you very much. Thanks for the introduction and thanks very much for the invitation I much appreciate it. It's very nice to be back here spent a year here about 20 years ago and Very much enjoyed it So I'm going to be talking about Ricci flow, but Certainly, although this is a big subject It won't be necessary to know anything about Ricci flow at all in fact The content of what I want to talk about is You can think of as being focused on a very simple partial differential equation the so-called a logarithmic fast diffusion equation and This is simply the equation du by dt is Laplacian of log u so this is for For Functions positive functions from let's say some domain in R2 You'll see a little bit more later. Okay, so this is a Very heavily studied equation Very Very elegant properties of the solution I would say really very very nice equation and The way I see the subject the reason for all these nice properties is that the equation is very linked to This particularly natural equation the Ricci flow equation. So let me just say a couple of words about that so The setup is that We want to work canal on a surface not necessarily R2 or part of R2, but Just take a just some smooth surface call it M So then a Ricci flow. I'll just write our F. So a Ricci flow is a one parameter family of Riemannian metrics So t of t let's say So t varying over some time interval So each point on the surface just an inner product just a way of measuring distances and It's course solves a PDE and the PDE is Dg by dt is Well in high dimensions it would be minus two times the Ricci curvature of g so Because we're working on services that simplifies very nicely To minus two times the Gauss curvature times g so k is the Gauss curvature. Okay? So an example maybe would be if you'd had M to be a sphere and You started the Ricci flow with a round metric then that would have constant Gauss curvature It would just be shrinking and it would shrink It would shrink to nothing in finite time This is going to be relevant even though it's a bit of a banal example It will be relevant in a second. So it would actually shrink to nothing and At time a half So what's the connection here between these two? Equations well, it's very simple. So On M you can pick special coordinates So called itothermal coordinates, of course So that g can be written locally as just some positive number times The Euclidean metric which let's just write it as a tensor like this. So Locally, it would just be Euclidean space where distances are formed by U to the half that would be okay So the area of the surface would be the integral of u locally And so the connection then is that G of t being a Ricci flow is equivalent to u solving This logarithmic fast diffusion equation. So I'm not really saying these equations equivalently Equivalent I'm saying this is being solved in each Coordinate chart, right so So part of the motivation for the new estimates that I want to Explain 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 a smooth Riemannian surface Okay, so we'll want to to start the Ricci flow with a rough object So you're all familiar with this sort of idea from other PD the rough object We might want to start with Might not be something of sort of Sobo left regularity or you know Something similar to that it might actually be that we want to start with a certain metric space But in reality, so sort of just to give you some morals of the story What that will amount to in many situations is that we want to basically start this equation with L1 initial data, so I'll come back to that in a second Before we do anything Rough Let's have a look at the smooth situation. So Although Ricci flow in two dimensions is by far the simplest situation for Ricci flow There's been a fair amount of development over the last few years and actually showing that the well-posed in this theory is Dramatically better than it is for higher dimensions. So there are lots of Contributions over decades, but the I'll also maybe say a little bit about the earlier results After the theorem the theorem in this form is due to Gregor Giesen and myself from 2011 and the uniqueness part will be due to me from last year And it says the following so let's suppose we have Let's Suppose we have a smooth Connected I won't write that down just to assume that smooth connected Riemannian surface G zero Well, I want to run a Ricci flow from that, but Because this is going to be quite a general result. Let me actually point out How general before I write anything else down? So we're not assuming that this is closed surface we're not assuming that It has bounded curvature We're not even assuming that it's complete Okay, so really just anything So suppose we start off with this Riemannian surface Then There exists a unique Ricci flow Let's call it G of t for t some time interval such that So the conditions are of course that we satisfy the initial data and the second condition is That G of t Is complete Obviously not at time zero because we're not assuming that but for all t after time zero, so it's kind of important here I'm not including zero so that seems like well Existence in uniqueness, but it's kind of a little bit weird if you think about it because this is really including This is dealing with an equation which typically has a lot of non uniqueness and it's dealing with some situations that you Don't normally Consider so I'll give you some examples in a second. Maybe I'll continue to say so without completeness You certainly don't know into here. You don't have uniqueness. That's all right for sure Yeah, so um, so so maybe just That sort of maybe prompt the discussion a little bit here. So let's say I started off