 That's right. So welcome to the last lecture by Professor Wiesken. Right. So I want to use the last lecture to give you some more further results and an overview in what directions these mean curvature flows could be useful. And one theme I thought might be interesting to many of you is how I can use mean curvature flow to prove some isoparametric inequalities. So this lecture is about applications and the isoparametric inequalities. And to fix ideas, let's do first the one dimensional case and show you the most complicated proof of the isoparametric inequality you've ever seen. It's not complicated if you know mean curvature flow, of course. So given an initial curve, gamma 0 S1 into R2. So gamma 0 of S1 is the boundary of our initial domain that we are interested in. And then we know by Grayson's theorem that there exists a solution gamma on some maximal time interval into R2, shortening flow. And this takes this complicated curve into it remains embedded. And it takes it into a small circle and then to a point. So gamma t converges to convex round circle, shrinking round circle as t goes to t max. And if you want to study isoparametric problem, we need to know how the length and area evolve. So l of t be the length of the curve and the a of t, the area of the domain enclosed at time t. And then we get d dt of the length. It's our old formula about the volume element. And d dt of the area is the integral of the speed along the curve. And this is, of course, the integral of the geodesic curvature in our closed embedded curve. We know that this is minus 2 pi. So in fact, we know in the one-dimensional case that t max, because the area is decreasing linearly. And then at the end, it is going to be 0. So the maximal time must be the initial area divided by 2 pi. And we know at the end, both of them tend to 0. So let's look at what the isoparametric distance, the isoparametric defect, if you like, does under the flow. This is what is supposed to be positive on the initial surface. But let's pretend we don't know that. And then let's just compute under the flow this is minus 2l times integral k squared because of the formula above. And from here, we get plus 8 pi squared. Now just use Schwarz inequality. And then you get that this is less than minus 2 times the integral of k ds squared plus 8 pi squared. But integral k ds is 2 pi. If you square that, you get 4 pi squared. So this is 0. In other words, this isoparametric defect decreases under the curve-shortening flow. And since l of t and a of t by Grayson's theorem, so here I use heavily Grayson, tend to 0 as t tends to t max. It must be, since it's been decreasing, that initially it was positive. So we conclude that at the beginning, we must have had this inequality, which is the isoparametric inequality. In other words, the isoparametric inequality in the plane is a trivial consequence of Grayson's theorem. Now it turns out you can try to do that now in higher dimensions. So let's just do the same thing. So we know that because of scaling, we should take power n plus 1 over n of the area of the surface at time t. Then we know there is this optimal constant that we expect from the sphere, from the round sphere. We know how much this should be times the volume enclosed by the surface. And now we just do the same computation. We just get here minus n plus 1 over n times the area to the 1 over n times integral. Now it's the mean curvature squared d mu. And from here, we get plus c0 times the integral of the mean curvature d mu over the surface. You see, we want to compare the positive term with the negative term. We hope that the negative term wins. So what can you do? You do Schwarz inequality just like before. So we estimate this from below by minus n plus 1 over n times the measure to the 1 on n. And from the Schwarz inequality, I get a minus 1 half. And here I get the integral of h squared d mu to the 1 half. Take this whole thing plus c0 times integral h d mu of this. The square root of this, estimated from below by integral h and the other square root stays here. Now you see, everything is fine. So it turns out that the x case, if you have convex surfaces, there's some kind of Minkowski inequality. Well known inequality that says that the integral of h d mu over the boundary of a domain is bigger than and the constant in the Minkowski inequality relates to the constant c0 by n over n plus 1 c0 times mn to the n minus 1 over n. So this is well known for convex surfaces. And then, of course, this means we can, using a gain holder, we can estimate integral h squared d mu to the 1 half from below by n over n plus 1 times this constant times mn to the, and now here I have to subtract 1 half. So I get 1 half minus 1 on n as the power. And therefore, these constants here obviously perfectly fit what stands here, and I get again that this is less or equal to 0. So I get again the isoparametric inequality because there's this theorem that I proved a long time ago that the convex surfaces shrink to a point just like in the one-dimensional case. So this gives you a proof, an immediate proof of the isoparametric inequality for convex regions in rn plus 1 since convex surfaces, convex surfaces movely contract to a point. Of course, these inequalities are well known. You don't need this particular proof, but you have now a new method, and you can hope to apply this method