 Okay, so let's get started with the last lecture, okay last lecture and we stopped ten minutes before right? Yeah, no no no Okay, yeah, some might say unfortunately Okay, so Yes, it is usual to begin your talks with saying who your collaborators were and I just realized that I've skipped that four times in a row so everything that I've you know anything everything that I've mentioned was Either someone else's work or it was joint work with Toti das Kalopoulos who was here last week Natasha Sessum, Tam Ilmenen, Juan Jovelasquez and David Cha Okay, so today I want to so this hour I'm going to talk about ancient solutions and It's The word makes me feel a bit old because there are all these things that people are calling ancient solutions that didn't exist when I was a Graduate student, so it's how old am I? So what is an ancient solution? ancient solution to mean curvature flow is It's a solution that is defined all T less than or equal to zero or More generally less than or equal to some time But the important thing is that the solution is exists for all negative times So what are examples of ancient solutions? If you allow yourself to think of non-compact solutions Any minimal surface is an ancient solution because it's stationary The sphere so SN minus so S2 sitting in R3 With radius square root 2 square root minus 2 or the cylinder With slightly smaller radius. These are all self-similar solutions that you may have seen in the first week They're solitons. They're self-similar solutions self-shrinkers and They they vanish so Usually one looks at these in forward times and they come out They show up as blow-up models for singularities of mean curvature flow Of course if you have if you have one of these self-shrinkers that is moving spy Just by similarity then you can follow it backward in time as far as you like So this thing is defined for all negative times T Any other self-shrinker that you have and there are many Also counts as an ancient solution. Why do we study ancient solutions? So one reason is One partial reason is that they show up as blow-up models for singularities of mean curvature flow That does not justify the ones that I'm interested in I want to talk about all convex ancient solutions Other reasons for studying ancient solutions could be and I'm making this up a little bit, but so there's one paper called Curve lengthening which by Three Japanese author One of them is called Wadati and they study curve lengthening which is curve shortening But with with a minus in front of the curvature in other words, it's a curve shortening going backwards in time Which is an ill post problem of course? But these things so you could look at velocity equals minus mean curvature That's an ill post problem because it's mean curvature going backward in time Right so if mean curvature make things automatically smooth So if you start a real analytic so if you start with a hyper surface that is only see infinity But not real analytic then there is no solution going backward in time So it's it's very ill posed but these things could come up as Approximations to higher order equations you could say you could And now here to keep it well posed. I believe you need a minus sign But as I don't want to talk about that so minus epsilon times this so there is a well posed equation Where for smooth solutions with nicely bounded curvature? This term is actually Bounded and the solutions that you get are as long as the curvatures are bounded are close to backward solutions to mean curvature flow So it might be worth Considering these So and Then convex mean curvature flow ancients the study of ancient solutions As I want to show you in a couple of minutes, I think it's just a very natural mathematical question Okay, so let me begin by what is known and this may repeat Things that you've seen last week. I don't know for sure So convex solutions convex ancient solutions So there is one so of course the circle so line That's the minimal surface the lines. It's still Circle its radius is square root minus 2t. Notice this minus 2t that radius is the same as the Radius of this cylinder because this cylinder is just that circle across a line Then there is What Matt Grayson and Steve all children have called the grim reaper it is a solution it has this form and Actually, there's an explicit expression for it. If you make it a graph like this it is y is minus log Cosine X what it does is it moves with constant velocity in this direction just by translation So what I wrote before you can think of backward Curve shortening as some approximation to some other well post problem This thing has another name in other circles Applied mathematicians know this thing as the Saffron Taylor finger and they have a There's some model for fluid flow where you have an unstable fluid and then at some point you perturb a little a boundary between Two flues you perturb it a little bit and then the perturbation is unstable and grows and the the simplest description that they have Is this thing but going in the other direction and it's it's it comes about as an ancient solution of mean curvature flow Okay, then there is the paper clip this thing which has an explicit expression and It's kind of a strange expression. So let me if you Take