with a Just a flat disc that's a That's a smooth Riemannian surface. So open flat disc so You could keep that as a flat disc That's a Ricci flow. It's not complete You could also write down the equation well, we've already seen that this is the equation That's a nice parabolic equation I can specify boundary data and look for as many solutions as I want the smooth up to the boundary So there's very extreme non uniqueness in that situation there is a unique complete solution starting with a flat disc So of course, it's not complete initially you can get to the boundary you get to infinity in a distance One so in particular in finite distance. So something has to change dramatically. These are all smooth Ricci flows So the solution blows up exactly so the the solution would blow up in the other boundary. So Here's your disc and then initially your you there would just be zero Sorry, it just be one and then for a little time it would have to blow up So that if you were to integrate Square root of u out to the infinity if you like out to the boundary then you'd have to get infinity So there'd have to be a certain rate of blow immediately To make it complete. In fact, this would have a geometric picture. This would be In a thin boundary layer around the disc that would be the hyperbolic metric of curvature minus one on 2t Peter are you saying that Your choice of boundary data is unique then and getting this solution. What's the boundary data here? It's infinity But different rates could be Uniqueness that's the whole point of the 2015 paper Always always unique. So we're feeding data in We're feeding heat in from infinity. Okay, so You're sort of ticking off in a style You're getting something that isn't just the flat disc. Well, that's not totally unreasonable, but There's only one way you can do it if you Didn't put as much heat in from infinity you would fail to get completeness the blow-up rate would not be Strong enough if you put more try to put more in that would fail because it's the more you put in the higher it blows up Then there's a damping you're getting in the equation Because of this non-inliarity here, so it might actually be instructive to look at the equation for V Which is half log u? And that satisfies the equation DTV is e to the minus 2v the plus in V So as V increases you're really slowing down the diffusion. So that is giving you then Uniqueness, but of course the theorem is not just saying if you start with a flat disc You have a solution it's saying you start with anything could be the worst fractal whatever boundary growth. I don't know what You always get a solution and always only one. I Guess my question is you're not allowed to put in twice as much heat on the left as on the right Still comes out Unless you willing to fail the test of completeness So in a sense it's giving you well-poseness this equation, which is always a bit of a conundrum A lot of papers on this I'll come back to in a few moments, but uniqueness is always a bit of a bit of a gray area You could in this situation right in this situation it would happen that You could for instance take Fixed boundary data equal to some constant they increase that constant But look that is a situation where you can actually make sense of a boundary. I don't care if you take a fractal set Okay, no boundary to you know, I don't want to talk about boundary that boundary data I don't want to talk about boundary in general. It's not even true That you get asymptotically you're conformed back to going to infinity. So it's not a question of sort of setting Everything to infinity on the boundary. That's just not true. Sometimes you have a You know on Euclidean space, it would just stay Euclidean space conformal factor would just stay one for instance What are the asymptotics of you at the boundary? Oh, so the In this situation That's what I was saying a second ago that it would look like in a boundary layer whose thickness you could say depending on time It would look like a Poincare metric scaled Homothetically so that the curvature was minus one on 2t That's for you. Sorry I'm plotting here you Sort of Yeah, so so I just mean the u corresponding to the hyperbolic metric so the hyperbolic metric would be corresponding to 2 over 1 minus Mod x squared squared And then just scale that so there's some value of uniformization of the boundary there's a You're gonna probably to give a different talk here, which is that the Ricci flow does uniformization for you in this case that's kind of Amazing to me that you start off with any metric on a on a surface like this that supports a hyperbolic metric and it will Flow to it in the sense that it will converge asymptotically if you divide them g of t by 2t To renormalize because everything is expanding in the hyperbolic setting setting If you just renormalize by dividing by 2t, it will converge smoothly locally to the unique Uniformization metric in the unique hyperbolic metric, but only smoothly locally because in general Maybe I'll write that down. So in general the the curvature Supremum is in infinity a little bit later or initially or whatever So it's not regularizing You to a hyperbolic metric Sort of uniformly a couple of words about the The t so the t is explicit the existence time so basically normally t equals infinity The there are exceptions, which maybe I'll just say out loud S2 obviously because S2 just around S2 shrinks to nothing and even a wobbly S2 Will shrink to around you know eventually become around the shrink to nothing. That's a theorem of Hamilton and Chow Which is sort of part of the dot dot dot and Also, if if you're Conformally the plane see if you're conformally the plane I don't