 So we had k is equal to 1 on 4 pi t0 minus t to the n plus 1 over 2 times e to the minus x0 minus x squared on 4 t0 minus t. And we had rho was equal to square root of 2t0 minus t times k. And then we had that d dt of the integral over the surface at time t of rho d mu with respect to the measure at time t was equal to minus integral h plus the normal component of the logarithm of rho. Because k is defined on all of rn plus 1, I can write this here. Or you can write k, if you like, because it's the same third rho d mu. And now, the question then is, and it was posed by you, what happens if the right-hand side is 0? Now, so the question is, which surfaces satisfy h equals minus gradient nu in the ambient space of log k, which is, if you take the logarithm here, it's just this term. So taking minus the gradient, the normal gradient, you get just here x, the inner product of x minus x0 times the unit normal divided by 2 times t0 minus t. This is the equation that a surface has to satisfy if the right-hand side is equal to 0. Well, let me show you. Of course, if you just stare at it for a moment, you find out that this is satisfied by the sphere. If you pick the radius right, depending on t0, and see a sphere centered at x0, or also by a cylinder with the right radius, if the point x0 is on the axis of the cylinder. So we have some solutions that satisfy this. There's the shrinking sphere, and there is the shrinking cylinder that satisfies these equations. So what is special about them? And it turns out they are all self-similar shrinking solutions. So let me show you that whenever you have a solution of this equation, you get a self-similar shrinking solution. Suppose the initial surface, mn0, satisfies the equation h equals, and let's take x0 to be the origin, x times nu. This is the position vector divided by 2 on t0. And then let's just shrink it. Then, so f0 is the map from mn into rn plus 1, satisfying this equation. And then I should write here the position vector f0. And then set f of pt to be, well, we should shrink this surface. So with what factor should we shrink it? Well, the right scaling seems to be to take this quantity to the 1 half, and take f0 of p. And then when we compute d dt of f of p of t, and take the normal component of that, well, then we get from here, we just get minus 1 over 2t0 minus t to the 1 half times f0 at p. And the whole thing is multiplied by nu. But we know what f0 p times nu is. This is equal to h times 2t0. This is h times 2t0. OK. We should rescale this in such a way that at time at t0, we get the same surface back. So I have to also divide by 1 on square root of 2t0, so that I get the same surface for t equals 0. And then here, I have to put in this factor times 1 on square root of 2t0. And then this is equal to, because of this equation, this is equal to minus 1 on, I have to be careful, f0 p. So I get, let's just write down what we had before, 1 on 2t0 minus t to the 1 half I had times 1 on square root of 2t0. And now I replace f0 nu by h times 2t0. And let's see what we finally got. What did I do here? Does this look right? I'm left with square root of 2t0 divided by square root of 2t0 minus t times a minus sign times h. This is just the mean, but this is the h of the initial surface of f0 at p. Now since I have rescaled this, by this factor, the mean curvature with exactly opposite factor is, of course, equal to the mean curvature at f of p of t. So we find that the normal component of the speed is equal to the mean curvature. So f solves mean curvature flow up to tangential difumorphisms. So in other words, this equation here characterizes self-similar shrinking solutions. So this equation star characterizes self-similar shrinking solutions. And it's clear that we cannot expect that this is true without allowing tangential difumorphisms, because if you're here on the cylinder, and the cylinder is shrinking, and you study this from this point x0 here, then, of course, here the shrinking in this direction is exactly the same as mean curvature flow. But of course, the shrinking from this point here, or from this point, it's only the normal component of the shrinking is giving you mean curvature flow. So that's why it's important that I have to multiply this with nu, only then you see the mean curvature. So this rescaled surface is, of course, not, it has very strong tangential component, as you can see in this picture, where you have the shrinking cylinder. Nevertheless, the flow, this monotonicity formula has the capability of singling out the self-similar solutions. And so the question is, are there more such self-similar solutions? So here are some other examples. It turns out for any equals 1. So just a curve in the plane, there was a whole family discovered. There's a circle, of course, which we just found already. And then there's a family of curves look like that. It's getting more complicated. And there's a two-parameter family of curves that all satisfy this equation. You see, when you write it down for a curve, this is just the curvature of the curve. So if you write it in