maybe in more general situations, maybe in Riemannian manifolds, assuming some curvature conditions. The first step, of course, would be to try in the high-dimensional case to extend this proof beyond surfaces that are convex. And also in Riemannian manifolds, convexity is such a strong assumption you want to do this in a more general setting. So the question is, can one extend this from convex to some other class, for example, to the class of positive mean curvature surfaces? That would be great because usually it's enough to prove the isoparametric inequality for positive mean curvature surfaces because if you have a general domain, you can always take the mean convex hull, which has non-negative mean curvature, and you could start, and that is then better in terms of the isoparametric inequality anyway, because it has the same area and has the, or even less area, and it has more volume. So you can usually start with the mean convex hull to prove the isoparametric inequality. And therefore the question is, can you extend this argument to surfaces of a non-negative mean curvature? Now, we know that in that case, however, we have possibly singularities. We could have this neck pinch. In the neck pinch, you have this long cylinder. So certainly if you were just to cut out this neck, if you could just cut out this neck and replace it with some cap, this would certainly improve the isoparametric situation. So if we could show somehow that only this or essentially this thing happens and we could extend by cutting out this neck, then we have a chance to extend this mean curvature flow proof of isoparametric inequalities to a more general setting. So the task will be try to extend the mean curvature flow beyond singularities. Very high dimensions. It is known that the singularities can be terrible. So if n is greater or equal than 3, singularities on sheets, which locally look like, say, S1 cross R2 or more general SN minus k cross Rk. k is bigger than 1. In other words, if you have more than one flat direction and you have really whole sheet, two-dimensional or higher dimensional sheet of spheres shrinking off, which can happen. I just can't draw it. We just don't know what to do, how to do the surgery. There is still a concept of weak solution that, for example, Professor Tony Gava told you about with level set approach. But certainly, the smooth approach that I'm choosing here and doing explicit surgery is completely hopeless in that case, at least for the time being. No technology for surgery available yet. So we have to somehow concentrate on the point where you have just one such neck and where this cross section here is like SN minus 1 cross R. And Carlo will tell you a little about how this works in dimensions greater or equal than 3. If you find a way to restrict to this case, what I want to do is concentrate on the case n equals 2. So the plan now is what can be done for two surfaces in R3 or in a general many-fold in 3G bar if the mean curvature is positive and the surface is embedded. And the initial surface, Carlo will tell you what you can do in higher dimensions if you have a stronger positivity condition to convexity for the surfaces, but there you don't need embeddedness. So here, I'm not going to speak about some. And this is joint work with Simon Brandlip. I think we finished this in 2016. And this is appearing as partially has appeared in Inventiones. And another part is going to appear in the Journal of the European Master Society. But this is. So we want to look at two-dimensional embedded surfaces in the Riemannian-3 many-fold and assume positive mean curvature. Now, this allows, of course, these singularities. And the first thing we have to do is we have to classify the singularities. We have to know exactly what singularities are possible. We could do this in the one-dimensional case. Remember, Grayson's theorem, we said the only possible singularity for embedded curves in R2 is the shrinking sphere. So now here, because of this picture, and because, remember, this degenerate problem, I was drawing this situation where the surface would develop a cusp. And here, you expect somehow this in the limit to be sort of this shrinking circle. And I said, here, you expect it somehow to be such a convex thing which is attached to a cylinder. And then, of course, you still have the shrinking spheres. So we want to prove that, essentially, these are the only possibilities. And to do that, one tool is something I just throw at you without proof. Carlo may say more about this. And this is a general theorem. This is joined for Carlo Sinistrari and myself a while back. So we're using a convexity result. If the initial surface, so in any dimension, has positive mean curvature, for any eta, there exists a constant c eta, which just depends on the initial surface. And maybe on the time interval where the solution exists. And on the ambient remanian manifold, so this is really quite general, such that the smallest principle curvature, or any principle curvature, is bounded from below by just a tiny negative fraction of the mean curvature, and then possibly this big constant c of eta. But the scaling invariant part in the inequality only has an arbitrarily small fraction on the right-hand side. So the negative eigenvalue that may be there is tiny near the singularity compared to the mean curvature, so that in