mean if the universal cover is conforming the plane sorry to diverge this way But if you're conformally the plane so if you're working an R2 Then you can have also Flows where everything disappears In a finite time so for instance giving another Cute example just take the sphere, but remove one point. So this is no longer Conform me the sphere. It's conform me the plane by stereographic projection. It just map it to the plane And it's no longer complete because you can get Outside the surface in a finite distance if you go towards this puncture So this says there's a unique flow and that what will happen is it will develop a hyperbolic cast straight away and On the other hand the bulb here would just sort of shrink to nothing In a finite time and you can analyze the asymptotics in fact Manuel was one of the people that did this sort of Analysis very is very very interesting stuff Let's not go that way right now then Okay, so Maybe I'll mention some more names Involved in this so in the closed case then the existence of a flow and uniqueness in fact is due to Hamilton from 1982 And the case where you're you have like completeness and bounded curvature then there's a flow due to she existence But it doesn't as last as long as ours in general I only got last until the curvature blows up whereas for us the curvature does blow up in general and you just keep going The uniqueness in the In the bounded curvature complete case is due to Chen and Zhu as an a shorter proof due to Koch file which I would recommend Existence also overlaps with existence theory for this equation Which is a huge literature, but I think the main results are the existence theory in R2 and We've got papers of Daskelopoulos and Del Pino De Benedetto and Dilla Vasquez estaban Rodriguez and Sure surely more more sides So it's a there's a big literature, but I think everything is subsumed into This result of which we use some of the the earlier theory to get this Okay So We want to talk about Flowing with rougher initial data okay, so As I alluded to a little bit earlier what you really need to do is try and flow this with L1 data So we have to talk about Trying to prove What's normally called L1 L infinity? Smoothing estimates, so what that really means is if you start off with L1 initial data Then do you get L infinity bounds at a later time? Depending on the L1 data and the time Okay, so you know for the ordinary heat equation So let's do L1 L infinity smoothing so for the ordinary linear Heat equation of course, this is trivial So if you don't have the non-linearity Then you know you can write you as a represent in terms of the representation formula that you learn in first year undergrad so you'd get a convolution of the heat kernel and And the initial data so immediately you can bound that by Then the size of this and the the size of this just using Hilda so You'd get a a bound like 4 pi t so in the in the To some power, I'll just work out in a second. So u0 and LP so this would be for ps1 it would be minus N on 2 so in general, I think it's at minus N on 2p We're just using Hilda So the key point here is that p equals 1 works, so I'll do it for all p But that's the key point So the of course extreme case is when you start with a delta function and it smooths out and that the reason I'm telling you this sort of Baby stuff is that that's exactly what fails for the logarithmic fast diffusion equation And the moral of the story is that if you start with a delta function Then you flow staying as a delta function Okay, because of the non-linearity so the way to view this Is to think geometrically again, so What are we going to do we're going to go back to this shrinking sphere Example here, so let's take so just to clarify here g Here is the round Units fear for this normalization to work Okay, so what I'm going to do is I'm going to take these nice local isothermal coordinates Just using stereographic projection And I'm going to write down my you so therefore I get a solution Okay, so that's some Maybe do that over here So the shrinking sphere reach you for you you can write down as So what I'm going to do is Give it some notation. I'm going to write you zero, but I'm also going to put an extra parameter in here and You'll see why in a second because so I'm going to modify this and it's just in a second so This would be the metric of the shrinking sphere But of course instead of just taking stereographic coordinates I could take stereographic coordinates and then just pull back the metric by dilation So that amounts just to changing this a little bit. I end up with a lambda here Lambda squared here in fact we saw that in the last Talk hidden hidden away there, but this is the metric of the sphere, of course so Lambs just some positive number so because it's totally obvious that This is a solution to the Ricci flow just by inspection. I Immediately get a solution here without actually having to compute the little plastic in a vlog of it It will just scale So this gives you the solution you lambda t so that would just be one minus two t you zero Lambda, okay, so why am I telling you this? Well, of course I Want to take the limit as lambda goes to infinity? and if you do that then you get this mass of four pi mass of area would just concentrate at the origin So user lambda would converge to the delta function and then the U lambda of t will be converging to not a nice spreading out Gaussian, but just still a Scale delta function, okay So in some sense diffusion in Ricci flow or logarithmic fast diffusion equation is happening relative to itself Should just think of it that way. Okay, so in particular it's not true that if you Give me the Oops If you