 I cannot see. Did you make me, do you want me to be a co-host just for safety reasons or? You are a co-host. I'm a co-host. Yes. Absolutely. Okay, so I think if everybody's ready, let's start. So welcome everybody to this basic notion seminar that we are having. And I'm very happy to introduce today Professor Giovanni Bellettini from the University of Siena and of course also ICTP, who will speak today about a very nice piece of differential geometry, namely the mean curvature flow of closed hypersurfaces. So the floor is yours. Thank you. Thank you a lot for the introduction. So this is a basic notion seminar. So I will try to be as elementary as possible and keeping the notation and the technicalities to a minimum. So let me first define what is mean curvature flow. And then I will discuss some examples. So the definition can be given in various ways. I will use the definition without the parametrizations, which may be simpler. So let ET be a family of solid sets contained in a REN. So for any T, which is a parameter, which is time ET is a solid set. And solid means say the closure of an open set. And I say that we say that T in ET or say T in the boundary of T is the same. I will somehow speak about evolution of hypersurfaces. These hypersurfaces are boundary of something. So for T say in some time interval, zero capital T, we say that this is a smooth mean curvature flow, which means that at the end the surface is the hypersurface will move by mean curvature if we define the following function. So this is the distance function with the sign is negative inside the solid set and positive outside. So this is my set, this is ET at fixed time. If this function is smooth, is smooth in a neighborhood in an open set containing all the boundaries. So which means essentially that the flow as a function of time is smooth and then also each boundary is smooth. So this is the following equation holds for any time. This is the PDE. This is the time derivative. This is function. This function is defined everywhere actually is smooth in an open set containing all boundaries. This is the gradient in a REN. This is the Laplacian in a REN. So this is say if you want the heat equation, this term here equal to this, but for the distance function with the sign and the gradient of D is the unit outward normal on the boundary. This is called mean curvature. Sometimes denoted also by H is a scalar mean curvature and is equal to the sum of the principal curvatures. Notice that this is a PDE, which is however valid on an hypersurface and this hypersurface is moving. Okay. So this is just valid on the boundary. So this is, this is called the normal velocity. So at the end, I'm saying that the normal velocity at each point is equal to the mean curvature. The mean curvature vector. So which means that if I have an initial set, say initial curve, so this is the interior, this is the boundary. It means that it means that at each point, the velocity here, the convention is that the velocity is where the curve shrinks if I am locally convex and here instead expands locally. So the evolution here, the normal velocity here must be proportional to the mean curvature here. Okay. And this must be valid at all, at all points of the curve. Okay, why I have given this definition. Well, this definition is you see there are no parametrizations. This is convenient in some sense. But there are other ways to define. So this is a remark. So there is a remark here, which says that there are other ways in particular. The flow can be defined and studied describing the manifolds instead that set of points using the maps immersion or embeddings of a given manifold using parametrizations. So I will not use probably here. This language, which is very convenient. For at least two reasons. Because it can be useful to treat immersions, not only embedding. So this means that my flowing manifold could self intersect. I will avoid here this case, but in general using immersions. This can be done. And another reason for which it is very convenient because it takes advantage of the covariant differential of the covariant language, covariant derivatives. Okay, so there are at least two ways to describe that this is maybe, maybe simpler, but maybe not not always the most convenient way actually. Okay. Why this, this problem. So another remark. This is this evolution is the fastest way to decrease area of the moving manifolds. So this means that in a sense, this flow, try to evolve this curve, say, as time is moving is going on. This tries to reduce the area of the flowing surface as fast as possible. This is not so easy to not so easy to make rigors, because typically, I mean, people says that this, this is the gradient flow of the area functional, which essentially would mean that this is sort of generalization of in. If I had the system of ordinary differential equations. This is the gradient flow of F, which goes trying to go towards a minimum of F in the fastest way. In this case, this is the analog of this but the difficulty are several because here we don't have say a vector structure. Maybe, and it's almost it's very difficult also to say what does it mean gradient of the area functional in difficult to make rigors also what is this for F area of some surface. This can be made rigorous