angular coordinates, this is just a second-order ODE for the position of the curve in the plane. And then you can always solve it for some initial conditions, but then the solution will just chaotically run around. And if you do it just right, you get a discrete family of parameters where the curve nicely closes up. And that gives you this family of curves, which was found by Operation Langer. And then I think it took about six or seven years until people discovered that Malin had studied this about 25 years earlier in the context of crystal growth and found these curves already. In higher dimensions, there is a whole bunch of these self-similar solutions which are axially symmetric. So I just draw the profile of these axially symmetric solutions. So here's the radius of these curves. And then, of course, you have the sphere. You have the cylinder. And then it turns out there is a embedded torus, sort of an oval-shaped cross-section, which was found by Sigurd Angenand. That's why it's called the Angenand Torus. Again, it's a matter of finding the right initial conditions for the resulting ODE in the axially symmetric case and make it close up. And then there are these pictures very similar to these ones. There's a curve like this. And then it turns horrible. You can have, essentially, there are pictures that look like that. So you have all sorts of immersed torus. And then there are some which hit the axis. So there is a huge variety of axially symmetric self-similar solutions. I should point out, however, these are the only embedded ones up to here. And so these guys are embedded. And these ones here are the only ones with positive mean curvature. So all these on this side, all this junk, either has not positive mean curvature or, in fact, it's not even embedded. So you would not expect these guys to appear as singularity models. They would be extremely unstable. So the really interesting ones are those three. And then there are others. I'm unable to draw them. But imagine a cube, punch a hole through in all three axes and make it into a smooth surface, what is left over. And then you get a surface with this cubicle symmetry. And in fact, someone showed numerically that you can solve this elliptic equation on such a surface. So there are embedded, complicated surfaces in R 3 that move under self-similarities and move under these shrinking equations. But again, you would expect that kind of thing to be very unstable. Now, there's another way of thinking about this. Because, see, this weight here is rather precise. But I come back to that later. We can, in fact, interpret these surfaces here as minimal surfaces in a funny metric coming from this Gaussian cone. And then this would be, say, the equator in a sphere. And this would look like the Cliffor torus or so. So there's a way of thinking about these surfaces as minimal surfaces just in a funny metric. But this thing here comes from a conformal factor. But the most important application of the monotonicity formula is that it allows you to understand the shape of singularities. And this only works if you do a rescaling procedure. So I have to tell you about the rescaling procedure for the mean curvature flow. And let's see. There's two ways to do rescaling. The first attempt. And you see, here, for these self-similar solutions, we have rescaled with this factor here. And we found this, we scared the initial surface, we found the solution. So now suppose we have a singularity. Assume that mean curvature flow forms a singularity that is similar, I will tell you in a moment, to spheres, cylinders, and all these pictures here. We should rescale, if this is the case, the right magnification rate that we should put on our microscope should be exactly this factor. Just we do it now the other way around. We want to rescale with this factor to the power minus one-half, s, t approaches t0. And t0 is a singular time. This is our old Tmax, a singular time. We have a singular time. And by the way, singularities can happen. As Carlo told you yesterday, recall, there are these singularities that come from convex surfaces. And we expect that if we train our microscope in the right way, that we see the shrinking spheres. And he also showed to you this argument with a barrier argument that you get a shrinking neck in here. And we expect that if we train our microscope on this region, that if we do this, we somehow should find the shrinking cylinder. This is the picture. And what we want to find out is how generally it's there. But if the picture is right, then we should rescale with this quantity. So the answer is to set given f from mn from 0 to t0 to rn plus 1, set f tilde of p and t to be this factor, 2t 0 minus 1 to the minus 1 half. We are blowing up. If t is very close to t0, this will be a huge factor times the f of p of t. If we expect the singularity at the origin, expect singularity at x0 equals 0, otherwise I would take here minus x0. But let's make things simple, rescale around the origin. Then it turns out best to introduce a new time parameter. And to get the constant right, s is