this so if as coordinate on your curve you take the tangent angle and if you specify the curvature k theta So if you know the curvature at each point as a function of the tangent angle you can reconstruct the curve So that's an exercise in differential geometry What was the formula so there's a formula that looks like this cosine theta minus Something like this. So I would put this in quotation marks, right? There's a formula. It looks like this And its derivation is so you can just substitute it in the equation the equation is kt equals K squared k theta theta plus k It's better if you write an equation for k squared And if you do that you substitute this and you find very quickly that this satisfies the equation there are I think I wrote this thing up the first time. I found it in a coffee shop somewhere there are Before that there is work by Galakhtionov who studied solutions Not thinking about mean curvature flow at all or any kind of curvature just thinking about quasi linear parabolic equations looked at solutions of This kind of PD Found a whole bunch of explicit solutions and this thing is actually a special case of that of the family that he found okay, so If you take that one and you interpret you take this formula and you interpret it what it looks like as As a curve it looks like this and what is it so what is what is the okay? So this is one times snapshot. How is it moving? Well in backward time This is moving in that direction and that is moving in this direction So in forward time you have these two two ends that are coming together And then in the end it becomes a circle What is the asymptotic shape of these? They are grim reapers or saffron tail or fingers depending on which direction of time you go in Then there is a theorem Tuscalopolis and sesame saying that if If CT their curves is a and now there are three conditions compact convex Three conditions compact now ancient. Oh embedded you might consider curves that have self intersections and Then you can find more Ancient solutions. So if you have a compact convex an embedded ancient solution Then CT is either It's one of these examples. So it's either a circle. These two are not compact. So they don't count It's either a circle or the paperclip. Okay, so then There are more examples of ancient solutions In the plane that show that each of these three conditions is necessary This so in a sense This is the best possible theorem of this type that can be proved So further examples and these are due to can you can you who in our thesis prove the existence of a whole bunch of other Ancient solutions so and instead of writing so I'll just draw I'll drop pictures of them. So One the simplest one was already found in the paper by wadati in 1994 I think They found it they also found the solution independently and they had the good idea to say well This is true for all values of C. I can translate this thing in time in particular. I can translate this thing by Pi over pi over two times I in time right in complex time direction. It's a formula You could it's algebra. You can just do it this hyperbolic cotangent then becomes a hyperbolic tangent Various science change here and you get a formula for another solution and this is what it looks like It's the ancient sign. So it's not a compact solution. Okay, so what are these? What does this look like? It's a periodic array of grim reapers One up one down one up one down and so under curve shortening. This is moving in this is moving up This one's moving up this one is moving down They all have the same width they're all moving at the same velocity and as time goes to plus infinity So this is an ancient solution you can follow it back in time But in forward time it you could also follow it forever and ever and what it does is it after a while It becomes flat and it'll It converges to the x-axis and by linearization you can actually show Which is not hard is that? This profile actually becomes sine a multiple of the sign it becomes almost the list So it becomes asymptotic to a solution to the linear heat equation So it's just the first foyer coefficient foyer term that survives. So that justifies doubly justifies the name ancient sine curve So what you found is that this is an x there's an explicit formula for this thing That's perhaps an accident there happens. So it turns out that there's no read no need to To take these distances Take these things equally spaced you can put them anywhere you like so you can take take any sequence of vertical lines With arbitrary distances between them then there is a solution to ancient solution to curve shortening is consists of grim reapers asymptotic to these lines So you have the you you have the freedom to put these vertical and so you can make an infinite array of these The only condition that you need is that the spacing the distance between the lines has to be uniformly bounded and uniformly Bounded away from zero, but then apart from that arbitrary So that gives you one sequence of choices and then the other choices is so these tips move pretty much with constant velocity Until they come here somewhere, but the far away By you know it used to be long ago that these things were down there And then they're almost grim reapers and they're moving with