and if you want to take to to have really a rigorous way to to understand this, this can be for instance, obtained from an algorithm by Algren Taylor and one in 1993, which is a sort of, which makes somehow, which gives a sense to this idea for when F is the area and uses a sort of implicit Euler scheme for the mean curvature flow. Okay. Examples it is always bad, very good to have several examples here in mind. So let me give you the basic examples. First example is is suppose that this is a minimal surface, which means a smooth hyper surface with zero mean curvature. So this is a static solution of mean curvature flow starting from Sigma is from this itself, because this is a zero mean curvature so zero velocity doesn't move it remains fixed and this is a solution which doesn't move. This is a very trivial example but very interesting because it could give barriers. Example to here suppose that you have an initial set, which is the interior of a solid ball in a rain. Okay. The solution a posteriori I mean a solution but at the end the solution is still given by shrinking sphere where r of t. So this is called where r of t is this radius here so this this fear as constant mean curvature. So it shrinks remains a sphere remaining the same shape and shrinks to a point here. You see, this is interesting because shrinking to a point can be considered an example of singularities. Of course a singularity should be defined but this is a moment in which the curvature blows up. And this moment is when t is equal to r square divided two minus one. So let me call this T max. So this, this evolution is defined up to T max actually. So T is from zero to T max actually from minus infinity to T max but for the moment, think that we started from our zero. And then we shrink up to T max and this T max. So the third expression of the T max here you get that and you compute the length of the second fundamental form, which is the sum of the, in this case they are all the same. If you compute this, this turns out to be a constant. T max minus T. So the length of the second fundamental form that is the square root of the sum of the squares of the old curvatures and this constant is one of a square root of two. In any case, you see that this blows up when T goes to the T max with a certain rate. Example two, example three is cylinder. So you say S one times R. So suppose that you start from say B or zero times R or more generally by you start from S M times R and minus one minus M. So you have a cylinder and the cylinder shrinks. This is non-compact by the way. So there is a serious difference between surfaces which are compact and many forms which are not compact. This is a non-compact case. One has to be very careful treating non-compact initial conditions. But at the end, if you have a cylinder, this shrinks remaining a cylinder and you have that this evolves with RT and this RT is the same as here and you just substitute this with M. And you get the same. So at some time there is a singularity and what you see is sort of the cylinder collapse to align to its axis. At some time and you have for similarly a blow up of the length of the second from the mental form. There are other important examples here. Other important example, there are very many other solutions. So example. For So these are graphs. They on R and minus one. So you can suppose that your initial condition is unbounded. So you have a graph over an entire graph over R and minus one. So you are still in the non-compact case. It's delicate, but you can at least have a you can at least say what model this using just one parabolic PDE non-linear PDE. And then you have with your graph initial. So the boundary of the initial condition is the graph 0 say given and this can be studied and it can be proven that under some conditions. There exists a solution and it remains a graph. So the property of being a graph is not destroyed by by the the the mean curvature flow. Okay. This is another geometric factor. Next, the next examples. Next examples are given by the so called the homothetically shrinking solutions. For example, five homothetically shrinking solutions. I don't want to enter here in the details. So this is a very big subject. Let me tell you that the sphere is a motetic why is a motetic because its shape remains same up to a rescaling and the same for the cylinder. So you can try to look for special solutions of. So you mean you have this PDE and you look for say special solution with time and space maybe independent. So special solutions. Solution where specials mean which are say is an homothetical a given initial condition. Several, there are several solutions. Actually, this is an important subject because understanding. Let me tell you one thing. So understanding how these solutions is strictly related on how this singularize. I have to define what does it mean but I as T that goes to T max in some sense. So this is for the moment, not rigorous because I have not defined what is the max yet. I don't have not tell you that there are singularities but that we just point out that the study of this special homothetically shrinking solutions is interesting by itself and also because it gives you information on how possible singularities may form for time slightly smaller than capital