Sharp local decay estimates for the Ricci flow on surfaces
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00:00
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Transkript: Englisch(automatisch erzeugt)
00:15
Okay, well, thank you very much. Thanks for the introduction and thanks very much for the invitation much appreciate it
00:20
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
00:42
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 Logarithmic fast diffusion equation and this is simply the equation d u by dt is
01:08
Laplacian of log u so this is for functions positive functions from let's say some domain in R2
01:21
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
01:42
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
02:04
The setup is that we want to work now on a surface not necessarily R2 or part of R2, but Just take a just some smooth surface. Let's call it M
02:22
So then a Ricci flow. I'll just write RF. 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
02:47
It's of 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
03:01
Because we're working on surfaces 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
03:25
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, but this is going to be relevant even though
03:40
It's a bit of a banal example. It will be relevant in a in a second. So it would actually Shrink to nothing in that time a half So what's the connection here between these two? equations well, it's very simple so
04:04
On M you can pick special coordinates So called isothermal coordinates, of course So that G can be written locally as Just some positive number times
04:21
The Euclidean metric which let's just write it the tensor like this so locally, it would just be Euclidean space where distances are deformed by U to the half that would be okay So the area of this surface would be the interval of U locally
04:43
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 the equations equivalently
05:05
Equivalent I'm saying this is being solved in each Coordinate chart, okay, right so
05:20
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
05:44
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
06:02
Might not be something of sort of Sobolev 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
06:25
L1 initial data, so I'll come back to that in a second Anyway 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
06:43
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
07:08
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
07:22
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 assume that smooth connected
07:41
Riemannian surface and G0 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 this is closed surface
08:05
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
08:22
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
08:45
The second condition is that G of T is complete Obviously not that time 0 because we're not assuming that but for all T after time 0 so it's kind of
09:02
important here I'm not including 0 so that seems like well existence and uniqueness, but it's kind of a little bit weird If you think about it because this is really including
09:20
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 them to hear you don't have uniqueness, that's all right for sure
09:45
Yeah, so 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
10:06
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 that smooth up to the boundary
10:21
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 a finite distance, so something has to change dramatically. These are all smooth Ricci flows
10:44
Exactly so the solution would blow up in the boundary. So Here's your disk and then initially your you there would just be zero I Sorry, it just be one and then for a little time it would have to blow up
11:03
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 have to be a certain rate of block Immediately to make it complete. In fact, this would have a geometric picture. This would be
11:22
In a thin boundary layer around the disk that would be the hyperbolic metric of curvature minus one on 2t 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
11:42
Uniqueness, that's the whole point of this 2015 paper Always always unique. So we're feeding data in Feeding heat in from infinity. Okay, so You're sort of ticking off in a star You're getting something that isn't just the flat disk. Well, that's not totally unreasonable, but
12:05
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 I tried to put more in that would fail Because it's the more you put in the high it blows up
12:23
Then there's a damping you're getting in the equation Because of this non-linearity here, so it might actually be Instructive to look at the equation for V which is half log u And that satisfies the equation DT V is e to the minus 2 V Laplacian V
12:44
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 disk 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
13:07
Guess my question is you're not allowed to put in twice as much heat on the left is on the right Still comes out unique. That's right unless you're willing to fail the test of completeness So in a sense, it's giving you well posed in this this equation
13:22
Which is always a bit of a conundrum a lot 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
13:40
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
14:02
Okay, no boundary to you. I don't want to talk about boundary boundary data. I don't want to talk about boundary in general It's not even true that you get asymptotically your conform factor 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
14:20
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
14:40
It would look like a Poincare metric scaled Homothetically so the curvature was minus one on 2t That's for you. Sorry I'm plotting here you sort of G
15:00
Yeah, so so I just mean the you 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
15:21
There's a You're going to prompt me to give a different talk here, which is That the reachy flow does uniformization for you in this case that's kind of Amazing to me that you you start off with any metric on a on a surface like this that supports I have a white metric and it will Flow to it in the sense that it will converge asymptotically if you divide them g of t by 2t
