 OK. So this point we have yesterday, what I want to discuss today is the behavior of putting all things together and try to describe what happens to the workflow passing through singularities and the asymptotic behavior. Before starting anyway, I want to mention a couple of mistakes that I did in what I wrote at the Blackboard in the last lecture. If you want to correct on possibly your notes, since the part on the Blackboard is not available, only the slides are available. Well, one mistake that at some point when I did the estimates on the curvature and derivatives, at some point I wrote this, that the time derivative of the integral of the jS derivative square of the j derivative in arc length square of the curvature could be bounded after using Gallardo-Nyrenberg interpolation inequalities by this guy, 0t integral of s t k square 2j plus 3 s xc plus constant integral of k square 2j plus 1 plus a constant. So this was a mistake. Actually, this is not there, if you correct on your notes. And also, this is my first mistake. Second, where this is more precise, yesterday in the scheme that I showed you, at some point I say that when you take a blow-up, you get the generate that I never defined. What is this the generate? What does the generate mean, actually? And actually, the generate simply means that it's the limit of regular networks, and the generate encode the fact that, for instance, this guy here that we saw, it's a very important shrinker that we call the cross or standard cross. It's, in a way, a degenerate shrinker. It's not a regular shrinker. It's a degenerate regular shrinker, because actually you can see it as limit of these regular networks with angles of 120 degrees when you make this curve collapse. So the generate simply definition means limit of regular networks that yesterday I never used this word. But in the survey that I uploaded, it's quite a nice way because it suggests that you get it as a limit on regular stuff. OK, this is just to make things more precise. Now, what I want to discuss today is actually how to restart the flow after the singularity. Now, yesterday we finished our description of what can happen at a singularity time or a curve collapsing. And in that case, you have bounded curvature. And that was the case where no original collapsing. So then you can use this theorem by, very nice theorem, by Tom and Ma and Andre Knaps and Felicius. That actually you can find a bracket flow starting from your non-regular networks that actually, from the analysis that we did, only contains the only bad points are actually four points forming angles of 120 degrees between them. So actually, the point is how to restart this guy. And I'm going to give you a very rough idea on how they can do it. But it means to this theorem, let me say only one thing. I think up to now you know what is a bracket flow. It is a variational definition of a flow by mean curvature that actually is defined by means of the formula, the bracket formula, which is actually an inequality in general. And this bracket flow that this theorem produces is a very special bracket flow. In particular, because it's almost a smooth flow. For every time interval, which it doesn't get to zero, we have a completely smooth flow. It's infinity curves, all the compatibility conditions, it's a flow like all the ones we consider in this lecture. The only point is at time zero. Only here, you need the concept of bracket flow. Only in the interval containing the initial guy here, which is not regular, because immediately this guy opens and becomes something like this. And opens in the direction of the small angles, actually. Moreover, since, as I said, the bracket definition of mean curvature flow is actually given by an inequality. That inequality allows a sudden loss of mass. If your network vanishes immediately, you still have a bracket flow. So actually, this bracket flow is a bracket flow such that the total length is a continuous function. There is no sudden loss of mass in this bracket flow. And the inequality in bracket formula is actually inequality. Sometimes it's called a special bracket flow. So it's the bracket formula, given the variation of the finish of bracket flow is actually an inequality and not an inequality. So it's a very special bracket flow. And actually, we hope that you can even make the find out other special properties of this very peculiar bracket flow in this situation. OK, what is the idea that they use in order to produce this bracket flow? This is a very nice idea, technically quite complicated, but the idea is not so complex. The starting point is that let's do in this situation, which is easier and actually, for this case, the most interesting. If you have a very special network like this, exactly the cross, the flat straight cross. OK, there is an almost explicit solution, the most explicit bracket flow that let this guy evolve. The idea is to open. But it is known that from a corner like this, there is a unique expanding motion by curvature, getting out from the cone. It's something that does things like this. It's an expander for curvature flow. But actually, if you consider this guy, forget about this line here, while it disconnects the network, which actually, we don't want that. It's actually also a bracket flow. If I take the two curves, expanding curves of this and this, it's