 So can you hear me? Well, I would like to thank the other organizers for having started the organization. I was contacted at the later stage, and I am happy to be here, to have collaborated to this event, and also to be a speaker here. So what I will talk about will have strong connections with the lectures of Huiscan, in particular. So I will also talk about mean curvature flow. We'll especially talk about the formation of singularities, the analysis of the profile of the solutions where singularities form. Mainly for a specific class of hypersurface is evolving by mean curvature flow that have some convexity, more or less strong convexity properties. Then I will present some part of the previous work with Huiscan, where we did some years ago a surgery procedure for the mean curvature flow, a way to continue the flow after singularities, which is suitable for topological applications of the flow. So before starting talking about some results, I would like to spend some word on the background of this theory. So how did mean curvature flow came in the interest of mathematicians? Because this is something I do not know in detail because I am not that old, but for what I know from the literature. As you have already heard in the previous lecture, one pioneering work in forming curvature flow is the work by Breckin, to remember the exact year, I think it was 74 or something like that. This was somehow motivated, I guess, it was clearly influenced by what was going on in geometric measure theory, minimal surface theory. So somehow mean curvature flow can be regarded as the evolutionary counterpart of the study of minimal surfaces. And at that time, there were a big theory going on, and somehow Breckin got the inspiration to go on in the parabolic part of that theory. But also, if you read the beginning of Breckin's work, there was some motivation coming from physical models, some physical phenomenon where some interface was evolving, and there were maybe some surface tension involved, and this creates some term involving the mean curvature. So there were also physical models. In fact, there was already a paper in the 50s by a guy named Mullins who considered, in the plane, solutions of the mean curvature flow, self-similar solutions. And he was motivated by a certain phenomenon of metals where some interface was evolving by mean curvature flow. Somehow independently from this, there was another line of research which led to mean curvature flow. And it was mentioned by Claudio this morning. There was a paper by Ilse and Samson. Again, I'm not sure about the exact year. I think it was in the 60s. About harmonic maps. They wanted to show existence of harmonic maps between two given and remaining manifolds. The idea was very roughly speaking, take any map and then let it improve. Let us make it harmonic by using the heat equation. We take an arbitrary map as a starting point for a heat equation. We studied the evolution, and they were able to prove that under some assumption on the sign of the sectional curvature of the target manifold that this heat equation for maps has a long time existence. And as time goes to infinity, it converges to a harmonic map. Having this as an inspiration, there was a well-known spectacular result by Richard Hamilton in the 80s who introduced the so-called Ricci flow. This school is a school on extrinsic flows. So flows where the extrinsic curvatures are involved. The Ricci flow instead is an intrinsic flow. It's the evolution of the metric of an abstract manifold for we will not focus on Ricci flow in this school. But it is worth mentioning it because there are connections between Ricci flow and Min curvature flow. Similarity is more than connections. And Ricci flow has provided a motivation and an inspiration for a lot of work of Min curvature flow. What I will talk about is largely influenced by work on Ricci flow. And Hamilton proved a result in Riemannian geometry which was this one. He proved that if M3 with some metric g is Riemannian manifold compact with positive Ricci, then it is different morphic to a sphere, quotient of a sphere by some discrete group. So the results of this kind are called the sphere theorems in Riemannian geometry. They are very important. And there are various kinds of results ensuring that some curvature assumption on the manifold implies that the manifold is different morphic or homomorphic depending on the cases to a sphere. But what was surprising at that time was the technique that Hamilton used. So until that time, the results of this kind were not proved using PDEs. Roughly speaking, techniques were, for instance, Morse theory or techniques inspired by Morse theory. So the hypothesis on the Ricci curvature was some hypothesis on the curvature could be used to study the behavior of geodesics of the geodesic flow to construct the formations of some parts of the manifold to prove such results. But Hamilton introduced a completely different technique. And he acknowledged inspiration by Ilz and Samson, but his idea was quite different. He said, well, let us take this metric and let us evolve by a sort of a heat equation. Heat equation should average out everything with time. So let's hope that it converges to something with constant curvature. And since the evolution he found that was the famous Ricci flow preserves positive Ricci, it had to be a metric with constant positive curvature. That is a quotient of the sphere. So he somehow constructed a continuous deformation from the