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The Poincaré Conjecture

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Well it is my pleasure to introduce the Leonard speaker reaches semis at each of them as those had professorships at Cornell University and I am proud to say send here wine and patently she's a professor at the Columbia University well have all the shot and Real here making that A heaven don't serial reachable let the of the most exciting developments of mathematics it is not an exaggeration to say that the reaches famine is about the heroes of this Congress Rich and met a toe thank you very much what we can deal with dealers Things get ready How but delay weather satellites Trying to get to screen out there there's some light Severed but can you see That OK I gave it While G past there was 40 years ago I was sitting injured Nielsen said For now when I he put the point trade conjecture up on the board suggested that Using PET need program And I said Well gee Hypothesis is the apology and the conclusions topology why would you expect proved to to be apology me said well the hypothesis is that you can't find any topology so that the biologist over what to do baby we can help them that I'm so that was a great idea and it goes Back at least EUR lobby Once had the idea of trying to to put a nice metric on 3 manifold he started with a simple case of the metric of constant scalar curvature and that was the you Monty conjecture that Brilliantly solved by exchange Well we're at San Diego Anyway It was 10 years later before I started thinking seriously about what you could do
A Eels and Sampson just a parabolic flowed through the existence of harmonic maps taken straighten it out and make it nice so I thought well maybe you could do the same thing with the mattress star with the metric and flow at try to spread the curvature out evenly over the whole manifold and I couldn't figure out how to do it for quite a while and then I thought well If you just Kind of Think of water Well what looked like he wanted 2nd order and parabolic causing linear and After you'd like to diffuse equally in all directions and all the components of the curvature of it turns out there's only 1 flow that has that property which is the reaching of wages PTT GATT the metric is minus flights Ricci curvature and this turns out to be be buried Nice very basic equation in geometry EU let Dick version is Ricci flatter tops the Ricci curvature of the hyperbolic version is the same thing will rent space where are signs vying showed that Can be used to describe gravity And I don't know if I The 1st person to write to 40 down Set on the Discovery Channel that 1st person to the steam engine Wasn't the 1 who 1st made a like something go round and round was steam the 1st figured out the powers something And so I figured out that you could actually approved in geometry with this and that's kind of however In order to have tackled the general case you have to allow surgeries because Equation forms singularities and the ideas Before it gets singular you go it Cut out the bad for and rounded off to continue the flow so Well let's put up a picture here What does the those sort of sort of thing that Ricci followed does 2 dimensions view of how above And asked to that has neck in well necklaces Nelson 1 and 1 from the point of view over money in geometry doesn't have any curvature of the neck opens up and it turns into around here and in 3 dimensions what I proved it as well you need to assume that you have
Positive Ricci curvature and while actually rounds The metric out to around 1 While it's doing this it straight down the . 0 3 expand together let but I succeeded in proving that with positive Ricci curvature under 3 manifold There's a constant curvature metric meaning it ceded the Spira quotient of So what But let us So right after this Yao suggested me that is the way to get rid of white But The Know what did it anyway that Yao suggested that in 3 dimensions of the neck You'd see a dumbbell shape thing is actually 2 that Those Intrinsic Romani incurvature and that shrinks the despair Now it's get a pinch but fold into 2 pieces So this can actually be used to perform at a connected some decomposition Pearson example Of a typical neck The other singularity that occurs generous neck If inch a because Which looks like this so you have I guess I'm a little else to pinching down but the 3 while on the other side of salsa inching down at the same time so when it Forms a singularity it's just that an isolated point with nothing left on the other side So searchers Richie following you start with many metric run the Ricci flow until the curvature gets too big And then you try to cut the curvature down by metric surgery and you get a look for for cylindrical callers An found all on 1 side if you assume that the folders from which you can already do the topology and see that's why like Dealing with this Face of simply connected and Prime because Then you don't have any topology And then you try to capital off so it looks nice again and the explain That this sense finite time on simply connected manifold after finitely many surgeries and and was seeing that it's actually the the 3 So as you think that there's relate Any sort of topology going on here All the surgeries that I'm going to to be doing Our what I call appendectomy see my dad was actually a real surgeons down when I was a kid a told me about taking out of season affect my parents treated me and my brother so thoroughly that the only thing he ever doubt that I didnt was appendicitis and so here you see All the topologies Over there on the left and there's this narrow neck Pitching but it's Escobar ball on the other side And I certainly think this