actually still a bracket flow, but it's not connected. So it's possible to perturb this expanding solution and creating an expanding network moving by curvature with angles of 120 degrees, which actually is still a solution of the curvature flow. And this guy is connected. This is OK for this special network, which is straight for a fork junction. So the idea that in the general case where the curves are no more two straight lines are curved lines, in a way you want to approximate the curved lines with the straight lines here. Because the blow up theorem tells you that the more you get close and close and look at things close to the origin, you actually find exactly the double cross that you have that. So the idea is that you get close enough here in order that the network is getting close to the standard cross, throw away what you have inside, substitute it with a standard cross, holding that ball, put inside this solution here, substitute it with this solution. There is a problem in connecting things. So you have to do it in a smaller ball, then you have to do some small curves connecting the outside, which is the original network, curved network with the inside, which is the small network given by this part here. You connect them, and then let it in the ball. Because after this replacement, that you replace the inside here with that guy there, the spanning guy there, then you let it evolve. This network is a regular network, so you can use the standard theorem for a regular network with smooth theorems. So you have an evolution for some time. And that's a delicate point. You want that this evolution doesn't develop singularities for some uniform amount of time. And then you do a sum r level with r, the radius of this ball, then you make it at the smaller levels, smaller levels, smaller levels, and you hope that all these flows, at every level you produce a flow, that all these flows sending r to 0 converge to some limit flow. If you are able to do that, then you hope that you find out at the end some curvature flow that actually is a good curvature flow for your original network. Because you are replacing smaller and smaller balls at the end, you can imagine the limit is like you weren't replacing anything, and you find actually a curvature flow for your original network. This can be done. Absolutely not easy. A lot of estimates in this procedure must be developed. New ones, the estimates that are shown to you are not sufficient. You need estimates on the time of smooth flow, and so on. You have to do this gluing operation in a very careful way, actually, particularly in the connection between these standard guys here with the original guys outside of the ball. And in this situation, it is a little bit easier to do it, but the generalized case is quite complicated. Because if you have five points, for instance, you have several possibilities to find an expanding network for a guy like this, for instance, but actually it can be done. The paper is quite tough to read, but the conclusion is very nice, it's quite technical. But this can be done, and this is more or less the, I'm not cheating too much, the idea behind the proof of this theory. So approximation with almost explicit solution when the network has only straight lines. Good? In this case, you only need this special case of the theorem, which is a little bit easier. But actually, in general, as we said, if we have a collapse of a region, yesterday we concluded that under what we call uniqueness of the limit assumption or conjecture, we have a limit which actually gives you some C1 network only, with a non-regular multipoint like this, that about the curvature, you can only say that the curvature is going like 1 over d at the point where d is the arc length distance, intrinsic distance to the multi-junction, which actually doesn't get in this theorem. This theorem asks for bounded curvature on the curves getting to the multiple junction. But actually, the proof can be extended to cover also this case. It's actually unpublished, and we check with Feli Schulze. So I don't know if it will be published or in a notes or somewhere, or maybe in the future in an updated version of our survey. But at the moment, it's not available explicitly. But anyway, this case can be covered, so you can also understand the theorem and restart the flow also in the situation where you have a collapse of a region. As I said, possibly losing the uniqueness. You remember my very first example that for this guy here, by symmetry reasons, if you have an evolution, you also have the rotation of your evolution is still another evolution. And we hope that uniqueness could hold at least for the generic initial data without so many very special symmetries. Anyway, even if you can extend the theorem and cover this case, we can anyway conjecture, there are good evidences to conjecture that also in this case, the curvature of the limit guy is bounded. Not the curvature along the flow converging to the network S of t, but the curvature of what you find at the end. So all these curves are actually bounded curvature. This is only conjecture, but I'm trying to convince you why it's a good conjecture. This guy actually come from, for instance, the collapse of a fiber regions with this curve. And the region R is collapsing to this point. Instead, the curves here converge to