give me the L1 norm and the time you can't get an infinity estimate. So, you know, of course U zero lambda and L1 is just four pi the area of the sphere But the U lambda t The size of that at the origin Let's write it like that. It's just one minus two t times four lambda squared. So this is going to infinity for As lambda goes to infinity with fixed t of course, right? So so so basic the moral is normally interpreted as so this is Normally interpreted as the fact that there's no L1 L infinity smoothing so Although the point of the lecture today is to actually prove an L1 L infinity smoothing estimate So you'll see whether Where the catch is in a second? So let me let me say what I'm raising why this is why this is actually a problem So, you know to go back to the motivation We want to start the Ricci flow for and quite a few reasons with rough data So what you would expect to do would be to take a rough data the sort of Time-honoured tradition approximated by smooth data run the smooth data But prove a priori estimates on the smooth flows from the smooth data and then pass the limit of the flows So that's the usual strategy So in some sense the estimate that you require To make to get the the right Ck a priori estimates on on the on the solutions is Exactly an L1 L infinity smoothing estimate apparently, you know once you have L infinity control on solutions of this equation then You can use the Georgian ash motion shower to just get any Ck Eston and and you're away If you consider the maximal solutions What you are saying is that you do this procedure of concentrating the initial condition you get an infinity estimates after uniform No, so I have not written down a claim yet All I've said is what doesn't work. So I'll write down a theorem in a minute and you'll see what we can prove But let me let me just say another thing that isn't is known which is that So the closest existing result is The LP L infinity smoothing for P strictly bigger than one so that is due to Not so completely clear, but there's a nice paper of the Benedetto and Dilla where they used a Georgie iteration to make this work and There's other work of that that cares which may have been earlier. I'm not sure Where he uses symmetrization techniques to make this work. So you You can symmetrize and then reduce to one dimension which is Then easier to handle Okay, so in some sense what this is saying you You know if you're an LP or a little bit more spread out somehow you controlled how concentrated you are and The moral you should take away maybe is that you know once you've spread out a bit Then you've got all the diffusion is happening relative to itself And then once you've spread out a little bit then it really gets going and you really get diffusion okay Unfortunately, this result is actually not very useful for applications because you just in Geometric applications, you just don't have LP control on your initial data, you know just basic The basic situation would be that you have L1 data, you know Many of the metrics you would end up considering there's maybe limits of smooth metrics with certain curvature conditions would end up Having singularities like a hyperbolic cusp for instance, you could approximate that Biometrics of curvature bounded below For instance and a hyperbolic cusp when you write it in coordinates would be an L1, but not in any LP Okay, so what we need to do is find a an L1 L infinity smoothing which by this cannot Exist in the traditional sense. So here's the idea. So normal L1 L infinity smoothing says You give me the L1 data and you give me the time and I'm going to give you L infinity bound Okay, which is false. What we're gonna do is you give me the bound that you want and You give me L1 data. Let's say and I'll give you the time you have to wait Okay, so it's a little bit twisted round and just by making that little twist around it actually Makes everything work. So let's write down a theorem and you'll see how it goes So this is the first there which is the sort of a Straight PD result and there's a little variation on that which is the more geometric result. So this is with How you in We proved it a few months ago. It's going to submit it soon so let's suppose we have a Solution to this logarithmic faster fusion equation and we're gonna we're gonna do it on the ball okay, so This here is the unit ball in R2 So suppose this solves This equation over here, so it's probably just easier to write it down and Let's say with initial data U0 in L1 of The ball I'm not making any assumptions about what happens as you approach the boundary of the ball You could be smooth up to the boundary. You could be totally crazy up to the boundary Okay, so what's the conclusion? So then Don't get distracted by this Delta is going to be a bit of a red herring. It's just to make it super sharp So for all K This is going to be you remember you're going to give me the upper bound I'm going to give you the time you have to flow for think of roughly The K or some scale version of K is being the upper bound so for any upper bound K and T Well, I have to specify the T So T such that So I'm going to say once you're at a certain time. We're going to get a good L infinity bound So what is that time? Well the time you have to wait is given by I'm going to take the The difference you zero minus K and just take the positive part Okay, measure it in L1 and That is the time modulo a Factor of 4 pi and then wait a little bit longer Depending on Delta So for all this and T such that that we have and then morally the estimate is saying We're now below the bound K although I'm going to put CK if you want you can replace key K by K over C And I'm also going