as the information were the curvature Scott that you won't cut it out throwaway The appendix witches Trivial it doesn't matter and then use an ice half and the idea is that you have reduced The information because you are Got rid of where the curvature was real big and replaced it with where it isn't nearly as big and so you say this is really designed not to take the apology which is already find it's designed to FedEx be Analysis of the effects the excessive growth of the curvature a distant boat was but a traitor and that's fine but so up The existence and uniqueness you start with any metric you light and then at least for a short time get a solution I so I was afraid of us If Very interested in the national inverse function there and I was very proud of the fact that I actually used it because this isn't strictly parabolic it's only weakly Parabolic in you lose derivatives and I thought 0 going our way when we have the learned national sir But actually 2 weeks later the Turks that showed how you could actually use regular there of the box So but it turned out that was great but it happened because everybody should read the paper That awareness Moser said I don't come back to to that idea later I proved global derivative estimates she won strong there was a student at Seattle did excellent work in Ricci float up managed to do a local version It's a standard sort of result for parabolic equations it's a little trick here in this case is the mantra changing but the idea is that if you have that Bound the curvature In a big parabolic cylinder then you get bounce on all the derivatives of smaller self and it's typical of Lipstick and parabolic Equations the smooth everything out down that really gives you lots and lots of extra control so for for classical geometry having Bounds on all the derivatives of the curvature is something that they think would be way too much but as soon as you're into the P theory you actually get it for free 1 of the reasons for liking a parabolic float instead of just the minimizing sequences they pick up so much extra regularity and here's another picture of that cylinder and the derivative that by the way the different powers Just according to this feeling well but next Basic ideas venturing out of And we now have lots of pinching estimates parabolic flows and We shall fight on did it for them Ricci scalar case the Chicago margarine and so on so in 3 dimensions the 1st result was the
Richie's positive that preserved or positive that Research but In 3 You get a very special sort of which is of pinching torts non negative curvature which says That Once the curvature is get very very day then there could be Very very negative curvature but it's not nearly as negative As the curvature is positive in some different directions at the same point that it's only diminishing by a factor of the Lord them so just to show you how it is as if the maximum curvature standoff 100 you'd expect the men among to West y about all the tender offer under which is about 110 square and so the minimum might be minus 98 so you see it still has a huge amount of negative curvature but it's very very small compared the positive curvature and that makes this lovely properties that what we do blow up elements they have not negative curvature of Which is a great thing that has been the techniques for doing this is applying the maximum principal to the diffusion reaction equation satisfied by the curvature and in order to draw a picture about this Reality Here is actually a sum of 2 squares that However You typically what you do is forget the diffusion and just study the reaction and since its homogeneous you project 1st acted this plane were aware the plus new plus new 1 and then you see that everything wants to the flow toward the curvature tensor best 3 accepted that hangs up on the cylinder but if it doesn't like actually go in the cylinder than it skirts around goes on in of the 3 straight down by this is kind of crude picture but here's a better 1 Love and story about my computer and he'll say these 3 wines here are low lines That triangle therefore Here are the As to process 1 curvature and the us 3 in the middle and everything is moving in for the middle of the tens that hang up on the cross 1 but it too Gets beyond even a little better But it comes on hinder us 3 gold from so The next step in the analysis depended on recognizing the importance of solid Assault on is something which moves without changing its shaky so in this case The thing is that the metric actually changes but the changes affected just quiet motion under a defeat workers That is reprogram authorization and the conditions for that as that the Ricci tensor is the leader of the Of the metric along a vector field it The vector fields gradient of a function of them The session of the function but you can also have expanding solid songs were you added Constant times the matchmaker shrinking funds streaking solar was with the going the other way and some pictures There is a steady solid gone there's this 1 which is quite Brennan opens like forever There's this two-dimensional 1 I call the cigar and hear the cigars cross or what did make it three-dimensional and You can also have solitons like coming out of total oral lower dimensional tone France are 1 for you And shrinking solid times like these as 3 shrinking to appoint arrest to cross bar streaking down to a lot of So but Solid And provide a model for studying various estimates there are typically what you find our Harnack estimates The whole theory of estimates Started with a very important feature of You have and Peter Leahy where they studied it for the heat equation and other diffusion acquaintance I thought about them in quite a while and finally realized could prove this nice in equality that For a complete solution Where not negative curvature you actually need not negative curvature operators hired dimensions you always There this in equality holding point lies and . 