this curve in the limit. And actually what is quite easy to conjecture is that the only problem with curvature are not coming from the five curves there. The only problem with curvature are coming from the region collapsing. That shrinking down, curvature must go to plus infinity. So you can expect that in all these situations, you can always separate the converging network in a collapsing part where the curvature is unbounded. Like I said, but then that vanish in the limit. And non-collapsing part, these five curves, where actually the curvature is bounded inside, up to this point. And then in the limit, since this curve, if you forget this and you know that the curvature on these five curves is bounded and take their limit, the bound on the curvature past the limit, so you expect that the five curves, which is all you see in the limit, has actually have bounded curvature. At the moment, we are not able to prove it, but well, it's quite a natural conjecture to expect that you can always make this separation between the vanishing part and non-vanishing part of your network. The vanishing part is the bad part where the curvature is exploding, but fortunately, at the end, it's gone. And the other part can be treated like we dealt with the case of bounded curvature. Today, I will state a lot of open problems and conjecture that actually we will be very happy if someone of you comes out with a good idea and let us know. So this is the standard, the easiest singularity case that I was discussing here, in my picture. So actually, this is perfectly coherent what the theorem of in my neighborhood, what it produced in this case, which the solution that is produced by that theorem can be analyzed a little bit better, so it's quite detailed. And actually, it's quite coherent with what we see in the simulation at the very beginning. So there is a collapse and an opening exactly with this central curve, exactly in the opposite direction of the collapse, of the collapsed one. There is also actually, in this case, hope that it should be possible to prove uniqueness that in general case is false. But in this case, there is hope to prove uniqueness of the evolution, at least in the framework of this theorem of a bracket flow. Which is at least, since when you deal with trees, these are the only singularities that you can encounter. So at least for trees, this would be very nice because at least for the motion of trees, you can conclude the uniqueness of your flow. So now after this analysis of singularities, what I want to discuss, OK, I have my singularities, I restart my flow, then I encounter another singularities. Maybe I have the same analysis, then I restart the flow, then I go on and go on and go on. What I am expecting is that since the flow is decreasing the area, the length, the total length of your network, and the boundary part now fixed, if passing singularity, your flow goes on for every time and doesn't vertically converge to a critical point of the length among the families of network with that fixed end point. And in doing this, we define this flow for every time. The only problem in defining this flow for every time, if the singularity times start to accumulate. So you have a singularity always faster and faster and faster. And this is a problem. It's, well, again, there are evidences that this cannot happen. We have a partial proof in the heuristic argument. We only miss some estimates in order to conclude that this cannot happen. What can actually happen, even if you are able to prove this conjecture, is that the number of singularities is not finite. So if they don't accumulate, you can always go on, go on, go on, and at some point, but still continue to find singularities, singularities. And then you don't have no clear idea if actually you can take a limit at infinity if your network doesn't stabilize. So actually the stronger conjecture is that actually the number of singular times is actually finite for every initial compact, at least for a three-like network. As you can guess, the collapsing singularity regions going away are surely finite. Because if initially there are finite number of regions, at some point if they start vanishing, at some point your network, if they all vanish, becomes a tree. And also because the Ilmane never shoots a theorem, it never produces regions. The expander here, that you glue in order to restart the flow, are always locally trees. There is no, in this case it's easy, but in general, no new regions can arise after the restarting theorem of Ilmane never shoots a theorem. So the number of regions during the flow is always decreasing. And starting from a finite number at the beginning, since your network is compact, at some point the number of collapsing of region singularities can be finite. Are the other singularities? The singularities where one curve is collapsing and then you have to open another equation. That a priori could be not finite. You can have an infinite number of such singularities. If instead such number is finite, well, at some point the flow becomes smooth, no more singularities. Become smooth and the evolving network converge up to some sequence, asymptotically, to something which is flat, connecting the fixed end point. Critical for the line function. Sometimes I would call it Steiner configuration. Why this happens, this is not so