to notice that if I make T really large then a different phenomenon kicks in You know, I might have a hyperbolic guide that's sort of expanding and making this big So I better add a T here, but that's sort of negligible for small time and C is depending on Delta Okay, so that's the The first theorem here So the four pi is I mean this is sharp. You can't do any better than this There are various ways of seeing that You can give an example. It's actually not the shrinking sphere So the shrinking sphere is saying, you know, you're like a Delta function, but only until time half Okay, and then you've got brilliant bounds. You're actually zero so That's not that that would give you a factor of two loss here in the estimate So they actually the sharp example is is a so-called cigar soliton metric of Hamilton Which is a little bit better. Well, I don't think maybe we should get into that now. How are we doing on time? Right Another maybe thing to point out is that there is no boundary assumption here at all Okay, so we're doing oh, so I better make sure for that to be true I better make sure that we do an interior estimate. Okay The notion of a solution is not an issue Everything smooth Yeah, just make Sorry Sorry, sorry, sorry in the theorem everything take everything to be smooth and Then you were in the nicer situation which has all the content you can then do approximations As you see fit So you even use error take to be smoother when I say it's an L1. I just mean If you integrate that smooth function you get something about it, right? Not that You're not too irregular on the interior All right Yeah, so there's no boundary There's no boundary assumption which is again something to do with the non-linearity of the Is to do with this non-linearity of this equation this sort of thing would be impossible We're getting local estimates here purely local estimates Which eventually can be applied to any Ricci flow in an arbitrary chart You couldn't possibly hope for lower bounds of the same form that would just work the wrong way around so Given any salute any initial data there would be a solution Which just disappears to nothing in a shorter time as you As you like so this is a complete contrast to that So let's quickly Give you a slightly More geometric form of the equation and then we'll say a few words about the proof so to see the more geometric form of the equation we have to consider the Metric I was talking about earlier the Poincare metric that you know the standard so I'm gonna put metric in In quotes here because I'm really considering the conformal factor. So as I wrote down before this is one minus mod x squared squared so that would have That would be a metric of constant Gauss curvature minus one Okay, so it looks something like this on the ball and if you evolve that under Ricci flow or just lift up so instead of Being multiplied by small and smaller number and shrinking to nothing. It would actually be multiplied by one plus two t would actually expand Okay, so that's worth bearing in mind So I have to think of a way of streamlining this a little bit so let's do a Slight variant on this theorem one There are two still with how you know and it says that Under the same assumptions here, but not this L1 here Assumption here. I'm going to make it a little bit different. So we're going to say if U zero minus The hyperbolic metric positive part is an L1 in fact any scaling for hyperbolic metric So let's take that to be the case So alpha is just some scaling so if if we're in L1 then More or less the same conclusions. So if t is big enough and now it's going to be the L1 norm of This Quantity that's an L1 of course So still divided by 4 pi And then just go on a little bit beyond then We get as a conclusion now on the whole ball That we're lying underneath C t plus alpha H so now that's existing on the whole ball not this is not even an interior estimate anymore So what what is this saying? Sort of running a little bit out of time But let me try and summarize so what it's saying is so you've got this hyperbolic metric Which evolves nicely by expanding and it's saying if you take any other metric even if it's got some really sharp points then Eventually this expanding hyperbolic metric will overtake The other one in some sense at least if you scale it a bit it'll sort of overtake Okay, so it's not something you can get from the tried and tested maximum principle. It's it's more subtle, but it's Yeah, anyway Okay, so I think We should probably say something about the The proof so why is it true? That for a definite time interval, I definitely don't have any Upper bound on my you but after this specific time then I do so why why is you know What's the mechanism that's making that work? That's what I try to want to try and explain so after After a shorter than this time, we won't have any l infinity control in this We won't even have any LP control on that. Okay, as soon as we ever knew that we had LP control on that Then the LP L infinity smoothing will kick in instantly and give us L infinity bounds immediately So for some reason there's no LP control for a definite amount of time and then suddenly you get This nice L infinity control So let me try and sort of summarize some of the ideas in the proof. So what we end up doing is defining a potential Which is derived? So let's call it psi zero. It's derived From Solving the equation Let's just restrict to the alpha equals one case Solving the equation as follows So this is now going to be on the ball and we're going to take zero boundary data for our potential So in the in the example you gave at the beginning The example you meaning the shrinking sphere and no because