4 and tension sector L This has a nice for a wary If you're solution goes all the way back to minus infinity you don't have the star over teacher and putting B 0 you have We are positive and that means or increases . 1 this is terribly important for the analysis of these ancient solution but also as this interesting corollary that The curvature non-negative and Riccio is 0 that bread or in the direction EU 0 will see that later The idea of proving Harnack customers You need some sort of positive for example of the curvature change signed than the positive stuff could cancel the negative stuff By hand But if our old hands 1 sign nothing can Council of Europe The maximum curvature Go away rapidly by cancelation of the year it can only brought down by diffusion that's White Keeps curvature from falling off to test the banned the proof in Balsam intense calculation because you have to define all of these Laughter In arresting sensors involving the 1st and 2nd derivatives of Curvature and they were found by looking at what vanishes on assault and then you get this Very nice in quality when you take In the imam of this plot operatic overall I call that Chase matrix matrix it satisfies this nice parabolic in and is my original paper I didn't have This extra per year I hadn't actually 1st women in the calculations are pretty bad I had my computer must duet and life thanks to Doria gold teaching mathematics and muffled found extra nights quadratic music and another creditor here involving 1st derivatives But I couldn't you You don't see it but when I wanna localized Karnak leader What Medicare is very important Early For So now we were all very excited by her The 1st thing that gives us is this Absolutely wonderful monks collapsing asked and this is something I've worked hard Quite a while and so I was totally thrilled to see that he actually done it and that actually looks Alex so here is Perelman's transport equation and this Isn't exactly how Pearlman wrote it down but it's completely Equivalent Perelman defines his fellow function a reduced length but this has nothing to do with the real well functions In terms of minimizing have from but they actually sold this Hyperbolic And now you might think they're gonna see some horrible hyperbolic PDP theory but it's only first-order in a single function so you can't solve it completely by
The method of characteristics and where would you find as well Perelman had an interesting argument that That way you might have found was that again You look At all the expressions that vanish on solid songs and if you look at the potential function of the solids on their own You can make a measure out of it the polls By the end joint The heat equation going back so What you're doing here is Runner reaching World War Up to a certain time and then you stop and you take another function and Saul opening equation backward and there's a complete duality between the parabolic equation that he has Harnack us But missed Transport equation which is really nothing but they transport equation for the lead yelled minimizing half a role for the Harnack that's so there's a whole wonderful theory here and I couldn't talk for the whole hour on this but then you wouldn't see and hear the rest so the characteristics Of the situation or found out how backward time we already sold 3 chief forward and now we're going back there that satisfies DVD tally bread at all And then you know how old calls on the characteristics in you could complete everything out with the and I take the silver operator in And it turns out along the characteristics that he notices Very nice evolution were this firm on the right that's where it is exactly the thing the benefit of its assault on with potential with potential function as and up Just equation Show the EL is less than their equal to 0 2 start and it remains the towering trees that is it You go back in So I don't know You get this In equality and payroll The from the path integral but rather complicated calculations I went over it and my myself are complete detailed twice Calculations completely correct up but this calculations a lot easier actually OK so they open up the next box Are
So Old each to get this Light tried to escape So introduced reduced falling where I take 4 Minus sent over to eat at a minus at all times the Romani and measures enacted Cynthia reduced volume for years and along the characteristics DD tell I have 5 years so it was last year equal to 0 start you decreases along the characteristics and it feels red Perelman's paper entered than this The animal reduced volume The property that the logarithm of Perlman center that is you do the animal reduced volume by by minimizing what you subject to the condition EL less than or equal to 0 down the skills that's lovely non collapsing on which says That gives I love metric that you start with is kind of nice to certain scale or not then subsequently you can show it doesn't collapse in a region where you'd controlled the curvature and that it would probably help if I just put up this picture here because it's Hearted breed the text quick enough hours might picture here we go again So We have this region where that Curvature is smaller than 1 over or where the ball of radius All are were worried that the volume of this fall will be bigger so if it is too small we want to get a car Production So you start by constructing a you where the corresponding satisfies this elliptic quality was the was 0 and then you could make You know Concentrated in this