equal to minus logarithm of 2t0 minus t to the 1 half. So as t approaches t0, this goes to 0 and minus the logarithm goes to plus infinity. You see, it's much better. We want to study now the asymptotics after we rescale. And to understand the asymptotics, it's better to make the time interval infinitely long. And it turns out this constant is just chosen such that if you now take the derivative of f, add the rescale thing with respect to the s variable, just do the computation. It turns out you get, from here it just scales right, the mean curvature vector of the rescaled surface. But then you get an extra term from this front thing. And this extra term is simply f hat at p s. If this is not there, this is the pure rescaling. If the mean curvature wasn't there, this is pure rescaling pushing the things up. So this is one way to rescale the flow. And notice that, of course, the fixed points of this new form flow, fixed points satisfy, in particular, that if you multiply with the unit normal, that h. So if you have fixed points in normal direction, so dds f hat normal component, if you want that to be 0, we always have to work modulate the tangential motion. So we only want the normal component to be 0. And then this just means that you get minus h, because mean curvature vector is minus h times nu. So you get minus h plus f hat dot nu equal to 0. And this is exactly the equation star as before. So we have indeed created, now, by rescaling a flow such that the condition that gives us the self-similar shrinking solutions are exactly the fixed points of this new flow. And then when you compute what is now, now let's see what the monotonicity formula is telling us in this setting. Well, if you said, well, let's just compute it, d dt rho d mu over mn. This is equal to 1 on 4 pi t0 minus t to the n over 2, right, times e to the minus f squared divided by 4 t0 minus t times d mu. Now, the d mu multiplied with this vector is exactly the rescaled d mu. Up to a 1 over 2 pi to the n over 2. This is just the rescaled d mu hat. And this thing here, over 4t0 minus 2, is just the rescaled e to the minus f hat squared over 2. d mu hat over the surface mn hat. So with this rescaling procedure, our weighted integral has just become an integral of the Gaussian, a fixed function which does no longer even depend on the parameter s or t. We've really gotten the Gaussian integral up to a constant. And then when you compute now under this new evolution equation, what is dds of integral mns of e to the minus x squared over 2 d mu, it turns out you get minus integral mns of mean curvature hat here minus f hat nu squared e to the minus x squared over 2 d mu. Mn hat of s. Well, mn of s, yeah, you can put a hat if you like. These are the rescaled surfaces. And the time parameter is now s. But now the time runs, note that s runs between the time when you had t equals 0. So this is minus log of 2t0 to the 1 half, which is some number, but it runs all the way to plus infinity. So if our rescaling produces a smooth limit, then this smooth limit must satisfy our crucial equation. So if we get a smooth limiting surface, then it must satisfy equals f dot nu. In other words, any such limiting surface, it must give rise a self-similar shrinking solution. So the question is, under what condition can we produce that smooth limit? Yes, this is the rescaled measure. Yes, everything is. No, no, no, this is just the, this is just, now I'm back on Euclidean space. This is just dx, if you like. This is just the ordinary, the induced metric on this surface. Yeah, I should write d mu s on this surface. Because this together times d mu gives this measure. Because the measure scales like distance to the n. And here I have rescaled distance with this factor. So the measure rescales with this factor to the n. And this is exactly this term here. OK, so the issue is, when do we get this limit? Well, when do we get a limit? When do we, if we can bound the curvature, right? So we need the curvature to be bounded. So what happens to the curvature under this rescaling? Well, the rescaling, the curvature scales exactly the other way around. So what we need is that the curvature, as we approach the singularity, behaves like the curvature of a sphere or cylinder. So need for this, that the soap of the second fundamental form, as M and T approaches the final time, that this is less than some constant divided by t0 minus t. As t approaches t0. So it's allowed to go to infinity, but only at this rate. Or another way of putting it, of course, is that the soap of a squared times t0 minus t. M and T is less than c less than infinity. If we have this assumption, if this is true, this condition here is called it's a type I singularity. If this condition is satisfied, we call the singularity a type I singularity. So the shrinking spheres, all these things that have been there on the board, all these pictures are type I singularities, because they are self-similar shrinking solutions. And of course, for the shrinking solutions, if they are self-similar, the second fundamental form clearly satisfies