than T max. For instance, here, this shrinks to a point like a sphere this shrinks to a line, etc. In particular, there is an interesting solution here. Interesting solution. Similarity which is called the Anganand. Taurus, which is sort of so you have your. This is your the solid torus is your smooth initial condition. And there is a special torus, which remains, which is has the same shape and this shrinks. Remaining torus. Okay. Notice that for fat torus somehow or a torus of the sort to one can imagine that here. Maybe the door is so fat that this whole whole maybe closes before that the torus disappear at all. Maybe at some time there should be something strange here on this point. And for a thin torus one could expect that instead here you see in torus means that here the mean curvature is very, very high toward the inside. And so this means that one can expect a normal velocity very quick toward the inside. So one could expect that before, if the torus is long but thin, one could expect that this total could collapse somehow to a to a curve here before disappearing. Okay, so all this kind of studies related to this to this could be related also to this to this problem here. So let me tell you now some result here. So the first theorem obviously, which is not trivial at all is the local existing. Say, if is smooth and compact, then there exists a unique mean coverage of flow, starting from, from this initial condition is smooth mean coverage of flow, starting from this initial condition. There is a positive time for some positive time. And actually, we can say also that we can we can arrange things so that there exists a maximal on a maximal time interval. So here means that you cannot go on after the max that you are you're not smooth anymore, I will be more precise in a minute. What happens to the second fundamental form as when I go to team max but this at least allows to start to the flow. So we are sure that there is a unique solution or smooth up to some maximal time. So here's the theorem is not trivial the first proof maybe is. I don't know, but I think it's gauge Hamilton 86. Then there is a proof by Evans. Prack, using a completely different proof of a completely different language 92 then there is another proof by whiskey. I think. Okay, so this is the first local in time existence result. Let me tell you a remarkable property of this flow. Which is which I will try to describe remarkable property of comparison property of comparison. Suppose that you have an initial solid set smooth with bound say compact with the compact boundary. Suppose that you want to do to smooth initial solid set so one inside the other say one is this, which is inside the big sphere I don't know so. So this is the one at the initial time this is two solid sets one inside the other, and then take their unique mean curvature flow. So for all these times these capital T is positive this is what it starts from me one this is more than starts from me to they are uniquely defined by this previous theorem one. Okay, then if initially one is inside the other. The same is preserved. For all. Times. This is actually very strong and important comparison principle is an important tool. And it's called comparison principle. So let me try to tell you why this this should be expected to be true. I will not do a proof because there is no time enough but I can at least tell you something which could suggest why this is true. Which could suggest. Why this is true. So let me draw a P and it's a simple picture suppose that we did we are in the case of the plane so we have the one and we have say it to and the one initially time zero is contained in it to suppose for simplicity. And the boundary of the one is at the positive distance from the boundary of it to just for simplicity. So we are not actually in this picture but we have say something like this. Okay, so you want. For simplicity. Now. We have a number of compact curves close the curve send. So that there are at least a pair. Maybe many pairs, but in this picture just probably one pair, which minimize the distance between the two boundaries along the flow. Take suppose this is one T to T for they they are once in its initially there they are one strictly inside the other the flow is smooth so for small time there, of course, still one inside the other. But what happens for larger times. So take a point a couple of points here, which realize the minimal distance between the boundary of. So let me call the of T distance. So this D of T is the distance so it may be realized by several pairs here in this picture. We have just two pairs probably. And the idea is to show that actually a statement stronger than this, not only they are remain inside about the distance is non decreasing. The idea is to show that the T. Is non decreasing, which means that for larger times these shrinks and this for this this quantity is maybe larger. Okay. Now, we have to be careful because this is a very simple picture but you have to imagine something more complicated with those torus inside the other. So I mean, this is just a very simplified picture. So the idea to show that this this is not increasing is to find a different I mean to show that the prime where it exists. Supposing it exists. To show that the prime is larger