region vanishes outside and you could make birth Total measure you very small then propagate held back by the transport equation which transports the the corresponding measure used back there you have a nice geometry so whenever You have this seen equality E L S 4 0 then be integral would you has a lower back And now by making traffic collapsed enough you could make the size of less this book Coming back Increases Forwarded increases in you get a car production and that's really all areas of proving the nice very very beautiful Very very useful Roussel might have Say I'm just so grateful pressure for doing this because in addition to really really needing it Or the point conjecture I had a plot by my fingernails forget non collapsing in a few special cases Now I never have to to worry about it again showed up So If the sea Here's a few miscellaneous facts that solid phones that I could refer to later the 1st cigar and it has a nice for the 2nd is that if you have a non-contact steady solar tongue with nonnegative curvature Its asymptotic volume ratio is 0 and Asymptotic curvature ratio is infinity except in 2 dimensions were 0 I should define these terms that asymptotic curvature ratio is the of the curvature kind of the sermon origins squared Volume is wins soup of the boy of footballer radius S Over Elster The aperture is the looms soup of them Find diameter of the spirit radius Over asks them This fine diameter is I don't really know the right firm but Tom element although the crude shall store manager but that's only case crude the talk in a certain what It is that if you're in love Not negative curvature case We're might really have 1 boundary component or it might get to it Weddings of product and so you take them Right Open covers of the boundary and you look at the end the of the summer of their diameter so Now we found the agents solutions that very important notion and the whole theory of parabolic equations that parabolic equations Make things better And an ancient solution is 1 which like me it's been around forever And get some things around forever has been getting better all the time you can bet it's already Pretty good So it turns out that there are many ancient solutions and they're very special when you consider a whole lot about em Also they come up a lot Only analysis of singularities when you do blowup limits and so we want study ancient solution What you can show about ancient solutions well I showed the curvature ratios If the curvature goes 0 goes to infinity the volume ratio V the aperture out Alpha 1st France The next resort Perelman is very very useful but he's out the money Ancient solution in 3 dimensions of the asymptotic volume ratio 0 New don't actually need not collapsing after this Because it collapses than the volumes 0 All these years that and he is hazardous to prove his very important fact that the collection of all agents solutions which earned on compacted normalize it so that the curvature you're origin 1 of the next Compaq and that the quality of sea infinity convergence on contracts so let me just Tried outlined how this goes there's another Jeremiah told you about which is the splitting there and the splitting their rooms says that if you have a solution of the Ricci
Float with non negative curvature and if after a while you actually have So long as their only in the curvature This is the curvature operator than the solution has restricted of all would like for example being Taylor and if you know that Ricci has 0 which is stronger than You know what a whole only reduces to end its what's the flat factor and approaches Basically you go go back to the non negative curvature being served to apply the strong maximum principal so this wedding theory has the same corollary Harnack that if the Curvature nonnegative everywhere and reaching some direction is 0 Then Rout or in that direction is 0 and here's a picture of the wedding and in fact you see that
Here when it's wet There are these directions along the flat factor were reaches 0 and in that direction the red or in the direction of thieves 0 So the contrast with the cold and the rest of Perelman's arguments you find all over the place and it's a very very lovely picture be cause the idea Parents But many Cases you show that something blows up What health and it Long has the property that you can find directions where reaching thereof but where rap artist namely the directions Going out away from the Cold War so it contrasted with the product well Heroin users Alexander on based on now I will grow clever back when Indian nationals through the existence of that the church didn't need it So there's famous story when Al sublime was a young boy found his uncle Pericles in his study hard work and he and his uncle said What are you doing pleased that I Studying how to give an account of my actions Soviet Union people or else supplied is replied to do better the study found not to have to give him so I thought That studied not Alexander also basins but found not Need the learn now dear Alexander faces wonderful things and we really do need 8 Version of weak solutions of the Ricci for higher dimensions but let's see if we can get away without it offered well It turns out that Gibbons Well Appropriate derivative estimates of the sort you need to do the call argument that you actually get a very nice quantitative result but if you have 2 points here that the ratio of the curvature it Peter the curvature it still is Bounded by the curvature a few times the distance PDQ squares on how that's a kind of a power loss the 6 was found gives you a very quantitative measure of the sort of control that Perelman gets that there must be some control out of the code argument and this is a more explicit