15:46
To renormalize because everything is expanding in the hyperbolic setting if you just renormalize by dividing by 2t It will converge smoothly locally to the unique uniformization metric in the unique hyperbolic metric
16:01
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
16:22
You to a hyperbolic metric Sort of uniformly that's a couple of words about the The T so the T is explicit the existence time so basically normally
16:41
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 and shrink to nothing. That's a theorem of Hamilton and
17:00
Chao which is sort of part of the dot dot dot And also if if you're Conformally the plane C 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
17:22
Then you can have also flows where everything disappears In a finite time, so for instance give you another Cute example just take the sphere but remove one point. So this is no longer Conforming the sphere it's conforming the plane by stereographic projection. It was just map it to the plane
17:41
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 the what will happen is it will develop a hyperbolic cusp straightaway and On the other hand the bulb here will just sort of shrink to nothing
18:00
In a finite time and you can analyze the asymptotics in fact Manuel was one of the people that Did this sort of? Analysis, sorry, it's very very interesting stuff Let's not go that way right now then Okay, so Maybe I'll mention some more names
18:22
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
18:42
existence But it doesn't as last as long as ours in general only got lost until the curvature blows up Whereas for us the coach it does blow up in general and you just keep going The uniqueness in the in the bounded curvature complete case is due to Chenin's you
19:04
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 Well, I think the main results are the existence theory in R2 and
19:20
we've got papers of Daskalopoulos and Del Pino and de Benedetto and Diller and Vasquez Esteban Rodriguez and Sure, surely more more science 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
19:40
to get this Okay So We want to talk about flying 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
20:03
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
20:21
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
20:42
So if you don't have the nonlinearity 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 get a convolution of the heat kernel and And the initial data so immediately you can bound that by
21:09
Then the size of this and the size of this just using Hilde so so you'd get a bound like
21:22
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 P is 1 it would be minus n on 2 So in general, I think it's at minus n on 2 P. We're just using Hilde
21:42
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 cases 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
22:02
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 nonlinearity. So the way to view this Is to think geometrically again, so
22:27
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
22:40
Unit sphere 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 I'm going to write down my you so therefore I get a solution Okay, so let's um Maybe do that over here. So the shrinking sphere Ricci flow you can write down as
23:18
So what I'm going to do is
23:25
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 just in a second so
23:43
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'd end up with a lambda here lambda squared here In fact, we saw that in the last talk hidden hidden away there. Well, this is the metric of the sphere, of course
24:05
so Lambdas of 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 class in a vlog of it
24:25
It will just scale So this gives you the solution u lambda t so that would just be 1 minus 2t u 0 Lambda, okay. So why why am I telling you this? Well, of course, I
24:43
Want to take the limit as lambda goes to infinity and if you do that then you get this Mass of 4 pi mass of area would just concentrate at the origin so user lambda would converge to the delta function and then the
25:06
u lambda of t will be converging to Not as nice spreading out Gaussian, but just still a A scale Delta function Okay, so in some sense diffusion in Ricci flow or logarithmic fast diffusion equation is happening relative to itself
25:26
Just think of it that way. Okay, so in particular It's not true that if you give me the
25:41
Oops, if you give me the l1 norm and the time you can't get an infinity estimate. So, you know, of course u 0 lambda and l1 is just 4 pi the area of the sphere But the
26:00
u lambda t The size of that at the origin Let's write it like that. It's just 1 minus 2t times 4 lambda squared so this is going to infinity for As lambda goes to infinity with fixed t of course, all right, so so basically the moral
26:23
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
26:41
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
27:05
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 this sort of time-honored tradition approximated by smooth data run the smooth data
27:23
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
27:46
Infinity smoothing estimate apparently, you know, once you have l infinity control on solutions of this equation then You can use the georgia nash mose and schouder to just get a ck Estimate and hand your way
28:02
You can see the maximum solutions What you're saying is that you do this procedure of concentrating the initial conditions you get an infinity estimate No, so I have not written down a claim yet
28:21
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
28:48
so That is due to not so completely clear, but there's a nice paper of de Benedetto and Diller where they used a georgia iteration to make this work and
29:03
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 can symmetrize and then reduce to one dimension, which is Then easier to handle
29:22
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 spread out a little bit, then it really gets going and you really get diffusion. Okay?