initial metric to the spherical metric provided by PDE. And a central tool in his analysis was the maximum principle, which will be the topic of this lecture. And somehow I think at that time it was surprising that someone could prove such differential geometric result basically by using repeatedly and in a clever way the maximum principle. But probably this is also the fascinating things about these flows intrinsic or extrinsic that they mix analytic and geometric features, both play a role. And let me also mention that, as you probably all know, these ideas were carried further in the following decades. And Ricci flow has been used to prove so one historical result that is the Poincaré conjecture, again with a program started by Hamilton and then to a conclusion by Perelman. And also the differentiable sphere theorem by Simone Brender and Rick Shane. And in the next lectures I will talk more about this technique that has been used for the Poincaré conjecture because what we will see is an analog of this technique for the mean curvature flow. But so I said this because this paper and later paper by Hamilton on Ricci flow has influenced a line of research on mean curvature flow. That is, shortly after this paper by Hamilton, there came a paper by Gerard Wiskern on mean curvature flow this time. And the result of this paper was the following, as you well know, result if Mn in Rn plus 1 is compact convex hypersurface. And if you consider the mean curvature flow Mnt starting from Mn, then it exists in a finite time interval with the finite. And as T goes to this singular time, the hypersurface shrinks to a point. And if one performs a rescaling which keeps, let's say, the area constant, it converges to a round sphere. So it's a sort of extrinsic analog of Hamilton's result. And the proof has some analogies and also some substantial differences with Hamilton's one. But in both results, the maximum principle plays an important role. So this is just to end this introduction to give some motivation. This is a first case where we have, so to say, an analysis of singularities. So typically, closed hyper surfaces evolving by mean curvature flow shrink. Therefore, they become singular in some way. But this case is showing that the singularity is not really something bad, because it's just a change of scale. If we remain at the same scale, the surface is actually improving. From a general convex, it's going to always the same limit. And the purpose of the later research is to consider more general initial data, so both in the richer flow and in the mean curvature flow, and try to understand what happens in the singularities. One, again, sees that when a singularity is formed, then typically there are only very few possible behaviors. And this allows to say something about the possible initial structure of the manifold. So eventually, in the richer flow, this has led to show that dropping the positivity of Richie and just requiring simply connected then the same conclusion holds without the quotient, of course. We will see a strategy of this kind for the mean curvature flow, where we are able to consider a class of hyper surfaces, which is more general than convex, although it has some convexity restriction. And showing that we can do mean curvature flow where we continue the flow after singularities using what is called a surgery procedure. And this allows us to say that the initial hyper surfaces had to be a certain structure. So this is what we are aiming to do. But let us start from probably the tool that is mostly used in this theory, that is maximum principle. And I guess that you should be familiar with some, at least, basic version of the maximum principle. So I will not start really from all the details. But I would like to explain in detail also some something which may be known to many of you. But let me just repeat it. And that is, well, you know that basically maximum principle says that if you have an elliptic equation with some restriction on the zero-order term, the maximum cannot be attained, elliptic equation in a bounded domain. The maximum cannot be attained in the interior. It has to be attained on the boundary. But something similar for parabolic equations, the maximum solution cannot be attained, is either attained at the initial time or at the boundary of the domain. We will consider this principle on a manifold. And if we consider, as I will do, a compact manifold, this makes the statement easier. Because in the Euclidean case, either you have all the space, then you have something non-compact which causes some additional difficulty for maximum principle, as Totti was recalling this morning. Or you have to consider the boundary also. But if we consider a compact manifold without boundary, the statements become particularly easy. So I will consider a manifold M, a dimensional manifold. G is a metric on the manifold which, in general, depends on time. For our applications, it will usually be the M will be a hypersurface evolving by mean curvature flow. And G of T will be the metric induced by the immersion at time T. So M and M will be compact. Then we have a Laplace operator induced by the metric. So it's time dependent. When we write it in coordinates, the coefficients will depend on time. So I will always write this way, but you have to remember that it is not a fixed operator, like, for instance, in Hill-Sampson work, but it is a time dependent operator. Then we consider a solution of this parabolic equation. Consider U function