formula so let's see an example of how this might come about That Probing this 1 of the asymptotic volume reaches 0 Found There are 3 cases And I had originally studied this in my formulation of singularities paper and I got it in the 1st 2 cases and I thought the 3rd case probably couldn't occur but I couldn't show 1st her code argument rules it out but what I managed to show it that if you're in this case Where the curvature times the assistance the origins Where his founded and the curvature times time Backers down there I showed Asymptotic volume ratios Positive and at least away from the origin of the curvature It is bounded above and below comparable to 1 over the distance where And now if I combined With some other stuff I get a contradiction and the basic motivation behind this Perelman would say well Since Opening like New blow it can get own in the neck contradicts But I wanted do the contradiction instead of from the heart so here soon The curvature Falling off the bounded above and below by 1 over distance Where'd you actually get you bound red or . by constant art of the 3 and since you the DRG the Harnack gives you this Very nice surviving equality which I use a lot of Does You also have the volume condition so Under these assumptions that's what you need to get this power and the way that you do it out the kind of show you how you can use these The ears There is off Very nice classical equality that As you couldn't control and a role Of reaching along and memorable Jia does with us The appropriate weighting of the distances by wall of the Jacoby & determinant of the exponential map from PD cheer and you can actually make this Jacoby and have a lower bound in this case because Of all the control you have over the curvature in the bowling than I take this little in equality ass I plug it into it This Did you integrate or overall are along the a few wall The curvature 1 and over the curvature of of the other than I play Koshy Schwartz to get out this thing you can control geometrically in your left with the other turbot isn't too If you're free and you just have to to choose the points x and Y ride along the chief because of Iowa if I went all the way to the and this would give an infinite saying so The show you the picture I'm really wishing I had 2 projectors right now but here's the point where the curvature big And there's a point where it's follow an awful lot and here you've got him somewhere around so we have plenty of volume so I can make this Jacoby and have a nice lower In I'm going Take X close the P and why close to but far enough away that I can't control everything and then Give Peckham alright plug it all and I don't think I have time to really do it many detail Prices college that Instead Not really drawing pictures it's a lot of calculated so Perelman then the years as this volume on collapsing Following way The 1st thing he wants to show is that If you know you have The volume of ball in an agent solution with Don Negative curvature than that actually controls the curvature out To a much greater distance from the radius of the ball and the proof is that You want Fear The point Cleverly so that you can do a blow up around here And detonation solution where Positive passenger volume ratio which we just saw can have so here's the picture you look at some radio like you could scale so are wanted NW would be the radius and You've got some volume you want about the curvature at this distance so I that this point where it's about its biggest it's going to be in such a way I can do a lot of women around it and when you blow up around here don't compare to the scale of this Will ball here goes off to infinity but because of its size getting bigger as you feel you have expanded the curvature control here and then this is getting bigger and further out but you see that the aware that you're going to get It's gonna have this positive asymptotic volume ratio and 1 thing I have learned from studying newspapers how much You can get in geometry out of least Boy especially in the presence of non negative curvature as well The next part of the argument of approving compactness is there to say
But you can actually found the curvature out to any finite In terms of the curvature of . Casey The Take a couple of steps that you use the previous argument to say Where you start by taking Other You have heck of a ball here you know the curvature hearing you want abound that furniture far away So you start by taking the ball of the size where they You take beggars radius are the maximum of the curvature over a ball of radius little Lawrence 1 over Where No I want show that that makes the curvature at the origin at least a fraction of that Well the idea was that This Maximum obtained somewhere here and there Brown Because We have these derivative estimates were will be because we have volume in here then We get Now the curvature To a bigger radius AM Harnack gives us a bounce back in time because foreign doesn't doesn't Going forward so it doesn't grow going back And she's local derivative estimates of controls on derivatives in Here so you can see the curvature is the good side please it'll be at least a fraction of that a fight back up in time and then we all know that We are making quality sites to be integrated over overpass Produced actual making qualities such as the curvature back accurately origin is at least a fraction of the curvature it too is at least a fraction of the curvature it in you get your ass and now the rest is easy because once you know that this fall here of the right size actually is controlled by the curvature of the origin of the previous estimate says you been bound the curvature at any finite distance from the ball which is now bounded by the