constant velocity. So that means that the height of this thing is The velocity of that grim reaper if you rescale it Grim reaper if you make it narrow its curvature becomes larger and it moves faster So the velocity of a grim reaper is Proportional to the distance between its asymptotes So these these grim reapers have all have different velocities depending on the width of the strips in which they live So this is velocity times time and then there is a a constant C So you can prescribe these constants in each in each strip you can also adjust this constant So there's a new the set of ancient solutions of this type is really flexible Right yeah, I understand the question I would guess so but she didn't look into that and I Haven't Haven't given it thought so but yeah, it sounds like that should be possible, right? So the limiting thing would be some curve like this if you right these C's if you make these large enough and Different enough than in the long run So yeah, there's there should that there should be a construction like that that works But I certainly have not worked out the detail and she she also didn't yeah, okay? So other examples that she constructed are Okay, so what is this show? These are ancient embedded solutions, but they are not compact and they're not convex so Okay, so she also constructed one of these so called I call these things trombones Okay, so you draw a couple of horizontal lines you number them and then you just go back and forth connecting them by by grim reapers Asymptotic to those horizontal lines and so each of these grim reapers so in backward time they move with constant velocity So this thing will move this point has coordinate velocity times time plus some offset and So what she showed is that you can pick an arbitrary finite number of lines like this you can prescribe these offsets And the one I drew is compact, but it doesn't have to be compact you can you can also make non Yeah, this one is not sorry. This one is compact. They don't have to be this one is convex In the sense it has no inflection points. It's not embedded, but it has no inflection points. This one is Not convex it has an inflection point here somewhere and so there's another one here somewhere Okay, so you can make ancient solutions of this type What this shows is that so this thing is it is compact It is convex it's not embedded Okay, and then there should be another one which is the ancient spiral and so so far there is no proof of those But I think so she was about to construct these things, but then she got a job in finance So there should be a solution of this type So there exists a self-similar solution that does not translate it rotates and so Matt Grayson and Steve Altjuler called this the Yin-Yang curve So it's an it's an an infinite spiral this ends. Well, okay So there's this thing and it spirals on forever. So what you could do is you could take these two ends and connect them by Grim Reaper Okay, so going backwards in time what this thing going backwards in time what this thing does is first of all It rotates like that Well backwards in time it will rotate like this right this forward in the time this moves in this direction this moves in that direction and then This will move inwards. So if you go backwards in time this thing will just Go out and fill up the whole space. So there should be a solution that looks like this This thing is compact embedded. It's just not convex because it has exactly it has one inflection point here and Another one over here. Okay, so that's for curve shortening I don't think so. In fact, I would Conjecture is a big word, but I would I would I would guess that The ones that she found are the only ones with finite total curvature so that they so ancient solutions with finite total curvature have to have All the possible limits that you can get out of those will be grim reapers And I would say that all of them have to have parallel asymptotes and they should be they should be of this type Should be in this list Okay, so are three and higher. So first there is who's const theorem cons convex ancient solutions so who's const theorem was The first theorem that I read on mean curvature flow It says that if empty if M zero is compact convex then empty Shrinks to a round point It is some point plus square root minus t times a sphere of an appropriate ratio radius So take any convex surface it'll shrink it'll shrink to a point And if you look really near that point and then magnified so that the thing becomes has a reasonable size It converges to a sphere. So this is and the year is 1986 I may be off by one or two years Okay, so this is a theorem that is it's the best possible theorem that you can prove here So actually he proved it in dimensions three and higher so for two-dimensional surface and then For some reason His proof doesn't work for curves in the plane because he used Okay, so let me let me not let me not explain why so Gage and Hamilton proved the analogous statement in the plane Okay, so It's the best possible statement. It basically says that no matter what convex thing you start with it always goes to a point So it seems like so what I wanted to talk about is the dynamics of Convicts ancient solutions so convicts the set of convicts