on a time interval. Consider parabolic equation. We could put more general elliptic operators, but for purposes Laplace is enough. Then we can make a coefficient first order. And then what I would, it is important for us, let us also consider some nonlinear term, F of U. Maybe we can let me write it this way. The same could hold also with the time and space dependence, but we, for simplicity, let me write it this way. So what I would like to state is a comparison principle. For a comparison principle, it is useful to, so this is some given function, and phi is some smooth, in general, possibly nonlinear function. And I use the Einstein summation convention. So repeated indices means that we are summing over i. And we can also consider the so-called sub-solution and super-solutions. So sub-solution, if we consider U of dt less than or equal to, and the super-solution, if we have greater than or equal to, then we have this statement. So let U2, respectively, sub-solution and super-solution of the equation, let me call it P like parabolic. And suppose the initial time, the sub-solution, is less than the super-solution, then the same holds for positive time. And well, I'm sure that most of you know the proof, but let me just recall how it goes. I recall the idea under, I make a stronger assumption, which makes the proof faster. I consider the case where all inequalities are strict. So I will write the proof assuming that here we have a strict inequality, and we have also a strict inequality here. Then we also have a strict inequality in the conclusion. And so how it goes, since we assume a strict inequality at time equals 0, so suppose the conclusion does not hold for all positive time, then there is a first, we can consider the supremum. So intuitively speaking, the first time where it does not hold, we consider the time such that the set of all t such that this holds on 0t, and we take the supremum, which is not capital T because we are assuming that theorem is false. Then there is first time t star such that u1 of x star t star is equal to p star t star for some p star in m. And at that point, we find the contradiction because we see that what do we have? We have that. So by continuity at that time, we have that u1 is less than or equal to u2 because at all smaller time, we have the strict inequality. So at t star, we have inequality with less than or equal to. So we have a certain point, possibly others, where u2 is this one, u1 is this one, this is p star. And then what do we have? We have that since they are smooth and they are touching, they have the same gradient. u1 dx i are equal to du2 over dx i at p star t star for all i. The function above has a greater Laplacian, delta u2 is greater than or equal to delta u1. And also, since this is the first time where they touch, it means that u1 is growing more rapidly than u2. That is, we have this inequality on the time derivatives at that point. And finally, since the two functions agree, the last term is equal at that point. Phi of u1 is equal to phi of u2. And then you find the contradiction because you see that if you compare, if you plug all the values here, you see that this is in contradiction with the fact that having a super solution and a sub-solution from this, you see that the left-hand side is greater than u1. The right-hand side is greater for u2 while the property of sub-solution and super solution says the opposite, that left-hand side should be smaller for u1 than for u2. And then you find a contradiction. And this argument is immediate because we have assumed strict inequalities at all these places. It may look that if we only have no strict inequality, then anything that we don't find a contradiction because we could have equalities everywhere. But it is possible to do some standard perturbation to replace our true functions with some perturbations that satisfies a strict inequality. To argue on the perturbation and then to show that by letting the perturbation go to zero that the original function also satisfy the inequality that would be in this case non-strict. So I hope that it was not too easy for you. And so let me show some corollary. An easy corollary is if phi of zero is greater than or equal to zero if we consider the equation like this. And let's say solution or also super solution of p is greater than or equal to zero at time zero. Then u is greater than or equal to zero for all greater times. Because we make a comparison with zero. So we take u1 equal to zero identically, which is, of course, a solution or a sub-solution because of this hypothesis. And u2 equal to u. And then we have that u is greater than or equal to zero. And an example where we can apply it is the equation of the mean curvature. If mt evolves by mean curvature flow, the mean curvature on m satisfies the equation we have seen this morning. Let me state it in a general Riemannian manifold while most of what I will talk about in the rest will be in Euclidean ambient space. Well, in this case, this is not really just a function of u alone. It also depends on, can be regarded as a certain suitable function of space and time. But you can see that the same proof works with space and time dependence here. So we can apply this corollary and show that h greater than or equal to zero is preserved by the flow. So this is one of the convexity properties which is invariant under the flow. And a remarkable fact is that this property is invariant also in a Riemannian ambient space. The other properties