curvature of the origin and that gives us 1st of 3 steps didn't they Proof But fact of the ancient solution was 2nd with crazy volume comparison that bounce the the curvature out infinity in the 3rd step in this case is easy because Harnack controls back now and this blow argument I've talked about limits so let me say a little bit about limits bountiful There's a definition of a sequence of manifolds converging to a limited and the topology of infinity with contracts for them That you couldn't find a sequence you you have sex origin the take a limited manifold and then you find a sequence of the amorphous so it's convergence of the were when you pull back The metrics are converging to your LONDON that trade in infinity and complex for which means it what's as close as you like it many derivatives as you like its biggest ever why this is a very nice notion of convergence for Partial differential equations because you have all the smoothness and 1 of the problems With Taking a weaker sort of convergence like that But In Alexander also They said is that you don't have control of derivatives of curvature So you have to kind of take only If that's the A lot of attention what's going on locally in your curvature and this is all but gone over very well But So a cruise liner in John Locke but if you stay in the smooth category Their advantage there is an example of limits of full of the sort of book procedure we won't be able to do where you're forming enacted here and there Lecturing you really exist Man detained at a constant size than the rest of the manifold gets bigger and bigger and eventually goes off to advantage and you your compacting spin converged Among contacting but in this Could be a nice round cylinder You also noticed the carrot these basic points in a fight that Points over here and get something that look like this But only went out in 1 direction so there is a very nice existence theorem which triggered told me he didn't pieces in the city and affinity with complex support case and that You get kind of compactness result that if you have On the curvature and all the derivatives 2 of any distance you want going to infinity and if you don't stop the thing from collapse than the limit actually exists and so the so the strong existence there and then You can't generalize it to reach flows and I felt that actually in the same conditions that you get a limit of 3 chief and advantage in the race for cases That's just having balance on the curvature in a parabolic cylinder by the derivative estimates you bounce of all the riveted as well So the sea infinity convergence of of natural and so here you see that Perelman's my collapsing estimate Plays a very important role in being able to take limits because it gives you the other thing you need an addition to controlling the curvature you have control the collapse as well But But It turns out that another important use of non collapsing fears On services and ever surfers It is not collapse and has no negative curvature And satisfies a derivative of the bidders isn't flat than it actually follows that compact and that's pretty easy to see Because if it were compact it would have to stretch out and wait And that there be A region that looked like that As 1 cross or 1 word would be flat So In fact from that you to stronger in Ucon so if you have a non collapsed surface that You can actually bound the diameter in terms of the maximum curvature and you can bet on the ratio of the maximum curvature to the I So Blue Now didn't know the argument summing up were gonna wanna be able to Bound curvature on 3 manifolds Out To infinity and there's a very nice result for low-volume comparisons that lets you do that and I think I'm sure owing get And I'm gonna go on to something else But I'll just mentioned that There's a very nice volume comparison you could use on annual alarm People I think they're very familiar with volume comparison long balls that you can get a nice resort on the radio 2 concentric annular also His Now Perelman have actually 3 lovely idea aroused and the 1st There's non collapsing result the 2nd as the compactness of ancient solutions and the 3rd as the conical neighborhood Which basically says that if you have a solution to of the Ricci flow that's nice to start an The curvature gets bigger than everywhere the curvature big enough you actually get it Looking Very not in the way phrases it is that it looks like a piece of an ancient solution Now
I went through all the You used improving canonical neighborhoods in using it And it seems to me that there for essential properties of the canonical neighborhood that are the ones that you need 2 years and the 1st is that you getting down on the 1st derivative in space and you have a lower bound on the 1st derivative and time None of us wants you have If you now the curvature of the You can control it out some distance in space sometimes backwards and then you get control of all the hire derivatives By the derivative at the case But the use of the Basic Law and then there's The non collapsed That is a nice property of Koenig neighborhoods and non collapsing is very important and finally there's this They have a relatively controlled aperture that Because Find diameter of the spirit radius Lambda is really last there if you take a point where the curvature 1 over our square in take the sphere of radius slammed the heart Millions Find diameter that is wonder to pieces It is no more than lamb da Matta keeps things from going off at all direction it basically confined at narrow apertures wondered to direction so But process Which It is quite clever and elaborate ears There is You really want a say Suppose it doesn't happen and then you take a sequence of counter examples and