equal than zero and two and one actually one shows that if you complete the prime. I mean it's not actually the prime because this the prime could not exist but this is sort of right derivative, but essentially the prime. One can show that, essentially, this is given by the difference. I mean this is these are two points which realize the minimal distance is not difficult to show that the tangent plane here and tangent plane here must be parallel. So this means that the normal is the same parallel they are parallel the normal. It is possible to show that this is equal to be one minus V two very one say V. I for I equal one two. So this is the difference of the two velocities. So but this is actually h one minus h two because they have the evolution equation so the velocity is equal h one and this is equal to h two. So why this should be positive or negative say. Well, the idea is that you think that now this is a point of this minimize the distance or if I make a translation here, rigid translation of this set and they put in contact. Let me translate this rigidly and put it in contact. So they have the same normal. So actually what I see locally here around this point is the following situation. There is a manifold here and another one which touches and locally. This is the largest and this is the smaller. Now, if I, I mean, the mean curvature is invariant under rotation and translation so I can do I can put arranged things that this such that this is horizontal. So this is somehow horizontal this point. And then and then if I compute the mean curvature at the point where I'm locally the graph horizontal graph like this I compute the curvature here. I mean curvature turns out to be the Laplacian of the function which described locally this is a graph. So at the end I have Laplacian of you one minus Laplacian of you to this only at the points where I am where the tangent plane is horizontal. And then this is a local I mean if I take the difference. This is the Laplacian of the difference and this is a local minimum and it's immediate to check that using minimality for a function that should be non non negative definite and you are done. So this shows that so this should be expected this is not the proof but maybe the idea of the proof. And at the end is that we can compare them in cover show of the set with the big with the mean cover cover show what another set which is inside at the contact point. Okay, so this is an important thing and then let me tell you. First, remarkable, I would say a difficult theorem. This is split into two because historically it splits into two. So, let me tell you your M three. This is gauge Hamilton. X. So suppose that you have you are in two dimensions so this is mean this is curvature flow shortening flow, a curvature flow of curves. If you have a convex say smooth suppose that the this is smooth and convex and bounded. And then, then its evolution flow exists and remains is convex remain convex for all time. So the max and at the max. At the max. The curve shrinks curve shrinks to a point. So nothing can happen before then shrinking to a point like in the case of the sphere, the curve shrinks to a point and once rescaled. It is close to a circle. The start from a convex curve what I see of course the evolution is not explicit so but it remains convex there is a magic point where it must shrinks and nothing strange happens before and here. What I really see is sort of almost a circle which shrinks and the circle is remember is a self similar solution. Okay. The theorem. Has been improved by remarkable. Theorem by Grayson now I don't remember exactly the year. Come here later. Grayson, who shows that actually. Grayson was able to remove this assumption. Suppose that now your curve is compact as mood. But not convex. So, this means that my initial curve. Can be very embedded, of course, but very embedded sorry maybe this picture makes not so clear it is embedded. So this zero. So what happens here is able to show that it remains smooth. Remains smooth. Of course the max is finite and all these cases are. So it remains smooth. And at some time at some tea. This is becoming convex. And so, once it becomes convex. So it's able to prove a say global existence theorem in the sense that. If I start for any smooth compact embedded curve. This doesn't do any, any nice, any nasty thing. I mean, it remains smooth at some moment becomes convex, then it remains convex, then it shrinks to a point that becoming a once rescale the circle. This is really remarkable because you see, you can imagine that the curve. Can be initially pass very close to itself, even if not self intersecting but. Can pass very close to itself. But still and one could imagine that at some moment that could be self intersections and this cannot happen by the theorem of Grayson so it is embedded remains and bad that nothing stage happens and this is the first. Now for this theorem there are more than one proofs. There is what the original proof of Grayson then there is a there is a proof by whiskey, which for proving this passes through some classification of singularities. So also this proof is difficult. I mean in some sense. Also this is not easy at all and then there is a more simple proof I think by Ben Andrews. Another proof. Okay, now