this. In fact, this is constant, because it's just shrinking. Now, if this is true, then we know that from such a curvature estimate, we get higher derivative estimates. So this means, of course, that the a hat squared is bounded by c, because the a hat squared is exactly this quantity. So in fact, this is the same. And then we know that if this curvature is bounded, we know how to do this. All the higher derivatives will also be bounded. So we have curvature bounds. We have gradient of curvature bounds. And we have, very important, we have some volume bound. Here, this is sort of the, it's a weighted volume. But still, it's a positive density everywhere on Rn plus 1. So this means if our initial surface is compact, this was finite, it is decreasing. So these surfaces cannot pile up. They cannot get high densities. So now we have no compilation, since integral rho d mu s over mns is less than this integral over the initial surface. The only thing that could still go wrong is that the surface disappears. And in fact, if I choose my point for rescaling here, if I'm stupid enough to rescale around that point, of course this whole thing will just shoot out to infinity and disappear in the never, never. And all these curvature estimates are useless, because I get nice surfaces, but they all disappear. But this is easily fixed. Either I recognize my mistake and I just translate it back to go through the origin. Or in fact, you can see that if you really approach this point with the singularity, then because you have a type 1 singularity, the mean curvature, which is the speed goes like 1 over the square root of this. And 1 over the square root of this is integrable, and therefore it cannot move arbitrarily fast, even if you push out with this rescaling, so you get a limit. So let me just write, it does not disappear by choosing the scaling point. And then you get, of course, whether you have a unique limit or no claims, but now with this control, you can show that you get a subsequence which converges smoothly to one of these solutions of the self-similarity equation. So let's collect that in a theorem. Any singularity of mean curvature flow can be rescaled, a self-similar shrinking solution of mean curvature flow. That's the first classification of singularities that you get out of the monotonicity formula. Those of you who have studied minimal surfaces, the comparison, you can view this as a parabolic cone. In minimal surface theory, you have a monotonicity formula. It's essentially the same one, except that H is 0, the formula, which tells you if you rescale a singularity, you see a cone, a minimal cone. Here, we see a self-similar shrinking solution of mean curvature flow. It's the parabolic analog of the minimal cone theorem for singularities of minimal surfaces. Can one classify these self-similar solutions? I have drawn this picture telling you that, in general, probably it's rather hopeless. What I can tell you is there's a theorem that I proved a long time ago that if the mean curvature is positive and if it's self-similar shrinking, then it is, well, first of all, you have Sn, the shrinking sphere. Secondly, you can have Sn minus k cross Rk, the shrinking cylinder. And the only other possibility is the very special case coming from the Arbor-Schlanger-Mullins solution. You can have one of these curves of Mullins and Arbor-Schlanger gamma mal, they're bad curves, cross Rn minus 1. And of course, these are ruled out. This case does not occur if you're in the embedded setting. There's no such thing in the embedded setting. So this I proved first around, I think it was around 1990, for bounded curvature. And I think it was proven by calling minicossi in the most general setting around 2016 or so, without assuming bounded curvature. Right, so this is the, all right, I think I should tell you one more. There's one more, very recent result, which is also very nice. So there was a result theorem by Brandle in 2017, I think, where he showed that if M2 is sitting in R3 closed, embedded and self-similar, and to diffiomorphic to the sphere, if it's simply connected, then it is the sphere of some radius. It's a remarkable result because he does not assume any curvature assumption. So he replaces positive mean curvature by embeddedness. And the fact, and the topological condition. He uses a topological condition and the sphere, and the embeddedness. I told you about this cube with the three holes punched through. That's an embedded surface, and it's a self-shrinker. At least it exists numerically. So this topology condition is absolutely necessary. And the embeddedness condition is also necessary, because we've seen all these other. But the engine in torus also shows you that the topological condition is necessary. And this gives some hope. If you cannot show that for these types of surfaces in R3, there's no type 2 singularity, no other singularity than the ones we've discussed so far. And we know the surface we start with a sphere, then the solution must exist until it has shrunk down to a nice round sphere. And that means you can