particular, when you rescale, the negative part goes completely away. Such that any rescaling of a singularity must be weakly convex, must be a weakly convex solution of mean curvature flow in Rn plus 1. Because even if you do this in a remanian manifold, during the rescaling, since the remanian manifold has bounded curvature, after rescaling, you just see the tangent space of the remanian manifold, which is Rn plus 1. So the singularity models will always live in Euclidean space, even if your surface happens to be in a remanian manifold. And of course, you see at least this is the first step towards this picture. So even though we start here with something only mean convex, if this singularity happens, after rescaling, you see the shrinking cylinder, which is weakly convex. And if we are in this situation here, we even get something which is strictly convex. And if we are in the case where we shrink to a sphere, also we are in the case of strictly convex. So this is just saying that we cannot have limits that are completely non-convex. Not going to say anything about the proof. It's based on analysis of the evolution equations for the curvature and the mean curvature and a lot of analysis at P estimates, Mosa iteration to do this. Now, what you can do with that in the two-dimensional case is the following. So suppose M2-0 is sitting in a smooth remanian-3 manifold and embedded and a positive mean curvature. Then the only rescaling of singularities that may occur, first you see S2 shrinking, the standard shrinking sphere. In R3, the second possibility is you see S1 cross R shrinking, shrinking cylinder. And the third one is a convex internal solution. I should say strictly convex here. Strictly convex internal solution, cylindrical end. So near infinity, it looks like number 2. So it's this picture. You rescale, you find the strictly convex part and the strictly convex part is attached to something which gets more and more looking like a cylinder. So that's a classification of the rescalings that you can get. And then you have to work a little bit harder to show that these rescalings fit together to give you a complete picture how the manifold looks. Those regions where the curvature is big. Say, if you close to a singular time, you can use this theorem to prove what is called a canonical neighborhood theorem. So from this theorem over there, I say something to the proof in a second. You prove another theorem again with Simon. I call this canonical neighborhood theorem. And this says, given the initial surface as described, embedded positive mean curvature in the remaining manifold, constant h0, some huge constant. This is just to indicate it's huge. Obviously, everything depends on the scaling, depending on the initial surface and the ambient manifold. Sorry, yes, we are in dimensions 2. The region, the mean curvature, is bigger than that number. In other words, the almost singular region. I'm still talking about a completely smooth surface. These surfaces are still below T max, but maybe very close to the singular time. And then we may have a region where the curvature is huge. And what I'm saying is you can determine a threshold for the mean curvature such that beyond this threshold, you know exactly how the manifold looks like. So it will be like this. There will be a part of the manifold which is sort of completely unknown to you. You just know it's embedded as positive mean curvature. And the mean curvature is less than h0 in this region. So this is sort of the nice region where h is less than h0. And then you have a region where maybe you have sort of such a neck. And then it may open up again to give you a nice region. So here you have another nice region. In here, use red. And here somewhere is the red. This is the region where h is bigger than h0. There may be also sort of bad regions which look like these cylindrical regions. But they end in this convex cap. And then maybe if the surface is disconnected, there may also be a region where you have a tiny shrinking sphere. And you may also have a region where sort of a cylindrical region of high curvature closes up in two caps. And you may even have the situation where such a cylindrical region closes up in itself. And you have sort of a shrinking circle. But the point is that in all these regions where h is bigger than h0, the rescaling of this region satisfies theorem. You see either the shrinking, you are either shrinking, you are either the rescaling under the microscope, this region looks like a cylinder, or a cap, or sort of. Yes. And you can make this quantitative. So I can choose this h0. So this is part of the theorem. It's a quantitative theorem given epsilon bigger than 0 and lambda less than infinity. So this is the picture of the previous theorem, in particular how close you are to a cylinder of length lambda in a huge norm. You can even prescribe the norm and see one over epsilon after rescaling. So given epsilon, you can fix the h0. So in other words, we can even introduce these parameters to make this quantitative. This will depend on the epsilon and it will depend on the lambda. They have a very quantitative theorem that tells you, you tell me how precisely you want to see these singularities. And I tell you how long you have to wait with your curvature, how huge the curvature have to be that you have that control in the red region. So