let me tell you another remarkable theorem here. Another remarkable theorem. So this is what concerns so you see in two dimensions for cursor actually singularities is the singularity of the of the sphere of the circle. In some sense. To see more complicated singularities in the plane one should start with the, as immersed and not embedded curves. So one should start with something like this say, or something like this. Of course if I want to flow this I have to abandon the distance function I should go into the language of parametrizations looking at the maps this is not a singular point anymore for the map. And it is, it can be proven this has been done by Angon and that this is so small if you if you arrange thing that this is sufficiently small then it shrinks very quickly and forms a cask. The cask and another kind of singularity but I cannot discuss this here. Second theorem. This is a remarkable theorem. The four. This is whiskey. 1984. Now suppose that I have in larger dimension and suppose that by initial boundary. My solid set is smooth and strict and strict and strict to convex. And then again. Exist is smooth. Quickly convex and again it leaves up to. Some T Max and after rescaling again properly scaling, which means that one keep keeps the area of this boundary constant essentially here after rescaling. And then it approaches to the it's very close to the to a to the ball. This one. So again what we see here we start from a convex set convex body say the theorem of which can say is that this shrinks it remains convex it does not do anything strange. And that's a moment. This is singularity. Time. Now the proof of this is different. Of course a strictly convex body surface as also is also mean convex clearly. Not only I mean strictly complex much more than, than this. And to show that this condition is preserved is not difficult. If I start. This condition to show that this is this is preserved is not difficult because it follows by some evolution equation. Some evolution equation for the mean curvature itself. And using this parabolic PD. This is sort of Laplace. This is called the plus Beltrami operator on the moving manifold. But in any case is sort of it equation for the mean curvature plus a nasty sort of term here. Um, but it's much more difficult to show that strictly convex it is preserved because this is the positive definiteness of the second fundamental format only its trace should be remain positive but all the second fundamental form should remain a positive definite. So to show that. The second fundamental form. Remains. Positive definite. This is more delicate one has to show one has first of all to find the. The evolution equation for the old elements of the second fundamental form which is a sort again is a sort of is a sort of something like this then there is minus. Sort of equation like this. So not only one has to find this which is ending down by whiskey. But also one possible way to prove is then once you have the PDE solved by the second fundamental form and all covariant derivative of the second fundamental form you have to show that you have to implement the maximum principle for 10 source by Hamilton. This is one possible proof. So it's, it's, it requires a lot of work. It's, it's whiskey 984. Yes. So these are various remarkable theorems. And now let me show you that the singularities in general appears singularity can appear. So, oh, let me tell you an example convincing example. Which has been proved in various ways. Now I tell you the idea. But this is an example due to. Gray zone. So, you see, we start from a convex object. This does not single a rise in the sense that it's singularized at the max, but at the max also the surface disappears. So it's a sort of not too complicated singularity. And the same for curves, even if they are not embedded, but not, but not not convex initially. But now Gray's on argued that suppose that you have they in three dimensions. So this is a sort of dumbbell. We're here. This neck is very, very thin. So here the the mean cover sure is very, very positive. Here is positive, but maybe not not large. But they have two balls essentially almost joined by a long and thin neck. And the idea of Gray's on was the following was that I want to show that this what could it mean this? Well, I could show that we have there is some time at some positive time. So let me call T max at the max. One possible way to say is to say this is that say the linear for as T grows to T max of the volume of the solid set. So this evolves in some way. But if I arrange things to be this very long and to be this very large this very long and this very large, maybe he's able to prove that at the max the surface is not disappearing. Something remains in the sense that the volume inside is positive. And the idea is the following. The idea is that here I can put a big sphere somehow inside and this sphere evolves with its evolution law. And we know by comparison that as soon as the flow is smooth the sphere and of the object outside the dumbbell this must remain inside. So it cannot happen. I mean this this dumbbell must leave for it for some somehow for a time sufficiently enough that this is still leaving. On the other hand I can put, well here there is some argument different from Grayson. Here