difficult to see. A version's part is the more delicate. Well, the fact that what we know, that the total area of your network, one of the very first computation, was given like standard curvature flow for curves by this guy here, by the square of the curve, the integral of the square of the curvature. So since if we integrate this, what you get, the area at the length at time t minus the length, well, length at time zero, minus the length at time t is equal to the integral between zero t of the integral of k square. And this is clearly positive. So if you send t to plus infinity, this is bounded clearly by L of zero. So we have a uniform bound of that integral. If you send t to plus infinity, supposing your flow is defined for every positive t, then what you get, again, an analogous argument that you have an integral for an infinite interval of some guy which is positive and this integral is bounded by some constant fix at the beginning. So which means that you can extract a subsequence of times going to plus infinity such that this guy inside is going to zero, which actually means that on this subsequence, your curvature is going to zero in L2, then you do some estimates and lower semi-continuous. If you are able to take a limit, the integral of the curvature squared is lower semi-continuous. So in the limit, it must be zero, which means that the limit is made of three segments without curvature. Moreover, if you are able to get as it is, because this guy controls the c1, if you have sequences where curvature is bound in L2, you have c1 convergence. It's one lock convergence. Then not only are made of a straight segment, but also they are made, the limit network keep the herring condition, the 120 degrees condition. So the limit guy, if you're able to take it, it's actually a regular network made only of straight segments, which is exactly what is called a Steiner configuration. This is up to a subsequence. Since here I struck the subsequence converging, so possibly change the subsequence, you change the limit that you get at the end. But actually again, what you can expect, actually, is that this limit is actually unique. So you have full convergence of your sequence to, again, another conjecture, but another expected one. If all this is what actually can be wrong, if this conjecture is not true. You could have a situation like this. These are quite easy networks, which is a tree. At some point, this curve here collapse, you get a full point, it opens in the other direction. Then after some time, it collapse again. And reopening, you get the same topological shape. So what you want to avoid is that this oscillation goes on infinite times. So oscillation of shapes. Again, here you have this guy that, a lamp type, collapse of this curve, opening, you get what I call an island. Then again, for some reason, this curve collapse, you get this guy again, it opens in another direction, and you got another lamp. And again, the oscillation is the one I like the most, oscillation between theta and eyeglasses. Again, notice that the corners here are 120 here, 60. And now here, instead, they are 120, 60, they switch. And then you get a theta again. We don't believe this can happen. And the only very speculative idea about how to avoid this is to find out some mixed combinatoric integral geometric quantity that decreases during the flow of the network. Because it's bounded below, decreasing, and decreasing on definite amount at every singularity, at every switching of shape like this. So hoping that in this way, if you find a quantity like that, you can, at some point, since you decrease at every singularity of definite amount, if it is bounded below, you cannot have a lot of infinite changing of shapes. But actually, we have, at the moment, very few ideas about the possible analytical shape of this quantity. OK, now I want to show you and collect also some problems that I mentioned during the course, and show you some other open questions and research direction. Well, the most important problem, actually, it's to be able to show the multiplicity of one conjecture that you saw enters in the whole analysis that I showed you. We really need this guy. We have a possible line very recently that we hope can be carried on in order to show it. Several, some years ago, we were able, actually, to prove it in a very special situation, a very easy one. It's a weak conclusion that if, for some reason, you know a priori that you don't have triple junction collision, well actually, the multiplicity one conjecture is anyway true. Or if your network has only one or two triple junctions, then you are able to prove it. And very possibly, again, it's quite technical, very possibly, it should be able, even locally, you only have two triple junctions. So the really bad situation that if you're able to exclude that, then you should be able to get the full conjecture in general. It's the best situation like this, where three triple junctions go to collide together at the same time. In a situation like this, that I'm going to make the curve straight, but actually think of this curve like you have some curvature. In a situation like this, the shape of your network, now I draw them straight, but think they are a little bit curved. And you have this three triple junction that for some reason are getting close each other. And after