that will be out by a factor of two So if you want one that showed that this is sharp Then you should take the cigar salt on and scale it properly So if you scale the cigar, so maybe that's sort of slightly different metric which geometrically is like a Half-cylinder which has been capped off And if you choose that correctly then it's a so-called steady salt under the equation a later time It would be isometric to the original solution, but it's not static solution in the sense that it'll be isometric via Not the identity So if you scale that in the right way then exactly what you say is true, so Yeah At that particular time it would That delta mass the way you have the way you would end up scaling at the delta mass would disappear okay, so I'm gonna have to sort of Paint this in broad brushes Lack of time, but what you end up doing is you consider a potential which actually a slight variant on this a slight variant which is sort of It's a quantity which you would consider When you're studying Caleb geometry in fact though we don't have to worry about that at all It would be basically this nice simple potential Plus Something that you can control actually We have to actually come up with a a particular instance of a so-called Caleb potential Because we're working locally, but we end up cooking up a potential Which we call phi I mean let's just be very rough which is sort of like this Plus something which blows up in a very prescribed way infinity, and then we let that evolve under the PDE Which looks a little bit like the logarithmic fast diffusion equation only the log of the plus in the wrong way around so you end up with this equation which is something that would crop up in Caleb geometry in fact and In some sense in this what you have to be very careful, which you don't normally have to do To make sure you have the right boundary conditions, which actually is boundary growth condition So then you look at this evolution, and then there's a harness inequality, which we could borrow from a Relative your recent paper of two guys from to lose goodge and Was very high here. I don't know you say his name and We adapted that what they did to our situation in the local case and it ends up giving you a Harnack estimate on a sort of evolved potential of this so for our purposes today I Can tell you what we managed to extract from that Harnack inequality, and That that will be enough I think so what we managed to extract is that a later time you at any time not at time Beyond some magic threshold at any later time we get an estimate Which is like e to the? Solution of this original potential divided by t so so this is going to be true on be a half so you don't have to worry about all the Derivation for today All you have to think about is this These two bits in boxes So can't get a simpler PD than that Can't really get a much simpler Estimate than that it's kind of not nice because the right-hand side here now Magically only involves the potential at time zero It's actually the H this hyperbolic guy right so I the way you prove it is to do this take this slightly more geometric Version right you can think of it. I mean locally it would be You know alpha H would be a little bit like K Right, so yours, but it's just convenient to be able to work globally and H actually It's like a K which blows up in infinity So what are we going to do with this estimate so Let's have a look at this PD So any got a couple of minutes left, but This quantity here by assumptions in L1 so We get an estimate here in two dimensions, which doesn't quite put this in w2 1 and L infinity if we were in L infinity Then this would be an infinity this would be bounded so we would already be bounded So actually, you know if you if you could get a w2 1 estimate offer You know off the borderline case for elliptic theory, then you wouldn't have This sort of shrinking sphere example, okay, so it's completely concrete on the other hand. There's an estimate of them which in this setting Normally be attributed to Brazies and now The super super simple very pretty indeed, so he takes a few lines through and it says basically If you have an equation like this where you have a plus, you know something is an L1 function then You get exponent. Well, they're various exponential interoperability statements as you all know For psi, but the one we need is the one written in in the Brazies mail paper and it will say So what it says is that if you have minus the plus and eta is F in L1 and Eta zero on the boundary then If your exponent P is Less than 4 pi over It would be the L1 norm of F then You'd get e to the eta in Lp Okay, so that's what we can apply here. So we're going to apply it So eta is not going to be psi zero. Let's just divide through by T may as well and Then this will be our eta and this will be our F. So if you unravel what that What that actually says Then it says that if just in one line If T is bigger than this magic threshold number then The exponential of psi zero over T Is in Lp for P bigger than one. That's probably the cleanest way of saying it So you have to get to that threshold time Then this quantity is an LP and that's exactly what you have the right-hand side of this estimate. So at that point You're an LP and the LP infinity smoothing kicks in so immediately an L infinity as well In fact, that's not what we do because the LP and infinity Smoothing estimates that are in the literature a little bit weaker than what we can deduce from our estimate So in fact, we go we just sort of do a double bootstrap here We get this in LP and then we just do the same argument again And get L infinity bounds and that works out Nicely so that gives you a