I could put say an Anganon torus around here. Okay. So I surround I surround the neck with the torus the Anganon torus which I know that it shrinks to something but maybe to a point but it does not synchronize before and then he has some time. So if I arrange things so that this this you see this neck must remain inside the hole of the torus as soon as everything is smooth. And so now you can morally intuitively believe this takes some time to shrink and if these balls are sufficiently large it takes more time to shrink and so this cannot disappear. And so what happens here at the end there should be some some some similarity before. So at the max before that the the wall dumbbell has disappeared and this can be made rigorous in various ways and it's been done by Grayson himself. So singularities appear. Okay. There is an interesting theorem here theorem by whiskey theorem. I don't maybe five or six five. And suppose that this is mood. Compact then we can detect some out in max in the sense that so T max is maximal time of of existence then the limit be converges to T max of the infinity norm of the second fundamental form of the manifold close up. So we can detect T max in the sense that the max necessarily something strange happens in the sense that if I compute the sum of the square of the principal curvatures then as soon as T converges to the max from the left this sum blows up. So this is quite I mean it's not easy at all. I mean it's not an easy proof it inspects the the various PDE sold by the in the sense in a sense why this should be true. Well if the second fundamental form does not blow up if say remains remains bounded somehow it is possible to show that not only it remains bounded but also all covariant derivatives also remain bounded in the same interval and then once you know that all derivatives of the second fundamental form remain bounded it is possible to take the limit of the manifold as T goes to the max you can go take the limit of the manifold you have a limit manifold this requires some work but there is a limit manifold for which you can restart the flow after start the flow on a small time interval T max say T max plus delta some positive delta and then this contradicts the definition of T max which was the maximal time of existence so and indeed I mean this is a starting point I mean then the story goes on a lot still big work by Wiskon and others but so the rate say the rate of the rate of so how quickly the norm of the second fundamental form one looks at the rate of blow up of the length of the second fundamental form as T goes to T max and then this through the allows to start to start the study of singularity formation like in the the Grayson example of the dumbbell study of singularity formation allows together with the so called Wiskon with the Wiskon monotonicity formula I cannot tell you what it is now I have two ingredients to start the study of singularity formation and try to classify how singularities look like and their relation with the homothetical shrinking special solutions that we have discussed in example three or four okay so now this is a very interesting subject let me tell you now I have three minutes to conclude by saying that singularities are maybe the most interesting phenomenon here so but in singularity here one has to invent the notions of weak solutions which means that solutions which are global defining time so they go beyond the first singularity time second singularity time etc so they are globally defined look for weak solutions so called weak solutions globally defined also in the presence of a singularity so where the Nincor Vashur is not defined point-wise say for instance globally defined for all times okay and of course coinciding with the smooth solution as long solution as long as it exists so there is a sort of extension of notion of flow which for times before the Tmax is of course the same as the classical flow but after Tmax it is still defined and it goes on the process goes on and of course I mean this is another subject which is quite huge try to so there are I think at least maybe 8, 9 or maybe also 10 different notions of weak solutions weak solutions what was given by Braque historically is first one is given by Braque 78 so the subject it's an interesting thing probably is to study how this the property of this weak solution and how these solutions are related one each other because they do not coincide in general after the formation of singularities so how do they relate each other unique I mean and so sort of things like this okay I think that my time is over and thank you okay so thank you very much for an excellent talk it was very interesting and also very nice to see an actual Blackboard talk first time for me in a few months so let's see if there are any questions you can write either in the chat or you can raise your hand in the participant list and they will call up on you one by one so let's see are there any questions okay so while I have a question? yeah okay so we have a question so Shal Faisal yes you're welcome to ask your question please so in the initial remark about the torus the defending on the initial picture the genus of the torus can disappear so what happens to the genus of the general genealogy surface what happens in