that's a very quantitative precise description of the singularities. And that's what enables you to do the surgery and restart the flow. Something Carlo will tell you more about. Now let me give you at least some ideas of how we can fit our previous results together to prove this theorem, what goes in. So the ingredients of proof, first of all, of course, the convexity estimate goes in. So we restrict you very much. Then you use the non-collapsing estimate. Of course, this implies that it remains embedded, but it remains embedded in a quantitative way. The non-collapsing estimate says that after rescaling, so non-collapsing, you have to combine these two. How? So the third step, you write down these two results and then you do the rescaling. And then these two estimates, one and two. So first, let's do it in two steps. One sees that you get either strictly convex or you get weakly convex. Now, if you get something which is weakly convex, you can see by the strong maximum principle, the rescaling, actually, if the rescaling is giving something which is only weakly convex, has to be weakly convex everywhere. If it is weakly convex everywhere, the smallest eigenvalue, one of the two eigenvalues, has to be 0 everywhere. But if it is 0 everywhere, you can see that you can integrate that direction for Binius theorem. And therefore, the surface has to contain a line. So if it is weakly convex, maybe I do the weakly convex case first. So weakly convex. This implies the rescaled surface is some gamma of t cross r. It must contain a line. And gamma t solves the mean curvature flow. And mean curvature flow turns into curve-shortening flow. Curve-shortening flow in the plane perpendicular to the line. But then you use, in particular here, in the non-collapsing estimate, there's the improved non-collapsing estimate by Brindley. So here the improved non-collapsing estimate, remember, showed the inscribed radius after rescaling is less than the mean curvature. That was the last result I showed yesterday, I think. After rescaling, because of this improved non-collapsing estimate by Brindley, mean curvature dominates smooth. But this means if we are, this must mean on gamma, must mean that the inscribed radius is bigger than 1 over the geodesic curvature. But this must mean gamma is a shrinking circle. Gamma is a shrinking. Proves that if we are in the weakly convex case, we get that this gamma t cross r is, in fact, s1 cross r, just a shrinking cylinder. So the weakly convex case is just the shrinking cylinder. We see. Now, the other possibility is that the limit is strictly convex. Now, if the limit is strictly convex, there's two possibilities. It might close up. If it's uniformly convex, then an argument similar to Meijer's theorem. Meijer's theorem is not quite enough to do it, actually. You have to work harder. But sort of think of Meijer's theorem. If it was uniformly convex all the time, you expect it to close up. And using our gradient estimates and everything else we know, we can replace Meijer's theorem to show that. So either it closes up because it's uniformly convex, ratio, the principle curvature is bounded below. But then we are in the case of s2. And that's the first case in the theorem. We have identified the second case in the theorem. We've identified the first case in the theorem, s2. The other possibility is it does not close up. But the only possibility that happens if it does not close up, it cannot be uniformly convex all the time. It cannot be uniformly convex all the time. With the smallest principle curvature, lambda 1 divided by h tending to 0 along some sequence tending to infinity. If this wouldn't go down to 0, it has to close up. But if this goes to 0, then we can rescale again around these points. And then we get back into this situation. Rescale again there. And I say rescale again there. This is an important point. You see, I'm now very, since I'm going out, I'm going away from, I found this strictly convex thing. Now I have to prove that there's a cylinder at the end. So I have to take points. Here's my sequence of points. They go out to infinity. These points are far away from the point where the maximum of the curvature is attained. If I do the rescaling, and I don't have a gradient estimate, I don't get a limit. So the limiting procedure requires the gradient estimate. And if I just have the simple gradient estimate that controls the gradient of the curvature in terms of the maximum curvature, it's not good enough to rescale here. So that's where I use the Harselhofer Kleiner estimate. Remember gradient A less than a constant times A squared. I don't write down all the conditions. But there was this interior estimate, depending on the non-collapsing estimate. Harselhofer Kleiner had the interior gradient estimate because they had the non-collapsing estimate. They didn't leave the improvement by Brentlett. But the non-collapsing estimate gave you this. This allows me to rescale points that are far away from the maximum. And then, because this tends to 0, I'm back in the case, get the cylinder again. I get a point where lambda 1 must be 0 in the limit. But then, there's a strong maximum principle, it must be 0 everywhere. And then, by the improved estimate of Brentlett, it must be, again, after rescaling S1 cross R. This is the third possibility, that I find something first, which is strictly