some time, they get close and close, you can imagine what you get in the limit, you get this guy. With multiplicity one and one, with multiplicity two. OK, if I draw them straight, they don't move. So it's not a counter-example. So if you want to look for a counter-example to multiplicity one conjecture, you actually have to put some curvature here around and hope that this makes your network shrinking down and getting in the limit is something like this. Well, we all believe it's not possible, but actually you're not able to exclude it. If you are able to exclude this case, well, actually this is the worst case. The case that possibly there is a possibility that multiplicity one conjecture is false. When three triple junctions locally go to collide, actually. If you're able to exclude this case, you are in good shape in order to prove the full conjecture. This is the most delicate situation that at the moment you are not able to deal with. Instead, when you have only two triple junctions, there is a geometric argument to prove multiplicity one conjecture, which is, in a way, pushing a little some geometric ideas of Hamilton and Wiesken because you are only two triple junctions. Well, let me do it in general for a network. Do this stuff. You take a couple of points, p and q here, the lines between p and q. Only if this line doesn't intersect other lines in the middle, consider the region that you get by the rest of the network and your line, call it apq. And then you consider this quotient, the distance pq. This is the distance in R2, exactly the length of this segment, divided by apq. The area of this region. OK, this is a kind of isopereometric ratio. We call it, this is also depending on t, because this is your network. Then minimize this guy among all the possible points pq on your network under this property that the segment doesn't intersect other parts of the network. And you call it q of t, it's a minimum of this guy. It can be shown that neither p, neither q, when you take the minimum, the minimum is realized by a couple of pq far from the triple junction. So you have this guy. This quantity, take the time derivative, q of t. If this quantity, this time derivative, is larger or equal to 0, well, if you take this minimum for a regular embedded network, for instance, for the initial network, this minimum is positive. Because if this minimum is 0, it's a minimum. It is realized. Means that p and q are the same point. If your initial network is embedded, this minimum is positive. If then you are able to show that this derivative is non-negative, then it's bounded below uniformly the real flow, always positive. And when this minimum is positive, your network cannot, at some point, go to touch itself. For instance, doing things like this. Why? Because otherwise, I take p from this side and q on the other side, and I get 0. So this kind of measure of embeddedness of your network. In particular, if this argument, q of 0, we know that it's larger than 0, and if this holds, so q of t is always larger than some epsilon to 0, it's another proof that embedded networks cannot lose embeddedness. Moreover, this quantity is scaling invariant. Look at this. If you rescale your network, you have quadratic factor here, quadratic factor here, the quantity scaling invariant. So if you rescale your network, your minimum is always the same. And if you take a limit again, your minimum is always the same. So that means that every limit of networks satisfying this must satisfy this. But then, suppose you can get the double line, the double line falsifying the multiplicity one conjecture. But this means that some part of your network is going to converge to something very, very close like this. And then here, there is some other part of your network. And also in this other side. But then I take p here, q here, and you see you can have an unlimited double, the double line means that these two guys get together, get close, and your minimum of these guys must be zero. Instead, since it's a limit of guy satisfying this, also the limit must satisfy this. And you have a contradiction. So all this stuff is to prove this inequality. It's a computation. Not immediate, but solid computation. Unfortunately, and we made a mistake in our very initial work on the subject. This guy here is larger or equal to zero only when realizing your minimum. The other part of the network bound in your region here contains at most two triple junctions. Which is actually true if in general your network has at most two triple junctions. So if you are able to show that when you go and get this minimum, the minimizing couple that determine the region of APQ on the other part contains only two triple junctions, one and two. Suppose that this region was not like this, but was this. Then this derivative is larger or equal to zero. Unfortunately, we try a lot, but we weren't able to have an argument to say that every time you take the minimum, you only have two triple junctions in the pink part here. So the only strong requirement that we made that ensure this is that the initial network at all has only two triple junctions. In that case, this line works. The recent idea that we are going to employ in order to multiply and conjecture is completely different because we gave up following this line because we didn't find any reason why these must