deform with mean curvature flow any embedded two-sphere in R3 into a round sphere with this PDE. This would be a complete reproof of PDE methods of Hatcher's theorem, a famous theorem in topology. So this is an open problem. Can we use mean curvature flow to give a completely new proof of this famous theorem in topology that embedded two spheres? You know, the Jordan Curve theorem, one dimension higher. At the moment, it still seems out of reach. But this is a hint that we may have a chance. The way Brentley proved this is he saw that the self-similar solutions, this equation here. So the main idea, there were many ideas, but the main idea here was that the equation equals f dot nu is the Euler Lagrange equation for the weighted area, for integral e to the minus x squared by 2. If you compute the Euler Lagrange equation for a hypersurface and you want to locally minimize this with this weight, you end up exactly with this equation. But this here is just the area with respect to a, and I shouldn't use hat, maybe I use bubble tilde, of a, this is just the ordinary area with respect to a conformal metric on Rn plus 1. You take the standard metric times e to the minus x squared over 2n. You take this conformal factor, then this area becomes this weighted ordinary area. And therefore, this here is the minimal surface operator in this metric. So self-similar solutions, minimal surfaces with respect to this metric g double tilde. And then you have the whole power of geometric measure theory, for example, to attack this and check whether you can really have a minimal surface of type S2 in R3 with this funny metric. Yes, I come to that in a second. Next thing is translating solutions, so yeah. I just have to remember your question. OK, so this is rescaling first attempt. So now comes rescaling second attempt. If there is a type 1 solution, there has to be a type 2 solution. So we are not that lucky as in that setup over there each time. So in case we don't know what the blow up rate is, we can still do something. Namely, we can do the following. We can just ask very generally, how can we rescale our flow? So given m and t, mn cross time interval into, yeah, let's do it in Rn plus 1. Don't want to rescale the ambient manifold, which we could also do, but let's do it in Rn plus 1. Then, and given time t0, where we want to rescale, and maybe a sequence of points or events, ti in mn cross 0 t0, can define some parameters lambda i bigger than 0, f f f hat, p, and tau. Introduce a new time parameter tau again. And here we take lambda i, multiply now with lambda i. And we take f at p. And here we take tau lambda i to the minus 2 plus ti. And we subtract f at pi ti. This is what we do. So where does this tau live? Well, the t's are these guys. So this guy is t, the old t. So the t was between 0 and capital T. Then this new guy tau must be between minus ti lambda i squared. And it must be from above less than capital T minus ti times lambda i squared. Time interval. And the lambda i here and the lambda i to the minus 2 here are just chosen. So I've just chosen in such a way that here that if you compute d d tau of f hat i of p tau, then the lambda i here times the inner derivative there just fits together with the df dt to spare you the calculation. You just get the mean curvature vector back again at p and tau. In other words, this is a rescaling. There I really rescale now in some sense more carefully. Here I've rescaled sort of just the space. And then I introduced a slightly stretched time variable. Here I don't try to do it all at once. I just take it each time at factor lambda i and rescale the space and a rescale time. And at the same time, I'm shifting the time a little bit such that my new tau equals 0 is at the selected times ti. And that my new value at tau equals 0 is exactly the origin. So I've shifted space and time to make it convenient to make the 0.00 in space time special. But I've got, again, a solution of mean curvature flow. I don't have an extra term. So in some sense, this is the cleaner, the fairer way of rescaling the thing. You have mean curvature flow again. In other words, I'm really just multiplied everything by factor lambda i, space and time. And now the question is, how do I choose? How do I choose the lambda i's? How do I choose my microscope? Well, I want to see something which is not trivial, but smooth. So I have to rescale sufficiently strong. But if I rescale too strong, I just see a plane. I just see a flat surface. So I have to choose the size just right. And the right thing to rescale seems to be just opposite to the curvature. So I can choose lambda i to be the soup of the curvature on the surface cross the time interval from 0 to, well, I want to go a little bit beyond ti. So let's go to t plus t minus ti over 2. So this is somewhere between halfway between ti and the end of my time interval t0. Sorry, I should take here t0. This is the point around which I want to rescale, right? Here this is the end of my time interval where I have the singularity. If I do that, then I get, first of all, that the rescale