we will talk about are more sensitive to what is happening in the ambient space. Since the result I've talked about is not a full force of the maximum principle because there is a well-known other result called the strong maximum principle, which I will just state. Strong, let's say, comparison principle, when it is stated in this way is more commonly called a comparison principle, says that under the same assumption as in the theorem that we have the strict inequality for positive times, except for the obvious case where the two functions are identically the same. So even if the functions are touching somewhere at the initial time, they detach immediately. And then we can also say that not only no negative curvature, mean curvature is preserved, but actually if you start with something which has just no negative mean curvature, it immediately becomes with strictly positive curvature. So let me now give some other less immediate application. So let us consider a function on a manifold which are the restriction of something which is in the ambient space. So consider MT in Rn plus 1, evolving by mean curvature flow. Let us consider some function on F from Rn plus 1 to R. And we can consider a restriction on F on MT. Let's call F as F tilde of Pt is equal to F evaluated as on the point of the immersion. Capital F is the immersion. And I want to choose a certain suitable function, small F, which give me useful information on what our manifold is doing. For this, it is useful to, since we want to show that it satisfies some parabolic equation, it is useful to have a way to compute the Laplace Beltrami of F on restricted on a manifold. So if we have MT, sorry, F tilde, oh, sorry, yes, actually, yes, you're right. I want to have a F which depends on the point in ambient space and time. And so let me, I recall your formula for the Laplace Beltrami, which is let me consider for a moment the case of constant F. Let me call it, if you have H from Rn plus 1 to R, and you have M on Rn plus 1, a hypersurface, then you consider H tilde, H restricted to M. Then the Laplace of H, I put this M to contrast it with Euclidean Laplace. What's the relation between the two? Well, you have an obvious difference on the Laplace Beltrami. You are taking second derivatives only in tangential directions. So you are missing the derivative in the normal direction. So you should take away the second derivative in the normal direction. But, well, naively, one may think this is all the story, but there is an additional term because somehow the fact that the surface is bending due to curvature is influencing the way the second derivative acts. So when you do the second derivative in tangential direction here, you are going straight on the tangent plane. When you are computing the second derivative here, you are bending. So there is an additional term which goes this way, which is the mean curvature times the scalar product of the gradient of H in normal direction. So this will be useful to make some explicit computation. And so in our case, let us use this formula. Let me make an easy example. Suppose that our initial hyper surface, where we assume that it is compact, let us take a sphere which encloses it. For simplicity, let us assume this is the origin. Let us call r0 the radius. I will show you that the maximum principle, the scalar maximum principle gives us an easy way to compare the evolution of this with the evolution of the sphere. Let us just define f equal to r0 square minus, I will call y the point in rn plus 1, a norm of y square minus 20. Well, let me change signs. So we have that it is negative at the initial time. Our claim is that it remains negative. And how we will show it? We will show by computing the equation satisfied by it. So df over dt, df tilde. So f tilde is defined in that way. So df tilde over dt has two contributions. It has one contribution just from the time derivative and then another contribution from here. So it has a, let's call it, let us denote by d, the gradient in the space components. So df times df dt, so the velocity of the point plus df dt. OK, what is the gradient of this function? The gradient of the function is just two times y. Y is the first entry of f, so it becomes f. And what's the speed? This is the mean curvature vector. So mean curvature minus mean curvature times the normal. And then we have df over dt, which is just minus 2n. OK, and let us also compute the Laplace of f tilde. So what's Laplace of f tilde? We use this formula here. We are in Rn plus 1, so the Laplace of this expression. So it is we sum n plus 1 times 2, which comes from the double derivative of a square. So we have 2n plus 1. And that's it. This does not contribute to the plus, but to the plus. Then we have to subtract a second derivative in the normal direction. Well, this again, that doesn't matter, which is the normal direction. This will always give a 2. Then we have this term. So we have minus h times the gradient of the function, which is, as before, 2f times nu. But if you compare the two expressions, of course, this goes away with this. You see that they are just, well, they should be the same, except for be the same. So I got a sign wrong somewhere. Sorry? Do I need to show you? Yes, sure. There's just a plus here, you guess. Yeah, I'm sorry, I mixed up. Yeah, sure, OK. Yeah, I made an inconvenient sign in the definition of f. So let's write it this way. Then we have a plus. Then the two things agree. Then it just satisfies the heat