have only two triple junctions. They have the minimal regions, actually. But anyway, if you actually have networks with few triple junctions, the analysis is the multiplicity and conjecture is there. It's with the theorem. And then you have a real complete description of what happens at a singular time. This was done, actually, by me and in our second paper by, well, putting together first and second paper with the Matteo Novaga, Tortorelli, and Magni, Alessandra Pluda made the full analysis for the spoon, actually, the one that I showed you some lecture ago. The other guys are more or less scattered around. We use these guys several occasional examples for the local descriptions of singularities, what can happen. Because anyway, even if they are so easy, the easiest network can imagine, they already encode a lot of phenomena that happens during a net-of-flow. OK, now I want to mention some other questions. Small and more important. Well, uniqueness issues are there in several points of this work. There are two kind of issues. Four regular networks, I recall, I said the natural uniqueness would be geometric uniqueness in the class of C2 in space, C1 in time networks, actually. We all believe it, but actually it's still an open problem, and the difficulty in trying to get such result is the lack of maximum principle. For non-regular networks, uniqueness is the uniqueness in this space of bracket flows. And at least in the nice situation of the cross that I showed you before, should be possible to prove on uniqueness in the theorem of Ilman and Neves-Schultz. In general, it's not, but we hope at least for the generic initial network, which means if you have an initial network that doesn't give you uniqueness, perturbing a little, you can have a network that gives you uniqueness in the restarting theorem of Ilman and Neves-Schultz, the short-term existence theorem of Ilman and Neves-Schultz. The generic initial network should have uniqueness. But let me say that we are very far in having such a result. Genericity results are usually very delicate, not easy to be proved. Another problem that I mentioned, this is not absolutely super relevant, but actually it's a natural question that's also still open also for several cases of smooth hypersurface from minkological flow, the uniqueness of blow-up limits when you do the whisking or other blow-up procedures. Another natural question that I mentioned to you yesterday, if actually type 2 singularity actually exists. We believe, like for curves, like for embedded curves, that they cannot have type 2 singularities, that in the motion of an initial embedded network, you cannot have type 2 singularities. Just to remember, type 2 singularity is a singularity where type 1 is a singularity like such that your curvature is of order 1 over square root of t minus t, type 1, and type 2 otherwise, which means that at least on a subsequence, you don't have no constant such that k max is bounded by a constant divided by square root t minus t. Thus, it does not exist such that it works like this. For curves, this is a good dichotomy between singularities because for curves, one already knows that the curvature approaching a singularity is for curves, smooth curves, is always larger than 1 over square root of 2 t minus t. This is no type 2 for embedded curves, embedded closed curves. For networks, we conjecture that these guys are not there, again, there are no type 2 singularities. Moreover, for networks, we are not even able to prove this, that the curvature getting close to a singularity goes to plus infinity at least with this rate, that also we conjecture is true, again, conjecture also for natural. Because actually for networks, the only we are able to show at the moment is that k max is larger than 1 over t minus constant, t minus t to 1 fourth, which is actually quite weaker. And the reason is that here is obtained by being a maximum principle, so it's a point-wise estimate. Instead, all our estimates on the curvature are integral. And taking integral and passing to a point-wise estimate from integral to point-wise, you lose a square. So possibly one needs to find another way to get to this. We also believe to this, even if all the two main conjecture uniqueness of the limit network when t goes to big t and the multiplicity conjecture are true, even with that, with what we know at the moment, we could not, well, possibly we could be able to prove this, but still the conjecture about type 2 singularity will still be there, actually. So it's not the consequence of not knowing enough about singularities. And actually, it was exactly what I was saying. In the case of bound of, this is possibly after the multiplicity conjecture, the most important open conjecture, the uniqueness of the limit network at a singular time in the case when a region is collapsing down. This is possibly the second main open problem. Another very large bunch of conjecture comes from the analysis, which means proving existence or non-existence according to fixing the shape, the shapes that I showed you yesterday in that gallery of Tommy Mannen, of actually of shrinkers, studying possible properties and hoping to get to some full classification. Studying on properties, well, there are several conjectures, means, for instance, I can mention a couple of conjectures about the space of