convex. But then, I can go along this strictly convex thing, and I can find a cylinder after rescaling again infinity. So this is a sketch of the first theorem. And then, you have to do this all quantitatively. And using the fact that you can rescale everywhere, you can create this picture easily. So the key thing is the theorem over there, and the fact that we are able to rescale everywhere, and then we can show that, unless we are in the region where the curvature is not so big, we can rescale wherever we want. And we get this thing. It's all contradiction arguments to sum up. It's all contradiction arguments. But given the epsilon, we produce this number H0. And then, we have quantitative control, and can do the surgery procedure, which I'm not going to explain in detail now. Rather, I'd like to show you a little bit what can then be done that eventually follows from this with surgery procedure. So the surgery procedure is the thing that either cuts off this piece. So the surgery, I'm just doing it schematically. It would just cut off here, and it would cut off there, and it would throw away this piece, because we know what it is. We know it's a tiny, arbitrarily tiny tube around some curve. We know what this is topologically. It's an S2. And we know what this is. And this one, we would cut off with another surgery. So in this picture, we would do three surgeries, one here, one there, one there, and throw away the pieces that we know. And this procedure improves the isoparametric behavior of the surface. And I claim, now, the theorem says that, again, Simon, using strongly the surgery procedure that Carlo and I developed in the higher-dimensional case, we can show that, given any initial surface, two-dimensional surface in the Riemannian 3-manifold, embedded h greater than 0, and let's assume the Riemannian 3-manifold is compact, it's sufficient if it is what they say mean convex in the infinity that the surface cannot escape at infinity. So given any such smooth initial surface, there is a solution of mean curvature flow interrupted by, finally, many surgeries, scales at a scale, are zero radius that you cut off comparable to this constant h0 to the minus 1. The area of M and T tends to 0, as T tends to some T max, which is finite. When I say M and T, M and T may consist of, finally, many disconnected components. Even if I start with something which is connected, already after the first surgery, when I cut off this neck here, I may have two disconnected components. So when I write M and T, I mean the union of all the pieces that are still there that I haven't thrown away. So either the area goes to 0 in the finite time, or what else could happen? Well, in the general 3-manifold, the thing could just get stuck on a minimal surface. Or M and T is smooth, or M and T exists on 0 infinity. And they exist sometime T1, bigger than finite T1, such that M and T is smooth. So no more surgeries after T1, Mn, I always write Mn. This is a 2 surface. And M2T converges smoothly to a smooth, weakly stable minimal surface. I draw some pictures in a second, weakly stable. So examples and remarks. So first of all, if you are in R3, T max is less than infinity, because even a complicated surface can be enclosed by some sphere which dies in finite time. So certainly the area will all go away in finite time. So in the Euclidean case, you can just have finitely many such surgeries. And the thing decomposes into these pieces. By the way, you can easily write down initial data of positive mean curvature. Take a thin, very thin circle, axially symmetric. And then you get this picture here, that you can have, in fact, in R3, a very thin torus. And if it is completely symmetric, it will shrink to a circle. So you can have this picture in R3. Now, in the Riemannian manifold, many things can happen. So for example, if the Riemannian manifold looks like this, if this is N3G bar, and your initial surface, see which color we use. Let's use yellow for the initial surface. So if the initial surface is something like this, then the mean curvature vector points in this direction. However, just smoothly, and slowly converge to this minimal surface in here. So this is Mn infinity. And this is Mn zero. Nothing happens. No singularity, nothing, no surgery. It just completely smoothly goes to the minimal surface in there, and that's stable. In fact, it could be weakly stable. If I had attached a cylinder here along here, then it would just be a weakly stable minimal surface. If I draw the picture slightly different, if I draw it like this, and there's no such sink in there, then in this case, I guess, if there's no serious things happening in between, the solution will just keep going, keep going, and end up here, the point, just like the convex solutions in Euclidean space. Let's make the three manifold more complicated. Suppose there's some topology here. Well, then if the surface starts moving, something has to go wrong, right? And well, what goes wrong is that at some stage here, you have sort of, yeah, I should use, yeah, maybe I use white or orange. You have a surgery time here because you have developed somewhere in neck. Here, you do the surgery here. And then the surfaces continue, you have two pieces, one piece here and one piece there. And now, depending on the topology, these could converge to two separate minimal surfaces. I wanted to