sort of idea of Proofs not not not that tricky. I was going to explain how you get the the nice LP L infinity smoothing it only takes a few lines But I think I'm out of time you'll have to read it in the paper. I'll stop there Thank you. Do you have questions or comments? For uniqueness you need a lower bound which you get from completeness Yeah, you're going back to the beginning theorem. All right You need a lower bound which you get from completeness is exactly what you say Exactly exactly it exactly gives you the lower bound that you need right because with that exactly How do you get the harness? How do you get the harness? So how do you get it so um You have to find the right quantity Which is a combination of psi and its derivatives Which then which we could borrow this is exactly the bet that we Borrowed from I'm not even sure what the first paper would consider this but this is the paper we got it from Once you've got this quantity you can look at the evolution equation For that but there you've got to be a little bit careful. You've got to make sure you're doing the geometric thing You know, there's no good looking at the evolution equation in r2 You've got to look at the evolution equation in hyperbolic space So if you do that, then you can apply the maximum principle You can show that that harness quantity by the way that we cook up Our potential infinity you can show that it actually vanishes infinity or you'd need in fact There's that it would be bounded or it wouldn't grow too badly Infinity but then the then you can argue that that particular quantity that you cook up is zero initially um and satisfies a nice Parabolic partial difference in the quality so it ends up having a sign And then when you consider the consequences of that You get this But you've got you've got to think of that you've got you've got to have that quantity at hand, right? Otherwise So you've mentioned a little bit about geometric applications going to save you Yeah, so, um so Well, there are several applications, but the one that i'm was um Alluding to earlier is right. So starting Ricci flow with rougher initial data. So What sort of rough data might you consider? So in practice when you're actually using Ricci flow, you might end up having to consider Uh data, which is a limit of smooth guys So a limit of smooth manifolds is just nothing in general But in practice you're trying to prove a theorem about maybe Manifolds with Ricci curvature bounded below for instance in this situation It would be Ricci it would be Gauss curvature bounded below that would be So if you take a sequence of such manifolds Then again, you don't have smoothness in the limit but there is a nice theory of conversions of such objects to metric spaces with lots of structure So alexandroff spaces or variants of alexandroff spaces, you know, there's a lots of flexibility in the actual theorem you write down so um So these metric spaces that you you end up considering You can view as ultimately By a work of people like rachetniac You can view them as reamon services with conformal factors you Do that have very low regularity. So how would you flow something like that? Well, you'd think oh, well It's a parabolic equation. So I can just flow it. But of course We've just seen, you know, delta functions don't smooth out. You don't get smoothing in general So you need to make sure that you can take this sort of l one type data And and flow it. So of course the way you do it is to smooth it Approximate it by smooth guys flow all the smooth guys, which by our theorem just will just exist typically for all time But you need estimates to pass the limit and the estimates you need are the l one and infinity smoothing estimate that we prove It what about in higher than the majority to float down the train? Is there any any hope of having something? uh, well There is I mean It depends which bit you want to extrapolate. So certainly the idea that you Try to consider metric spaces that are limits Of smooth guys with various geometric conditions Because you know, maybe the goal is to prove results geometric results about Manifolds with you know, Ricci curvature down below and you know volume growth of a certain behavior or whatever So, you know in an argument in a contradiction argument, you know might be trying to prove something about Such objects. So you'd end up saying well if if it's not true, then you end up with a sequence where the Ricci curvature is, you know um banded Has a uniform bound etc etc, and then you'd end up having to pass to a limit and then Consider the Ricci flow with that limit. So this is something that Is that does actually work. So mile simon is the pioneer in that sort of line of work Works instead mile simon So, um, yeah, so on the other hand there, there's still really a lot a lot to be done Yeah, that's kind of interesting subject So, thank you Yeah, right so, um, but this is a much more general statement which I was sort of alluding to Um, when Sergio was asking his question before so the theorem says the following You give me so I already gave you a theorem that says you can always flow any surface If you give me a surface which Well, look at its conformal type. It's either Has the universal curve of hyperbolic space or the plane or s2 by the uniformization term if it's hyperbolic, which is almost every case If it's in other words if there exists a hyperbolic metric Then our theorem always says you have existence for all time And it says but then there's another theorem which says if you divide by If you divide g of t by 2t, so scale down Then that will always converge smoothly locally to the hyperbolic guy Right, so yes, that's the answer