then you try to take a limit of remember there were those conditions and each counterexample ears But 1st time when 1 of them fail so up until a certain time they all hold And then 1 day down that was a certain curvature level so then you up the curvature level and try again and look at the 1st time it fails in if no matter how high you push up curvature level you still fail to get this nice that figure above it then You get a sequence of counter examples and you try to take a look and there's 3 steps the doing the limit the 1st is to say you been bound to a finite distance than that you can't bound to an that of the and then that you can't bound all the way back time So In order to donors Heroin users arguments on limiting am I found that you can replace the With the sort of inert role of off Harnack estimate that I was showing for you just have to take care of some turn so the 1st step 2 Backed by a Harnack estimates And work a bit harder in there that Member I told you there is that extra quite erratic termed the muffins and that's exactly what you need to introduce a barrier and localized So you can then show but if you just have a solution In opens that opened parabolic cylinder and if you have a ban on the curvature about And below were you thinking the bound below is very small compared to the boundary of which is what you get from pinching then you can balance these where by well constant case carried Ricci and that And why no 1 will face These so here you see that the negative curvature gives you a little where're here but if there's only little negative curvature it's a little error and up This is going to be the start of thing that you can use and then Health Before the idea With derivative bands you can replace these maxima and minima in the hole Most cylinder why their actual values of the Center for is so Then what happens is that You try to repeat the previous argument that I have showed you in the B positive case only now we don't have not negative curvature we get a little bit of negative curvature Typically happens if you're forming and there was now the idea that in terms of the curvature and the dust and fee from June we want control the curvature it And the way Perelman does it is to say you take the limit of counter examples and as the curvature blows up the building of 0 from pinching and this limit is and use the splitting together contradiction but if you work a bit harder and use the past The site from the local Harnack then you actually show that you can control the curvature From the curvature accused by sort of itself With the fact that the further away from Judy wanted go The more you have had shrink down 4 0 in order to be able to control the year And so here you see very clearly how everything in place That error in the curvature and what happens in the back The idea of this You Repeat they ask that you have the floor Where trying to integrate reaching 80 times the cutoff function over the Aegean does it and you get in Like Dell The time are and in order to handle this I 1st played the point he very nice according the maximizing Bader and I think the point acts along that Offer to maximize gamma and then I have these 2 Quantities beta gamma and it turns out that because of that aperture Condition you can't compare Don't Don't damage Satisfies a barrier estimates there down the beta is continuous and about what that means That The gamma satisfies barrier estimate which is like this Then Lord gamma is bounded by a constant depending on this proportional distance study you wanna go kind 1 plus Delta Gamma now remember I can make the smaller sigh light so no matter what this constant is here depending on the distance W I wanna go if I make those small enough I get a barrier And so Bader satisfies the barrier also wants to show the beta and gamma comparable for which you need this aperture condition and Them Bader Here's depending on your choice of the proportional this stub you but it depends continuously you clearly control Bader when W. small So there W goes to infinity you can keep control but you have pushed the double 0 a study goes to infinity that A very nice controls about For that Control Infinity is again us The bowling comparisons but to control the curvature back in time of use A lower bound on argued teeth and so what you do is you assume that there's a nice result that simply says assuming that you have this sort of derivative control that You can show that Solution if you have solution defined pretty bigger than 0 you actually the derivative asked give you control or even back 0 which then let you continue back that just can be done my sleeve y Bounding Tee times Maximum curvature And you get down bottom below from the derivative estimates And that lets you take out a sequence which converges Type 3 blow up the terminology that's a sequence where Find the maximum curvature retains its Max and I had showed already in Karnak that if you get any quality the strong maximum principal makes it a steady solid evidence of how I don't generalized that