general to a genus of the now this I don't know what happens in general to a genus I don't know I think well of course you again you need the notion of weak solution because if you imagine that the holes now disappear in a sense that if you have a fat torus say then this singularized at some moment here and then maybe after now here you need the notion of weak solutions because you want to go on in order to answer to your question so what happens depending on which solution you choose maybe okay this becomes a sphere and then it shrinks so I think that here you need the notion of solution this so in general I cannot tell you there is a there is a study on this but I'm not completely sure so I studied by Brian White which I don't know I don't remember now the results but as far as I know maybe it's Brian White which studied the partially maybe this problem but of course you have to select the notion of weak solution so for this maybe he was using sort of set theoretic solution or also viscosity solutions maybe but I cannot tell you more than that right second question you define this distance function I think in theorem 3 can you please draw some picture to show that it is not in general smooth yes the distance function has a sign in general the distance function from a manifold of course is not smooth exactly on the boundary of the manifold and this is a problem no so if you take just the distance just the distance of the point so f of tx just the distance then this is of course say you have this curve here so this positive here this zero here but the point is that I want a PD exactly on this manifold here where this distance is not differentiable you see because locally here what you see locally here your positive here your positive and locally you have a corner the graph of these exactly a corner on your curve so exactly there where I want to have a PD exactly on the moving manifold the distance is not smooth but now so this is no I cannot use it but I suppose that this part here make it negative okay make it negative this means that instead of having distance positive everywhere I have negative positive and zero and now it is now it is possible to show that here in a neighborhood of the curve the this sign distance function which is now positive and negative and zero on the manifold is smooth as smooth enough in a sense that smoothness of local smoothness of a manifold here is equivalent to say that this distance function now with the sign you see when I am inside this is this is zero so the distance is negative when I am outside this is zero so the distance is positive it is zero on the boundary so it is possible to show that a manifold is smooth if and only if this function here now fixed take a fixed for a moment there is a neighborhood of your of the of your boundary in which this is smooth so this is equivalent and so this is essentially a way to describe your manifold without parametrization just as a level set the single level set of a single function globally and this function is very very nice because it is a distance function so where it is smooth so in the neighborhood of the manifold you have the a-icon equation and so this a-icon equation is essentially a way to extend the normal vector field keeping length one not only on the move on the manifold but also in a small neighborhood and once you have this extension of the normal vector field in an open neighborhood all curvature invariance all curvatures can be described instead of using parametrization using the derivative of the distance function for instance this is equal to this is a theorem so on the boundary the mean curvature is the Laplacian and the advantage of this that this Laplacian is a standard Laplacian in a so what do you do you do the Laplacian in an open neighborhood and then you restrict on the manifold and the restriction is equal to this stage and so on second fundamental form can be described by the action of the distance etc etc right thank you okay thank you so there are two other people who have questions so the first question is from Sog Gatumov so if you can unmute please and ask your question let's see if he is here okay otherwise maybe we wait for that and I will let Hamza our own diploma student Hamza honestly ask a question please okay thank you for the talk can you hear me yes sure okay so my question is whether we are these results obtained are considered only for manifolds embedded in the acridian space or the results are still valid when changing the metric and the second question is in this case whether changing the metric I mean whether there is a surface of R3 that may shrink to a singularity for a specific metric but not for another one I mean whether changing the metric may let us avoid singularities in some way okay what I know is that this so first of all differential geometers usually do not use often do not use the distance function they prefer to use parameterizations so they look at the whole map instead of just the image of points but this is an advantage that they can treat also the self-intersected case the mean differential flow has been studied by whisker and others also sitting in a Riemannian manifold so and there are there are a whole literature on short time existence singularity time, team max, etc. also in a Riemannian