use red for the limit. So this could be the M2 infinity. Yeah, it's number two. So this can happen, breaks up, goes into two. And of course, I could have put some tentacle here, then you could have one piece being stuck on a minimal surface. This keeps going, and then the other piece could contract to a point, but also be possible. And one other piece is important to notice that, yeah, let's do a application to general relativity. So in general relativity, you have these three many folds which have stars. Here's a star. And this is some N3g bar, where here near infinity, it looks like R3, so asymptotically flat. And you should think of this sitting in some Lorentzian four many fold. And it's a space-like slice. And here you have a black hole. So this is a horizon, sigma 2h. And in the simplest case, the mean curvature of a horizon is 0 in this three many fold. And here you have some star, which means a region of positive scalar curvature. And then you can do the following with this theorem. You could start with some huge sphere near infinity. So m2 0 is the boundary of some huge sphere around the origin in this asymptotically flat region. And then you let it flow. And then at some stage, there's my orange here, at some stage, a surgery will be necessary because it reaches this point. And then after the surgery, these guys here will contract in finite time to this point. And these other guys will take infinite time until they reach this horizon. So you can use mean curvature flow to sweep out asymptotically flat three many folds, the exterior region. So mean curvature flow can sweep out the asymptotically flat region and asymptotically of a space-like slice, a Lorentzian manifold, which models in isolated gravitating system. I should make one remark, I should give credit that I didn't do that yet, examples remarks. And one remark that I didn't make for this long-term existence, that for some time, for some finite time, after that everything is smooth and no more singularity can happen, for this part, we use ideas from Brian White. You see, Brian White has pushed another concept of weak solutions, the level set approach, to its limit. And he has shown that in this situation, you can also have level set solutions. He doesn't have quite as much control on the singularities. He just has an estimate on the house of dimension of the singular set. But he can also prove that his weak solution is going to be smooth near the final minimal surface. And we prove our result by proving that our solution, after a long time, is very close to Brian White's solution. And that's how we get our smoothness for a long time. So we should give a lot of credit here to Brian White. So the theorem here doesn't assume any assumption on the curvature of the three manifold. So for example here, it is known in general relativity that you would like to consider three many faults with non-negative scalar curvature. But for this result, that you can start near infinity and you move until you find the outermost minimal surface, you don't need the non-negative scalar curvature. If you have the non-negative scalar curvature, you can conclude that this limiting surface must be a two-sphere and not a torus or a higher genus surface. So only at that last step, where you show that the horizon actually is a two-sphere in this low-dimensional case, you would use an energy assumption on the Lorentz in many faults or so. Here, this is completely general. But what we can say is, since we're always cutting necks, in other words, we're reducing genus, what we can say is that the genus of this limiting minimal surface, so the genus of FM2 infinity, is certainly less than the genus of the initial surface. It's not so easy. What you cannot do is argue that each time you do a surgery, you reduce the genus. Because sometimes you might cut off a trivial sphere. If you have a neck here, and you just cut off this sphere, it doesn't change the topology. So just looking at how much surgery you had to do doesn't tell you how much the genus has dropped. But you know that the genus will never go up. And I want to give you one more example that can happen. Now, this three-mini-fold could have one of these things that you need in spaceship enterprise. There is a three-mini-fold that looks like this. And it has this bridge here. It allows you to go from one part of the universe to the other part of the universe. And then, of course, your initial surface has to do some more surgery here. And then the surfaces would move in here, and there, and there. And it could happen that there is just one minimal surface in this whole bridge. These minimal surfaces, you would have two pieces, two separate pieces of your solution, which approach the same minimal surface inside the three-mini-fold from two different sides. This can also happen. But this is the worst that can happen. Because there's, again, a theorem of Brian White that you cannot have this would contradict, essentially, the monotonicity formula, density formula, that you cannot have three solutions of mean curvature flow coming arbitrarily close if they are embedded. Sorry, more than three. You can have at most three, one from the left, one from the right, and the minimal surface in the middle. You cannot have more than three. That's the result of Brian White, very general result for minimal surfaces and solutions of mean