surfaces have bounded curvature. And then it is bounded by 1 on mn cross the time interval up to a little bit beyond this ti. So I go to tau is equal to t minus ti times lambda i to the 2. So if I put 2, if t is less than ti plus this, this is less than lambda i squared up to t, if I subtract ti from this, I get t0 minus ti over 2, t0 minus ti over 2. I get the bound on this set, which in turn is, oops, I forgot lambda times lambda i squared, of course. Sorry, i squared. And I get that f of pi ti, if I put ti in here, f of tau equals 0, f of pi ti just goes out to 0. And I also get that the second fundamental form, so I've shifted everything to 0. And if I want to, I could have rotated the thing also. So you could think of the surface somehow sitting around the origin in our n plus 1. And at time, somehow we have rescaled this thing, it looks like that. And since these guys are bounded, I can also get, after rescaling, I get all the derivatives of the AIs bounded by some other constant depending on n. And therefore, I can use, since I've translated everything, and again, since I have area estimates, it is not so hard. You have to work a little bit, but it's not so hard that, again, you get a subsequence which converges, as i tends to infinity, get a limiting mean curvature flow. Surfaces m infinity, n of t, or tau, I should call them. And they solve mean curvature flow again. And on what time interval? Well, what happens to this time interval? Here's this time interval. Now, since we run into a singularity, we know at the singularity, we can pick the events such that the second fundamental form goes to infinity. That was our theorem in the first lecture. If we have the singularity, then the curvature has to come unbounded. So the lambda i's tend to infinity. The t i's go to capital T. So this thing here certainly goes to minus infinity. So we certainly get a solution that has existed for all negative times. About here, now, all of a sudden, it makes a difference whether we have a type I singularity or not. You see, this lambda i, we choose the maximum of the curvature on intervals that get closer and closer to the singular time t naught. So if we have a type I singularity, then we know that these guys remain bounded. So we get that this is less than some t max, less than infinity, if the singularity is of type I. But we get a solution on minus infinity up to plus infinity if the solution is not type I. Well, if it's not type I, people call it type II. Why is the curvature, well, the way I've chosen here, this interval here, see, this interval also, if it's type II singularity, this, the 1 half doesn't destroy this. This has to go to infinity. So I get my curvature bound on arbitrarily large time intervals. So I really get a completely smooth solution of mean curvature flow, which exists all the way from minus infinity to plus infinity. So that's the advantage of the second approach. We can still do type I, then we get a solution that exists from minus infinity to some finite time. Well, this is the convex case, type I. You have this convex surface, you rescale, and you get a sphere that comes from huge radius, time minus infinity, and it contracts to a point at some finite time. Or if you have a cylinder or a neck pinch, you have this neck, and you rescale here, and you see the cylinder coming from huge radius at time minus infinity, and it converges to this line at a finite time. So this is this picture, finite time. Turns out there are some other singularities which are of type II and which can be rescaled to have a solution on infinity time. What could that be? Examples. So example of type II singularity. First example is if you have a double loop, and if you let that shrink, what happens is that the small loop shrinks faster than the big loop. And you develop a cusp rather than a neck. This is in the case n equals 1, a curve in the plane. And when you put the microscope on this thing here, what you see is a solution is y, and here is x, and here is pi over 2, and here is minus pi over 2. And you get a solution that is asymptotic to these two lines, and looks like that. And it just moves by translation. Infinity 1 moves by translation. And in fact, you can give a formula. You get that x is equal to x at t is equal to x at t and y is equal to minus log cosine y plus t. So it translates with speed 1 from right to left. And therefore, of course, it exists on minus infinity, less than t, less than infinity. You get a solution for all time because you get a translating solution. The second example is something like this happens is if you try to do this neck pinch picture that we had before, but now you do it in such a way that the bubble on the right is not so big, but also not so small. If this was a very small bubble, then it would just sort of shoot inside, the surface becomes convex, and then we get a singularity like this. If the bubble on the right hand side is much bigger, then we get a nice neck pinch in the cylinder like that. But if you choose the size of the second bubble just right, then what happens is it gets