manifold and of course maybe there it could happen I mean not necessarily true that maybe you should disappear in finite time it could probably happen that you go toward a sort of geodesic I am imagining that suppose now that you suppose that you flow an initial curve which is here on the boundary of this surface by mean curvature and then you flow and then maybe it happens that not necessarily you can shrink you could go maybe toward a sort of geodesic curve there so yes so the answer at the end is yes there is a big literature I think on the mean curvature flow in a Riemannian manifold in general I don't know if I can answer it to you but thank you okay so did you have any more questions Hamza or that was it the question was I mean in this case now whether a surface for example the sphere can shrink for example the intrinsic matrix we have seen that it shrinks to a point following the associated mean mean curvature flow now if we consider the general case whether it depends on the choice of the matrix this fact or not I mean whether it's always the solution is that it shrinks to a point may depend as I told you that this does not give you an answer no? I don't see how I mean in this case we will not have the same result as before I mean in general this probably could evolve toward this and remaining remaining here now trapped here ah okay let's see okay thank you okay so finally going back to the question from so again to most of microphone is not working but I will ask for the question instead so the question is does the weak solution suitably defined exists globally in time yes as I said there are there are several notions of weak solutions with which ranges from definition from geometric measure theory to purely PDE's definitions one very popular is the viscosity solution which is a notion of weak solution just maybe uniformly continuous function which is for which essentially this PDE one studies this one so this is the general non-linear PDE so one can give a notion of weak solution not necessarily see to actually only lip sheets to this PDE geometrically each level set of this possible disputative solution evolved itself by Ming curvatures not not only one level set but all together they evolved by Ming curvatures so there is a big theorem here existence and more over more difficult and uniqueness uniqueness of a viscosity solution for this Ming curvatures flow problem and this is goes back to Evans and changing agotto in 93 and also Evans the year I don't remember sorry maybe it's 91 Evans at the same time famous result for this notion there is a global existence for all times uniqueness which is much more difficult in this case and this gives a solution which coincides in a proper way to the classical solution as long as the classical solution exists so it is just one example there are actually historically black in 78 was the first to define a notion of weak solution completely different from this using very full theory so geometric measure theory is a sort of okay very full theory gives another notion of solution for which uniqueness has maybe is less but in any case there is another theory of weak solution and then the story goes on as I said starting from black passing through changing agotto Evans then Ilman the Georgie and others give several different notions of weak solutions okay thank you same participant also the second question which is the notion of mean curvature flow safe for space like hyper surfaces in a Lorenzian manifold so the Lorenzian manifold what I know is a sort of analog I mean in a Lorenzian manifold I mean with the Minkowski metric say what I know is that the analog of this problem is in this case now time has a different meaning they are not any more parabolic problems but are more close to hyperbolic problems and what I know is that sort of geometric similar problem is a minimal surface equal to zero where now the mean curvature is in the Minkowski space and so I mean time and space they are somehow the same role and so they speak more than like minimal surface in the Minkowski space rather than mean curvature flow so the analog as far as I see of mean curvature flow is minimal surface but in Minkowski case this is what I want to say and it is not far to some which is not any more velocity equal mean curvature but this is sort of acceleration equal mean curvature times a factor which I think if I am not wrong makes the equation Lorenz invariant but this is a way to see, to look at this is mean curvature of the time of the time slice all is like to say h equals zero in time space there is a big literature on this I think they are called classical strings ok thank you very much for that answer so I think these were all the questions we had for now so on that note I would like to thank again the speaker Professor Bellatini for a wonderful talk and also the camera crew the technical staff for making it possible so I am for all the participants for being here and so before we leave I just want to start a small poll but before you leave I would be grateful if you could answer it takes just 20 seconds and otherwise I hope to see you see you next time so thank you all thank you Professor Bellatini and if you are a diploma student you can also stay if you have some more questions I think this is ok