29:45
Unfortunately, this result is actually not very useful for applications because you just in Geometric applications you just don't have LP control 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 as maybe limits
30:04
Of smooth metrics with certain curvature conditions would end up Having singularities like a hyperbolic cusp for instance, you could approximate that by metrics 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
30:23
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 an infinity bound
30:43
Okay, which is false. What we're going to 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
31:03
So let's write down a a theorem And you'll see how it goes So this is the first theorem which is the sort of Straight PD result and there's a little variation that which is the more geometric result. So this is with
31:21
How again We proved it a few months ago and I'm going to submit it soon so let's suppose we have a Solution to this logarithmic faster fusion equation we're going to we're going to do it on the ball
31:43
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
32:04
Use it 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?
32:26
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 the time you have to flow for think of roughly
32:43
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
33:01
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 u0 minus K and just take the positive part
33:22
Okay, measure it in l1 And that is the time modulo 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
33:43
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
34:02
You know, I might have a hyperbolic guy 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 first theorem here So the 4 pi is I mean the this is sharp. You can't do any better than this
34:23
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 a time half
34:42
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, but I don't think maybe we should get into that now how we doing on time
35:07
Right Another thing 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
35:32
Everything smooth Not finished today Yeah, just make it Sorry
35:41
Sorry, sorry. Sorry in the theorem everything take everything to be smooth and Then you're in the nicest situation, which has all the content you can then do approximations As you see fit So you even use here or 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
36:04
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
36:22
Is to do 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
36:41
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 of it So let's quickly Give you a slightly
37:03
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 can put metric in
37:24
In quotes here because I'm really considering the conformal factor. So as I wrote down before this is 1 minus mod x squared squared so that would have That would be a metric of constant Gauss curvature minus 1
37:41
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 1 plus 2t would actually expand okay, so that's worth bearing in mind
38:04
So I'd have to think of a way of streamlining this a little bit so let's do a slight variant on this theorem 1 theorem 2 still with how you in And it says that
38:22
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 0 minus The hyperbolic metric positive part is in l1 in fact any scaling for hyperbolic metric, so let's take
38:43
that to be the case So alpha is just some scaling so if if we're in l1 Then More or less the same conclusion. So if t is big enough, and now it's going to be the l1 norm
39:05
of this quantity that's in l1 of course So still divided by 4 pi and then just go on a little bit beyond then
39:21
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 or not. This is not even an interior estimate anymore. So what is this saying?
39:41
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?
40:03
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, right, so I think we should probably say something about the
40:29
The proof so why is it true? That for a definite time interval, I definitely don't have any
40:40
Upper bound on my u 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 want to try and explain so after after a shorter than this time We won't have any L infinity control on this we won't even have any Lp control on that
41:01
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
41:25
so what we end up doing is defining a Potential which is derived, so let's call it Psi zero. It's derived
41:41
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
42:02
potential Excuse me, so in the in the example you gave at the beginning So you give your response we mean immediately after the lifetime of the compact solution The example you mean in the shrinking sphere
42:22
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 salt maybe that's sort of slightly different metric which geometrically is like a
42:42
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 you'll be isometric via Not the identity
43:02
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 um I'm going to have to sort of
43:21
paint this in broad brushes Lack of time, but what you end up doing is you consider a potential which has your slight variant on this a slight variant which is sort of It's a quantity which you would consider When you're studying Kala geometry in fact, although we don't have to worry about that at all
43:41
It would be basically this nice simple potential Plus Something that you can control actually We have to actually come up with a particular instance of a so-called Kala potential Because we're working locally, but we end up
44:00
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 logarithmic last end of the wrong way around
44:23
So you end up with this equation? which is Something that would crop up in Kala 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
44:42
condition So then you look at this evolution, and then there's a harness inequality which we could borrow from a Relatively recent paper of two guys from Toulouse good and Very high, I don't know you say his name and
45:03
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
45:22
From that Harnack inequality and That that will be enough to think so what we managed to extract is that a later time you at any time Not a time beyond some magic threshold at any later time we get an estimate which is like e to the
45:41
Solution of this original potential divided by T. So so this is going to be true on B. Aha 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
46:05
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 is to do this take this slightly more geometric
46:27
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 yes, but it's just convenient to be able to work globally and H actually
46:41
It's like a K which blows up infinity So what are we going to do with this estimate so Let's have a look at this PD. So I only got a couple of minutes left, but
47:00
This quantity here by assumptions in L1, so We get an estimate here in two dimensions, which doesn't quite put us in w21 and L infinity if we were in L infinity Then this would be an L infinity this would be bounded, so we would already be bounded So actually you know if you if you could get a w21 estimate off a you know
47:21
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?