A Ricci Cabler flow and Jan long in sushi Showed that If she'd times are Max attains its maximum than it's actually expanding And now you get a contradiction because the expanding solids Did bigger than 0 But all the derivative estimates like we did before may be But now Let me say a few words about the surgery That I find and epsilon neck the B 1 where the metric in space climb after dilation and perfume worthless and is Absolon closing caterer derivatives of the round track on a section of link Ellen back in finals where and I discussed the sort of surgery In a paper I wrote on reaching plant for manifolds with positive isotropic curvature and Perelman is glad to see quoted in years The number of those results in his surgery arguments Apple and the surgery procedure that you can use it as you said 3 different levels for the surgery there's less It out Which is when you reach a cat who have to the do surgery and Boris the level you wanna cut it down to And can get a level below which you will have Except Found there what you show it By taking these large and inappropriate proportions you actually get only a finite number of surgery Now In his papers After bridge Actually elects the curvature go to infinity and form for and what he says he was interested in doing is proving the existence of a weak solution Continuing through the singularity pennies Given us a lot of ideas for doing that but Just do the surgery part You actually don't need to do anything so elaborate you can't simply England Do the surgery According to a fixed procedure like this Without the curvature ever becoming infant you consider the case where it becomes infinite Together with counter examples but the actual surgery Things find them But Then you have to to check out on a number of things such as that Pinching estimates survive surgery and a number of other estimates survive surgery and a Trickiest part verifying that the the non collapsing survive surgery here really like the bank My very good friend South I on for explaining a lot of this to me That You don't have to prove the non collapsing us And that down to the smallest scales it suffices to get it down to a scale arbitrarily small compared you're initial scale and you can then fill in the rest of the non collapsing where you repeat the canonical neighborhood argument for us Surgery solutions of the Ricci flow the argument for a finite surgeries is kind of interesting that you imagine that no matter how big you choose Please Things You still reach a point where you can't do the surgery successful you can remove all excess Curvature to cut it down to the level you want and you say so that that can't happen no matter how big I push these parameters so you try over and over again in you keep Failing and then you get a counter examples and you look at the the limit of the range and then you try to Analyze it So what you 1st do is take the limit as it blows up and then you take the lowdown limit going far away from that and that down Ltd So what Sosa product Because you can blow down a long a minimal geodesic going to infinity and use the splitting fear and then you try to extend back in time from that and as you go back in time what you find is that There's not You look at the maximum time you In its then this slowdown of the blow up to them The curvature can't go back To infinity by that little curvature Found I showed you the follows the derivative estimates you of and it can go back into a surgery either because that would violate led to a positive curvature here where the 0 here so That have so actually you Can rule out that and the only remaining possibility is that best While Down the blow up as a product that goes all the way back in time and then the only remaining pieces to show that this cross around you actually don't need the roundness up until this West Point where you wanna do the surgery Before that having abounded of liquidity is quite enough and see that it gets round European use actually Gates 2 results Perelman use The honest services to different entropy involving the curvature You can also use the ice Parametric Horatio where you look at all Curves of Lake All dividing it into areas 81 and 82 you take the imam overall cursive squared times 1 over a 1 plus 1 overreaching then it turns out that the size of parametric ratio increases under the Ricci flow and it's strictly increasing unless you're wrong around here so now if you have an ancient solutions to the Richie flow into dimensions which is uniformly Kaplan on collapsed that's been around forever But backwards when that would have this ratio constant and Vendors You see it's actually the spirit so that gets I read love with Finite surgeries and finite time and all you need to say that the case for it simply connected that All you need this is my last slide by the way So now there's a nice their by calling a minute Coxley and another version of it by Perelman but the polling that I can't see theory seems to involve less technical difficulties The wall surface theory you look at a map of degree 1 and 2 of the 3 manifold which the biologist Tellus exist And looked at the Ecuador eels mirrors and their corresponding area and now you do and Min Max And it turns out There's a nice met in the Ricci flow which is actually the same 1 I use The He goes to infinity case show the incompressibility hyperbolic pieces were you show this Decreases at a fixed rate and I can't go on forever so now there's regularity of the Minimax itself But minute Foxy told me that It was based on Resolved But out finally said Yes I looked at it Yeah I used it so I think I'm pretty happy about this This way you actually get approved The . 3 conjectures so I I think I'm about Rice this anyone see this all working at an enormously grateful it Christopher for finishing it off and I'm row fat OK thanks a lot bye bye-bye
Konstante
Punkt
Linienelement
Mathematik
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Skalarfeld
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Metadaten

Formale Metadaten

Titel The Poincaré Conjecture
Serientitel International Congress of Mathematicians, Madrid 2006
Anzahl der Teile 33
Autor Hamilton, Richard
Lizenz CC-Namensnennung 3.0 Deutschland:
Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen.
DOI 10.5446/15954
Herausgeber Instituto de Ciencias Matemáticas (ICMAT)
Erscheinungsjahr 2006
Sprache Englisch

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