services is it's a nice set We have a flow on it curve short mean curvature flow. What is the dynamics of that flow? this theorem shows that the dynamics is really boring because everything just goes to a point to Right the fate of every solution is the same. There's nothing interesting What I want to explain so I wrote with Toti and Natasha, we wrote a paper that is kind of long and Has a lot of steps and it's about one particular kind of ancient solution the ancient holes of certain type So instead of doing that in a great detail or any detail. I wanted to show where it fits in the In the larger scheme of all possible convex ancient solutions. So if you have So before talking about generalities, which I'll do in a second again. Let me go back. Let's look at a few more examples So one question you can ask is there an analog? Is there an analog of the paper clip in for say for two-dimensional services in our three at the paper clip is this The thing that I just drew these two grim reapers that that run towards each other and then become a circle And so the answer is yes, it turns out the end there are various answers depending So the answer is yes in various different ways depending on what you mean what kind of analog you're looking for so one analog is So this is yes number one So this is a construction by Brian white Which then later was done in more detail by? Robert Hasselhofer or Hershkovits and this is the solution that then that Daskalopoulos says him and I Looked at So he came up with the following construction So there is no explicit formula for a solution to be curvature flow. So we're out of luck there He said let's do the following thing I'll take I know what a cylinder does a cylinder just shrinks to a point So here's my cylinder it shrinks to I said point I meant line a cylinder shrinks to a line It's it's rotation axis in finite time I'm going to cap this thing off So what he said is I'll take an infinite cylinder and I'll just take a finite segment of that thing So here's the finite segment so the my initial surface has radius R And it has length L Okay, and now this thing is going to shrink to And I cap it off. Let's say just with flat discs, right? So it's It's a log Under mean curvature flow. What does this thing do? It's a convex surface. Who's concessing becomes a sphere, okay? But what happens what happens in the meantime before it becomes a sphere because so For a really long time if you're just looking at this central piece of the surface If this L is really large if you're just looking at the central piece of the surface, it looks like a cylinder, right? So if you and It's a heat equation. So it has infinite speed of propagation But for the first amount of time the the effect of perturbations far away is always exponentially small So you would think that so there's there's some sort of localization This thing is it's like a cylinder So for a while it'll just do the same as if it's a cylinder and it'll take a while for this to come in and spoil the cylindrical nature of the thing and make it more round, okay, so Okay, so by taking limits, so assume the surface. So This is M zero Which becomes which shrinks to so which gives you a solution empty and this thing depends on L and R And this thing shrinks to a point. So let's say that M MT LR shrinks to a point as T converges to capital T So it shrinks to a round point. Let's say the origin And this the time it takes depends on L and R The larger you take the initial surface the longer it will take for it to shrink to a point So what Brian White did is he said I'm going to take a limit of these things But first I'm going to translate them in time. So I'll look at M tilde LR T is MLR I'll just subtract this capital T LR So I'll translate it in time so that the the point the time at which it shrinks to to a point is time zero Okay, so now I have a solution to mean curvature flow that doesn't exist for positive time But it exists for not all negative time, but as I make L and R larger for longer and longer time periods Going back in history Okay, so then take limit that before So you let L and R go to infinity and in the limit you get a solution you get an ancient solution and These are all convex things so in particular their mean convex things. So At the time that Brian White was doing this There was not that much theory that you could apply by now There is a lot of theory and this conclusion that you get a limit in this case is something that you can you can Look in your notes from last week and just Draw the conclusion So this is an ancient solution and now how do we know that this is not a sphere? Okay, a priori it could be a sphere if you let L and R always be if you choose L and R always to be equal Then what you're just doing is you're taking a finite, you know short log and you're just making it larger And this thing will it'll become a sphere So you have to choose L and R so what you have to do you always have to choose L L over R you have to let when you're choosing L and R you have to let R go to infinity But L should be larger much larger Okay, and so if you so for a suitable choice of L and R going both going to infinity L much larger than R You get an ancient solution that is not a sphere Okay, so and then he had a