47:40
Normally be attributed to Brazil's and now The super super simple very pretty indeed so he takes a few lines to prove and it says basically If you have an equation like this where you have a class you know something is an L1 function then You get exponent. Well there are various exponential integrability statements as you all know
48:03
For psi, but the one we need is the one written in in the breezy's mail paper, and it will say So what it says is that if you have minus laplacian eta Is f in L1 and eta 0 on the boundary then?
48:27
If your exponent P is Less than 4 pi over it would be the L1 norm of F then You get e to the eta
48:42
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 0 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
49:04
what that What that actually says then it says that if just in one line If T is bigger than this magic threshold number
49:27
then the exponential of psi 0 over T Is in LP for P bigger than 1 that's probably the cleanest way of saying it So you have to get to that threshold?
49:41
Time then this quantity is an LP, and that's exactly what we have the right-hand side of this estimate so at that point You're an LP and the LP L infinity smoothing kicks in so immediately an L infinity as well So that's not what we do because the LP L infinity Smoothing estimates that are in the literature a little bit weaker than what we can deduce from our estimate
50:04
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 our idea of The proof is not not not that tricky. I was going to explain how you get them the nice LP
50:26
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
50:54
Yeah, you're going back to the beginning theorem all right you need a lower bound which you get from completeness
51:04
Exactly exactly exactly gives you the low bound that you need because with that exactly So how do you get it so
51:21
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 bit that we? Borrowed from I'm not even sure the first paper with consider this, but this is the paper we got from
51:41
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 There's no good looking 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 on that quantity by the way that we cook up
52:03
Our potential infinity you can show that it actually vanishes infinity all 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 and satisfies a nice
52:22
Parabolic partial differential equality 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 quantum. You've got you've got to have that in quantity at hand All right, otherwise more tricky So you mentioned a little bit of the geometric applications games a few yeah, so
52:45
So Well that there are several applications, but the one that I'm was Alluding to earlier is right so starting Ricci flow with rougher initial data So what sort of rough data might you consider?
53:01
So in practice when you're actually using Ricci flow you might end up having to consider 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 banded below for instance in this situation
53:22
It would be reaching it would be Gauss curvature banded 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
53:40
Alexandra spaces or variance of Alexandra spaces, you know, there's a lots of flexibility in the actual theorem you write down so you So these metric spaces that you you end up considering you can view as ultimately
54:00
by work of people like Rochetniak You can view them as Riemann surfaces with conformal factors you That have very low regularity. So how would you flow something like that? Well, do you think? Oh, well, it's a parabolic equation. So I can just blow it but of course We've just seen you know, Delta functions don't smooth out you don't get smoothing in general
54:20
so you need to make sure that you can take this sort of l1 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 of the L1 and infinity smoothing estimate that we prove
54:43
And what about in Heidelberg and Ricci flow in Heidelberg? Is there any hope of having something? well There is I mean, yeah, it depends which bit you want to extrapolate. So certainly the idea that you Try to consider metric spaces that are limits of
55:03
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 banded 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
55:23
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 Banded Has a uniform bound etc. Etc. 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
55:47
works in search miles Simon so 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
56:16
Yeah, right so um, but this is a much more general statement which I was sort of alluding to
56:22
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
56:43
Has the universal cover hyperbolic space or the plane or s2 by the uniformization term if it's hyperbolic which is almost every case If it's a in other words if there exists on 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 2 T
57:04
So scale down then that will always convert smoothly locally to the hyperbolic guy, right? So yes Thank you