stuck. It gets stuck in here and forms again a cusp. And when you point the microscope, this rescaling procedure that I explained at the tip of the cusp, then you see again something like this looks like a parabola, grows like a parameter. It looks like a parabola, grows like a parabola, but is not a parabola. But it moves by translation again. So this is a bowl moving by translation. There's one thing I didn't stress. It's inherent in what I wrote down. You have to pick a sequence of points. I've shown you so far in all the, is assuming I have a sequence of points which actually goes to the point where the curvature is maximum, sort of a sequence of points sitting at the tip, sitting at the tip here. There's more subtlety, which I didn't have time to explain. But if you don't choose your points right, of course, you could see something else. If you choose your points where you rescale and do this procedure somewhat differently, then you might rescale it a little bit next to the tip. And if you do that, then you see the cylinder again, which you see in the ordinary neck pinch. And of course, this is the cylinder that is attached to this parabola. If you go on a parabola far out to infinity and rescale, you get a cylinder. So this rescaling thing here is not a, it is extremely helpful, and it gives you a lot of information. In particular, you get these self-similar solutions and you get these translating solutions. But it still depends a lot on how you do it. And you may see there's no uniqueness whatsoever. So you have to be extremely careful how you pick your sequence, what kind of limit you get. But there are some theorems. For example, there is a theorem by Richard Hamilton, which says that an ancient solution, which is convex, and where the maximum of the mean curvature is attained, or the maximum of the second fundamental form, is a translating solution. Sorry, not an ancient, an eternal. An eternal. These are all ancient solutions because they started ancient times, but only the second one here is eternal. These names have all been made up by Richard Hamilton. He loves giving these names, and I think they make sense. And so in order to speak, this theorem applies in the two pictures I've shown you. You get a convex, of course, you have to prove that you get this convex solution here, or you get this convex solution there. There's a lot of work to do to prove that. But you can do that, and then you get this solution. And of course, if you pick the points right, we can make sure that the maximum is attained here or there. And this theorem then says, yes, we got a translation solution. And now you have a complete picture. Not a complete picture, I said there are subtleties. But now we have sort of a chance to understand singularities or rule out singularities. Because if we want to show, for example, that the flow has beautiful properties and doesn't have singularities, now instead of analyzing mean curvature flow step by step and controlling every single little piece of the complicated surface, this theory says all you have to do is concentrate on the singular times. Just assume you have some singular time, and only then you look at the microscope, and then you just have these two possibilities. Either you get a type I singularity, and then you know they are self-similar. This reduces your problem, for example, in the positive mean curvature flow, to just study these spheres and cylinders. You have a type II solution, and then you just have to analyze the translating solutions. And how that works, I'm going to show you in the next lecture improving Grayson's theorem. So we will combine these techniques and a sketch of proof of Grayson's theorem. So next time I'm going to prove this theorem by Matthew Grayson. If gamma zero is an embedded curve in R2, is embedded, then mean curvature flow, which is also called, in this case, curve shortening, deforms gamma zero to a tiny circle. So you've got a proof, an analytic proof of the Jordan-Koff's theorem. Any complicated, no matter how complicated, curve in euclidean two-space can be deformed with this flow until it's nice, small, and convex. And shrinks to a point exactly. In other words, the way we're going to prove it is I show you that the only singularity that can ever happen is this one. I'm not going to analyze what this curve is doing, how it is winding in R2. Just saying, if something goes wrong, something will go wrong because it's enclosed in some circle, which dies eventually. If something goes wrong, this is the only way it can go wrong. And therefore, just before it went wrong, it must have been a tiny circle. And I'm done. So the whole thing of Grayson had a completely different proof where he really followed the curve and he counted how many turning points there are and so on that had a beautiful way of analyzing that. It was a fantastic paper. But you can't do it this way. You don't have to do all that. You want to be lazy. You just look at the singularity and say, this is the only singularity that's possible. Therefore, it must have unwinded. OK, that's for tomorrow.