shrinkers. Well, one, what I found the nicest is the fact, a strong fact that the possible shape of shrinks are finite, which is due to Tommy Mannen that also has a partial argument, but not a full proof. But for instance, there are sub-conjectures of these. For instance, if you take an unbounded shrinker, we saw that the unbounded part must be half lines. And a natural conjecture is that you cannot have more than five half lines going to plus infinity in a shrinker. This is related to the fact that regions with less than six edges contract with six edges keep the area constant and the more than six edges they expand. So actually, in a way, this is reflected on the number of unbounded part of a possible shrinker. This is another open problem. We would like to have better estimate in the restarting theorem of Ilmar and Neves-Schultz related to that construction that I mentioned you before, because actually, the existence of that bracket flow is performed at the end, taking a lot of sub-sequences in all these constructs. Construct, replace, glue, then take a sub-sequence. At the end, you get something converging. And in several points, you take this sequence losing or making a rough estimate. So this means that you don't have so much control on the behavior of the curvature when you restart. The point is that you start with something which has bounded curvature, possibly the limit in the first case, standard 120 degrees. You have this guy. This guy has bounded curvature. And then you restart it, opening in some way, by means of ENS theorem. And if you look at the curvature function, this is for big t and this is for t larger than t. If you look at, for instance, the maximum of the curvature, a big t, we say that it's k max bounded by c. And immediately after, because of the construction, the maximum of the curvature goes like this, comes from plus infinity. Well, not going to 0 or something. But there is a coming from plus infinity doing. If you imagine what's happening here, I told you it's like, for instance, in the stress situation, that immediately there is a shrink and expander getting out from this. And if you imagine that looking at things back in time, this guy is contracting to become a standard cross. But the angle here is 120. Angle here is 120. And it must contract to an angle of 60. So the curvature, in order to do this, must go to plus infinity, back in time. Forward in time means that immediately for t larger than t, the curvature is coming down from plus infinity. And then going on till the next singularities. What we would like to have, that it's missing now in their work, to have some precise control in the way this curvature is coming down from plus, and more precise control, quantitative, the way it's coming down to plus infinity. This is in the framework of showing that singularities cannot accumulate. If you have information on the way that curvature is going down like that, well, you can hope to have better information. You can hope to have information on how long after t your flow must be again smooth. How long? You like to have a bound from below, depending on this behavior, on the time of existence of a smooth flow. In order to say that next singularity is too far to accumulate, actually. But to do this, you need really to push, to get really inside their proof, and get better estimates in some parts of the construction. Final problem that I want to present, what are generic stable singularities? Again, in very special situations like the one that I showed you at the beginning, we are super symmetric. It's a symmetric pentagon shrinking down and producing a five point. But you can ask yourself, OK, I modify a little bit one of this curve, breaking the symmetry. It's still true that you get a five point, or maybe my modification here makes this curve vanish before the shrinking. So you get the four region. That means that here in the liberty, you no more get the five guys like this, but the four line guy. So in a way, you're asking yourself, how much are stable these singularities I see in the flow? If I modify a little bit my network, I still see the same change of the problem, the same structure. And this is a very natural question, because if you reduce the number of stable singularities, that means for the general network, you have very few kinds of singularities, or less singularities, better understanding of your flow. Actually, it's easy to see that you can analyze this fact by looking at a blow up. And what can be seen that if you get a blow up, which is a line, or a cross, or a circle, which is a very special network, well, if you modify your network, as I said before, you get a gain, another singularity, and again the same type of blow up. Other conjectural stable blow up are the bracket spoon, the guy's called the lens, and this guy here, which are all shrinkers. No more with an higher number of edges. So this is a conjectural, again due to Thomas Mannen, that these guys are the only dynamically stable shrinkers. So if you have a singularities with a blow up doing like this, you perturbate your network, you still see the same kind of singularity, which also means if they are the only ones that you have a region of more than three edges collapsing down, in the blow up you see a full region in the shrinker. If you perturbate four or more, you perturbate a little what is