more detailed description Hasselhofer and Herschkiewitz later did the same construction and they gave more details of what the solution looks like In particular they Their construction so I drew it here for something in R3 The construction that that is written in by both of them is in RP Cross RQ with the same kind of symmetry that I've been talking about these last couple of days They produce surfaces with those Symmetry that are not spherical under their ancient solutions convicts ancient solutions of that time So now you can ask a more detailed question What is the asymptotic shape of these surfaces as T goes to minus infinity this particular ancient solution and so the answer so there's a formal asymptotic answer which which is There's formal asymptotics. This was so if you see formal asymptotics for the neck pinching for a whole bunch of other things it's sort of like It's not a very short calculation But it's like like our calculus students do calculus problems. You've seen a hundred problems and here's another one That's like it. So we do the calculation and you find an answer so the answer is It's an ellipsoid An actual ellipsoid where this radius is square root It's the cylinder so in the center for going back in time If the thing looks like a cylinder what I said here if you look at the center part It always looks like a cylinder. So the radius in the center is minus 2t and So I was taking to the R3 case and the long axis so it's rotationally symmetric in this direction And the long axis is this times log T So this follows from a formal match the asymptotic expansion with meaning there is no proof that Such a solution exists or if a solution exists that it has to obey these things. There's it's just a calculation right so no proofs so what with What we did is What we first proved is every solution has these asymptotic expansions now. I say ancient oval and So symmetric so in this work we assumed rotational symmetry every Symmetric ancient oval satisfies The asymptotics I say it's an ellipsoid. It's there's a little complication if you look at the tip of this ellipsoid if you If you calculate the curvature of the ellipsoid at this point it turns out that the mean curvature has is the Should coincide with the velocity of the lip so it's off by a factor So there's there's an extra complication and you have to magnify this thing and what it looks like there is so the ellipsoid comes in like this This part you have to cut out and you have to replace it by what is called the bowl soliton So it's the analog of the grim reaper It's a rotationally symmetric surface that translate with constant velocity and it looks like a paraboloid But it is not exactly a paraboloid. So there's So it is it is an ellipsoid except that the two tips where it is a more complicated thing okay, so Every symmetric ancient oval satisfies these asymptotics and then theorem 2 Is a uniqueness theorem namely if you have two solutions that have the same asymptotics, right? So this theorem says that the solution that Brian White constructed and the solution that Hasselhofer and Hershkovitz Constructed they have the same asymptotics. They could still be there because they use different procedures different limiting procedures They could have been different solutions. So this theorem says that no, they are the same solutions There is only one such solution one such ancient solution that going back in time becomes really long like a cylinder and forward time Has to go to a sphere So up to translation in time and this theorem we also proved assuming symmetry once we were almost done with this see one brendler and joy Came up with a paper and they proved they proved That all translating ancient Non-compact ancient solutions are rotationally symmetric and the techniques that they used could also be used here And so he told us that that might work here. So we figured out that You can do a first step here if you have if you have a soliton That is like an ancient oval that in backwards time converges to and a cylinder Becomes elongated and like a cylinder then it actually has to be rotationally symmetric after which the previous stuff that we have here applies okay, so So the proof of this is kind of long and has many many steps And I want to so I'll say something about it I want to draw the bigger picture that I started about here The first thing to do is to look at So suppose that empty is an ancient solution. I want to talk about the dynamics of ancient of convex Solutions to mean curvature flow not necessarily compact convex solutions. So if empty is an ancient solution then Then in many circumstances you can prove that it converges to So what I'm going to say now none of these are theorems. These are so these are ideas and these are theorems that that are sometimes true and sometimes Still need proof and sometimes they're not So Any ancient solutions going back in time are asymptotic to self-shrinkers for example The solutions that I was talking there in backward time if you only look at a compact space interval So you shrink it down by a factor squared minus t What you get what you get to see is the self-similar shrinking cylinder So this tells me that instead of looking at if I'm going to look at ancient solutions. I shouldn't