equation. It satisfies the heat equation. Then, in particular, 0 is also a solution. Then, by comparison, we have f smaller than or equal to 0 at positive time, which means that y square is less than or equal to r0 square minus 2 and t. Yes, this sign is right. Thank you. Which means that this is just the radius of the shrinking sphere, which we have seen this morning. This means that mt remains inside the shrinking sphere. And in particular, this shows that it cannot exist when the right-hand side becomes negative, because we would have a contradiction. So this, in particular, shows that the maximum time of existence is less than r0 squared over 2n. And also that, for t less than the maximum time of existence, the evolution of our hypersurface is contained in the evolution of the sphere. And this apparently implies that this hypersurface shrinks to a point, because since the sphere shrinks to a point, it seems that this stays inside the sphere, and it seems that it should also shrink to a point. But you should keep in mind that these properties hold under the assumption that we have a smooth solution. So we only have proved that it stays inside the sphere as long as it stays smooth. It could become not smooth before shrinking to a point. So this is the first application. And there is a more sophisticated application where we can show the occurrence of a different kind of singularity called the neck pinch. I told you that a Wisconsin's result says that every convex hypersurface shrinks to a point. One could think maybe all hypersurface do the same with the main curvature flow. They shrink to a point. But it's easy to convince ourselves that this is not true, and there was this example that was already conjectured at the beginning of this theory. This consider M0 made like this. M0 hypersurface in Rn plus 1, greater than or equal to 2. We have here maybe we can make it longer, a tiny little tube. Since this is not a curve, but it has some tiny neck, then it has some big positive curvature. So here, h is very large. Instead, here, h is not large. So intuitively, you expect that for a small time, this part of the hypersurface will not move much. Instead, since here, the curvature is very large, and the radius is very small, a tiny amount of time will suffice to make this neck shrink. And so to have the behavior which is called the neck pinch. And for some time, this was just a conjecture. Then some rigorous proof came. And in fact, there is a rigorous proof found by Klaus Hecker by a very simple argument, which is just a bit more elaborate than this one. You can take a function of this form. You take this function here. It is possible to show that by computation similar to the bit longer, but similar computations, one can show that it is not really a solution or the heat equation, but it is a sub-solution. So again, if it is negative at the initial time, it stays negative. What does it mean that it is negative at the? Yes, this on any main curvature flow, without assumption on the hyper surface, we then can obtain this that if let's consider 1 as a reference value. If f is smaller than 1 at time equal to 0, then f is smaller than 1 for time greater than 0. What does it mean? F is smaller than 1. This means that if we denote the, well, you can write it in this way, this means y1 squared plus yn squared. Then we have 1 minus, we have minus, OK. This means that this is less than 1 minus t. But this is the equation of hyperboloid. Let's, to fit with that picture, we call n plus 1 this direction. And all the rest will be the other ones. And you see that for fixed yn plus 1, you have something hyperboloid. So this means that the hyper surface is contained here inside. So this shows that if the radius of this hyperboloid is become smaller and smaller, when t is equal to 1, the hyperboloid becomes a cone. So if you start with a hyper surface, which is contained inside this hyperboloid with radius 1, then it will be still contained inside this hyperboloid. And if, on the other hand, you make these two pieces large enough as to contain spheres that shrink to a point in a larger time than 1, then this means that the surface cannot go all on one side or on the other. And then somehow it has to shrink at this central point before the two other parts have become singular. And therefore it shows that there exist solutions that do not become singular by shrinking into a single point. This prevents it. OK. And I'm sorry that I was much slower than expected because I also wanted to say something about maybe just let me mention something that I will tell in more detail in the next lectures. That is, we have said that positive mean curvature is invariant under the flow. This is a property which means that the sum of the principle curvature is positive. It is sometimes called the mean convexity. But one could be interested in maybe a more common property. What about convexity itself so that each single curvature is positive for mean curvature flow in Euclidean space? Because this property is false for, in general, not true in Riemannian ambient space. Then convexity is equivalent to Weingarten operator positive semi-definite. So we do not deal with a single function. We deal with a set of functions. Then there is a maximum principle for tensors due to Hamilton, which can be applied to the equation satisfied by this that shows that this property is preserved. I have no longer time now, so I will talk about this in the second lecture. So thank you for your attention.