curvature flow if they are embedded. And you see, it can happen. So for the physicists, of course, they would separate the thing here and treat it as two different ends as seen from infinity. Now, last few minutes, what does this have to do with isoparametric? What can you do about isoparametric inequalities now? Well, it turns out you can now try to use this technique, for example, in such asymptotically flat stream manifolds and say something about the isoparametric behavior of these asymptotically flat stream manifolds. And those of you who know these things in general relativity, there's a concept of mass for these asymptotically flat stream manifolds. Sort of measures how big the stuff inside is. So there's this famous positive mass theorem. If this three-manifold N3g bar has a scalar curvature of g bar greater or equal to 0, and the thing is asymptotically flat, the mass, due to Anouwitz-Deser-Miesner of this three-manifold, is defined by the physicists as some constant by the two-sphere near infinity. So you take huge spheres near infinity of some derivatives of the metric, dGij dxj minus dGii dxj times the normal d sigma. Think of this like a Newtonian, a flux integral, like the metric is the Newtonian potential, and this is a normal integral over the normal derivative of the potential. You know in Newtonian mechanics this would give you the mass, the manifold turns out this flux integral near infinity is under suitable decay estimates of this metric through the Euclidean metric is a geometric invariant that replaces the Newton flux integral. And the positive mass theorem that this thing says that if the scalar curvature is greater or equal to 0, this thing is non-negative with equality if, not only if, N3g bar is equal to r3 with the standard metric, and this positive mass theorem is due to Schoenen-Yau, around 79. And you see this flux integral, at first you don't see that it is independent of the coordinate system. And there's a lot of work to be done. And it uses first derivatives of the metric. So it's not a nice quantity to work with. And positive mass theorem is very, very hard to prove actually using minimal surface theorem and so on. It turns out you can prove this theorem using inverse mean curvature flow. And you can do more using mean curvature flow. If you combine inverse mean curvature flow, which Toti talked about, and mean curvature flow, then you get an isoparametric control on very large regions. Remember, Toti said something about certain integrals being monotone and so on. You can use the inverse mean curvature flow to get a lower bound on the integral h squared, integral of the mean curvature squared, on any outward minimizing surface in the Schoenen-Yau fault. And that was the key step. Remember, at the beginning of this lecture, when I proved the isoparametric inequality in r3 for convex surfaces, the key point was to have a good lower bound on integral h squared. Now, it turns out you can use inverse mean curvature flow on each of these slices that you get from mean curvature flow. So we take mean curvature flow starting at the big yellow surface near infinity, and we flow inverse. On each of these surfaces coming from mean curvature flow, I can use inverse mean curvature flow to get a lower bound on integral of the mean curvature squared. And then I use that lower bound to prove an isoparametric inequality. And that isoparametric inequality in the end turns out to characterize the mass. So the theorem I want to finish with is that you can characterize the mass of such a 3-manifold as the isoparametric deficit. So you take the limb-soup domains of the boundaries of domains tending to infinity. So take larger and larger domains. And then you take the extra volume that you can put in compared to the Euclidean thing. And the Euclidean thing, we have the isoparametric inequality would say that of the boundary to the 3 over 2 is always bigger than 6 square root of pi times the volume. So in Euclidean space, this would almost be negative. But if you are in this manifold and the mass is positive, if the mass is positive, you can do better. And you just have to rescale the things. You have to put a 2 here and divide by the area because mass scales like area. So you have to put in the right scaling. But then you get this thing. You can characterize the mass by how much the manifold differs isoparametrically from Euclidean space. And this is the one result that I could explain what I'm doing to my teenage children. I said, if you have a trampoline, then if it's flat, nobody on the trampoline, then the isoparametric inequality tells you that with a rope of length 2 pi, you can bound an area equal to pi. But if you, my child, stand on the trampoline, you have this tent. And then there's a mass on the trampoline. And then because of this tent, with the same rope, you can bound more area as just pi. And if I'm standing on the trampoline, it may get the black hole. But anyway, you can get more area. And it is exactly the same thing in general relativity. If you have a heavy star bending the three manifold that slices through spacetime, then you can put more volume into the same area than you can do in Euclidean space. And you can, in fact, use the isoparametric defect. How much more volume you can put is an exact measure for the mass in general relativity. And I think that's a good point to stop.