look at At curvature mean curvature flow I should look at shrinking mean curvature flow Where I do the substitution. I said m t is squared minus t times n So I let this n vary and I call this That's my new time variable tau and then n tau It evolves by Velocity equals mean curvature That's that's the highest order part plus one half x X the position vector so anywhere on the surface of n and new is unit normal there. So this is the This is our shrinking mean curvature flow So what I really want to look at is the dynamics for solutions to this equation Almost really So fixed points for this For this dynamical system are things with velocity zero So they are surfaces on which h plus one half x dot new is zero these are self-shrinkers and I'm only going to look at convex surfaces and in one of his first papers on mean curvature flow who's can prove that there's There's a finite list of these and they're very simple They are the sphere the cylinder and higher cylinder. So in our end Sorry our n minus k times s Okay, and the radius of sk is square 2k. Okay, so these are generalized cylinders, right? So there's a whole sequence of these so for each 4k Okay, so let me so what do these look like? There is sn There is r cross sn minus one In the theorem that we these terms that we prove we didn't use the symmetry that so the symmetry that we used is It's the ones that these have so this one has our s o 1 o 1 cross O n symmetry, right? Then there is our two cross sn minus two and so on The last one would be our n minus one cross s one, but I also like to include this one our n cross s zero In this so these with the appropriate radii these are these are all the same self-shrinkers. So what is s zero? Sn minus one is the boundary of bn so s zero is the boundary of b1 It's the boundary of the interval minus one plus one So it's the set with two points So the the zero-dimensional circle consists of two points And the radius that we have to give it is square root 2k So it's two points, but they are at distance zero. So it's a double point So the way to think so if I have to draw these self-similar solutions Then I can't because they are not three-dimensional, but in higher dimensions, but we can make cartoons so we have This this is sn the next one is So the next ones all look like this where this axis is rk and This sphere is sn minus k or our n minus k s k Square root 2k and the last one is Rn Times a double point. So I think of that as a it's a double plane Okay, so all these are self-similar solutions To shrinking mean curvature flow. So these are fixed points for the flow The flow is nice. It has a It's a gradient flow and it has a Lyapunov functional. There's a quantity that is always decreasing Along the flow what quantity is that we do He leaks out all while I erase that which thing would be which quantity is decreasing along This renormalize the curvature flow begins with H It's not Hamilton named after someone you saw last week. Yeah, okay, who's good? It's the who's can monotonicity formula. Yeah So there are a couple of ways of writing who's can's monotonicity formula one way is To say that it is that the following quantity h of a surface and is So there's a pre-factor 4 pi over n over 2 integral e to the minus x squared over 4 D mu over n of tau. So who's can's functional is this along a solution to Right so the the if you write the who's can monotonicity formula for mean curvature flow It's a more complicated formula because you have to divide by time here But because in this situation we have rescaled by time The the expression simplifies a bit this thing is monotone along solutions for mean curvature flow That's that's completely equivalent with who's can's monotonicity Formula right so we have D It's less than or equal to zero and it is only equal to zero on on shrinkers okay, so this quantity is decreasing on any convex ancient solution and It will be constant on these things and I have written these things in this order because So if you write the Yeah, so let me know any qualities are exactly the wrong way around sorry so this means that for The shrinking the curvature flow you there cannot be a solution that starts here and ends up like that Right they have to you can start here and you can end up over there And who's can's theorem says that if you're compact and convex you always end up here okay Could you have a solution that has this as a limit? And so the answer is yes But you can not if you start as a compact surface because if you start as a compact surface Then who's can's theorem says you always end up being a sphere so There are non compact initial data that end up here for example It's a fixed point you could start being this and then you would never change But you could also by Grayson's theorem you could take any any curve in the plane and then Do a cylinder you know construct a cylinder on this on this curve How does this evolve a mean curvature flow? Well the extras Cross are that I have done is irrelevant. You just evolve this curve by curve shortening it because it becomes convex It becomes a circle and then in the end shrinks that circle shrinks to a point by gauge Hamilton Now you remember that we have the cylinder over this thing and so you see that in the limit It converges to this to the cylinder it provides a lower bound so So there is such an upper bound and so it's one of the one of the steps that we had to do in our proof So it's fairly easy because if But it's not it doesn't come like this So if you have a convex set you just cover it with certain with a finite number of caps Each of those is a graph and if it's a graph you write out the the who's can functional for a graph and you find that it's You find that this you can bound the contribution of each each cap by this integrated over our end Yes, it gives you an upper bound Not the best upper bound so I think the best upper bound is this guy Okay, so let me show you more solutions So the solution that I've shown you the the ancient oval that I've drawn before you could think of that as a connecting orbit between this fixed point and that fixed point because it starts out being like a cylinder and then it It comes in from infinity and then it becomes rounder and rounder and converges to being to being a sphere one or two years ago, there's a three people Burney Langford and Tina Aglia They constructed so I remember the question does there exist an analog of the paper clip in in R3 And I gave you yes number one. So here's yes number two so let's go back and I think this is a better yes because What so if you what would you be your first or one of your first ten guesses for What the analog should be one of the things that you could do and you that you could hope as you could say Well, I take I take the paper clip put a vertical axis here and I swing it around like that so what you get is What you get is an a being Dutch I like to think of so it looks like this So they call it a pancake because it's it's if you go very far back in time. It looks like this It's very flat. So it's like a really flat pancake and it shrinks to So the edges the edges are like grim reapers This edge has the shape of a grim reaper and Therefore it is moving inward the top is pretty much flat the bottom is flat being Dutch I would draw it in yellow and I think of it as a cheese It's an ancient cheese that is shrinking Okay, so there's an ancient solution so this is what you would get if you take the paper clip and you rotate it around an axis What you get is not exactly a solution to mean curvature flow It turns it so it's there's no explicit formula for this ancient solution So what they what they managed to do is to construct a solution that pretty much is like this so So theorem by those three people There exists an ancient pancake Okay, so in forward time, what does this thing do it it just it obeys Husken and it becomes it's it's it's convex So it just becomes a sphere In backward time, what does it do? Well, it just fills out so the cross section seen from aside the cross section is is this This width doesn't change you could assume that that is one And so here I'm drawing solutions to mean curvature flow not to rescale mean it mean curvature flow So backward times this thing just fills up the whole slab between plus and minus one or a slab of width one width one If you do a rescale mean curvature flow then as you go back in time You have to shrink the thing by a factor square root of time So on the rescale mean curvature flow This distance is much larger than square root time. So this still becomes a large distance and this distance becomes One over square minus time so it goes to zero so in backward time this thing converges to a double plane So they constructed this thing and within the class of rotationally symmetric solutions. They show that this thing is unique So and it is so that so they're the Burney-Lankford-Tinaglia Solution is a connecting orbit between these two the reason to believe that in this case Rotational symmetry is not necessary that there might be non-symmetric versions of this thing Let me not go into that Okay, so What is the complete picture? So we don't know yet, but Should have it should look like this So Burney-Lankford-Tinaglia could produce the old pancake. Let me call it the pancake Solution that converges from here to here There is so this is in R3 and the cylinder is so the only self-similar solutions How many fixed points are there for the for shrinking mean curvature flow in R3? the generalized cylinders are s0 across a plane so the double plane s1 cross Cross R and asked s2. This is s2. I'm sorry, right the exponents have to add up to 2 so white and then Hasselhofer-Hershkovitz and then with Toti and Natasha we studied this particular thing So this one we know is unique within the class of rotationally solution symmetric solutions This one is unique There is a connecting orbit from here to here and that one is easy to come by Namely you take the paper clip cross R Okay, so you have a shrinking paper clip like this Build a cylinder on that thing. Oh in backwards time This thing converges to a double plane in forward time it shrinks to a cylinder Right, so we have so now the total dynamics that we have is three fixed points and connecting orbits between them and It can't be that this is everything because the space of connected of convex surfaces And now let's assume that they are all reflections symmetric so that we know that they converge at the origin And we we allow we mod out by rescaling so that we know we can always be sure that they converge that they If they that they converge at time zero and then so which is important for the rescaled In curvature flow the space of surfaces is connected, right? So everything should be this thing has to be a deformation retract of the whole space because under a flow everything in forward time has to If it has to converge to one of these things again, this is not a theorem. This is an idea But but it sounds sounds Like something that could be proved so That means that this is not these are not the only solutions. There have to be other solutions like this So this thing has to have a two-dimensional unstable manifold this thing has a one-dimensional unstable manifold this thing is an attractor So what does that mean? There are there should be solutions that in backward time converge to a double plane in forward time they They converge to They behave like the paper clip cross R and they converge to a cylinder, but then once they get to a cylinder they suddenly become They discover that they are compact, so how would I make one of those I would take the paper clip are so this thing I would Give length R and then I would make Take this take the product with Not our but a really long interval, okay, so how is this thing going to shrink so in backwards time you so this is an ancient solution that is asymmetric It is In backward time it is it's like a really long rectangle and it converges to a double plane in forward But it is convex And I'm sorry it is compact because I've made it bounded and convex and closed So it's compact in forward time what it does is it this will shrink This is the first thing that shrinks And so it becomes a cylinder, but once it gets close to a cylinder You start noticing that the surface actually has it's not a complete cylinder It's a thing with finite length Which is the initial value that Brian White used to construct his ancient oval and then it starts So after that it would have to Do this right so there should be a whole family of solutions like that and then in higher dimensions you You can imagine how the picture goes right now you can you can take this picture and multiply it with R Then you get a whole bunch of solutions in higher dimensions And this then has become a cylinder and then somewhere down there There's s3 and there should be more connecting orbits, right? So the I think an interesting thing to look at would be to complete this picture What does the complete flow look like? um Okay, so I have more stuff here There is so there is the proof of the theorem with Natasha and totie Which has very many steps Somewhere in the in the proof This foliation by self shrinkers comes up So I just want to state that fact, but I think it's Friday and everybody is ready for a break. I think So I'll just take this fact and stop so somewhere in the and I so I'm saying this because I think so what we What we constructed are self shrinkers. They're not complete self shrinkers if you so if you look at self shrinking surfaces of rotation What do they look like? Well, there's the there's the cylinder. This is one of them These are solutions to this differential equation One more time UX X plus divided by one plus UX squared Minus X over to UX Minus one over you. Oh plus you over to this differential equation. So the cylinder is one of them The sphere is another And then you can so there are other surfaces that start perpendicularly here. What do those look like? So you can make computer picture of these fairly easily They behave like an ellipsoid This part is like an ellipsoid, which is part of the reason why this asymptotics why this thing is sorry Not this one. The other one is like an ellipsoid They're not complete once you follow. So for a long time until you reach a certain particular number here So in three dimensions, this is roughly five Everything is fine. You can make a whole bunch of these and you get a foliation If you take one of these and you go beyond five You have to give up that they are graphs because they want to do this Right, you get strange pictures like this They are they become curves that self intersect and it becomes Becomes a mess or really pretty depending on what colors you put in the picture But certainly not nothing embedded So this part turns out to be useful because this is a foliation and these are minimal surfaces Not for the standard area functional, but for whose cons functional and so if you take the unit normals to these surfaces They satisfy divergence of not of the normal because that would just give you H, but e to the minus x squared over four Yeah, this thing this is zero because if you calculate this divergence You get exactly how it's gone you get that divergence turns out to be exactly h Plus and this allows you to compare the whose confunctional of any surface that you have like your solution with say that of the Cone or with one of these shrinkers and this This helps you deal with the various estimates that you that are needed in the proof Also, these special solutions can be used as barriers for ancient solutions. So I Think I should stop there So I'd like to thank the organizers for this really great week. I hope everybody got something out of this Certainly worthwhile