going to happen that possibly one of the four edges collapsed before the collapse of the full region. And you get something like this guy. Again, there are possibilities because actually you have actually to compute the stability of these guys, which is actually dynamical, but you can do it with a computation, not so easy. The difficult part is the only part. It should be possible to prove that these guys are stable, that they are the only ones which are stable, could be more difficult, with the only doubt on these guys. This could be unstable. These two guys, we are almost sure that they are stable. There are some doubt, some that doesn't doubt, we doubt a little bit, computation not completely easy, that this guy may be unstable. Which means actually, as I say, the conclusion that generically, for the generic initial network, only a region with three edges can collapse. If a region wants to collapse generically, it must before reduce the number of visages to three. Then it can collapse, to three or less than three. Final remark, what are the possible future? Very far, in my opinion, to try to extend all these kind of analysis to higher dimensions. So the most natural guy to study is the cluster of bubbles. And in particular, the easiest one to test the ideas, which is simply a double bubble, which is the two-dimensional analog of the data network, actually. And what is the difficulties here in generalizing this analysis to the two-dimensional situation? Well, even from closed curve to closed surfaces, in R3, there is a huge change in difficulties, a huge change in the results that you can hope. A smooth, embedded, smooth, closed curve stays smooth, becomes convex, gets round and round, and then shrinks down to a point. It's absolutely false for a surface, can develop singularities, and you want to continue the flow. Sometimes you can break into pieces, and so on. So you can imagine that also here, there is a huge gap between one-dimensional and two-dimensional situation. Moreover, here, there's also the point that in the network case, you have two levels of objects, curves, and triple junctions. If you deal with the two-dimensional interfaces, you have surfaces bound in your balls, lines where three surfaces can arrive, and points where four surfaces, where four regions can arrive at this point. You can imagine something like the angles with the carbon atom, actually. All inside the tetrahedron, if you connect the regular tetrahedron, if you connect the vertices to the body center, actually. So you have three levels and three levels of equation to study. Your system becomes much more complicated. And at the moment, well, there are several approaches, the one of phallics by means of, well, generalized approach by means of minimizing movement, the approach by bracket flow, by Tonegawa, that you possibly saw in the previous week. But in the world of the smooth approach that was the one that we tried to follow as close as possible for the networks, actually, I'm aware only a couple of papers, these papers by Deppner, Garke and Kosaka, that they prove short-time existence for very special interfaces, not for the general case. And uniqueness similar to the TRMS that I showed you for regular networks, satisfying compatibility condition, the TR are more complicated. And another paper containing also estimates, more quantitative, again, only for special initial interfaces by Felicius and Brian White. Special initial interfaces means that the initial bubble, cluster of bubble, must only contains not possible that four regions arrive at the point. You only have two levels. We have surfaces and a curve where three surfaces arrive. But not a point where four regions are concurring, which actually means that more or less you are studying the double bubble situation. But in the smooth approach, I'm only aware of these two cases in this special, there is no theory and reason for the general case that would be very interesting to have. OK, just to conclude, since I have just very few minutes, I want to ask you a question. I'll let you have an open problem, a very elementary to say open problem, that we are not able to do it. So possibly you are younger, more energy, you are able to find out a good argument. We will be very interested in having an elsewhere. It will be very interesting. It will simplify some parts of arguments in our work. And I want to state it absolutely independently of the network flow. It's a technical, I'm opinion, nice question that could be useful for us. Well, if you tell you what is known, if you take a closed curve, C2, in the plane, well, it can be proved. Well, if you assume that the curvature is smaller than 1 for your curve, it can be proved that the area of your curve is larger than pi. It seems innocuous, but let me tell you, it's absolutely non-elementary. If what is interesting can give you a reference to some proof, there are several proofs around. In the star-shaped case or convex case, it's easier, but in the general case, I'm not asking anything about the curve. Simple, C2 curve, closed curve. It's absolutely smart proof for the general case. But I suggest you to try to do it by yourself. More precisely, if you're in this situation, the interior contains a disk of radius 1, which actually implies this. So if you have this, you have d1 is contained in the collet, which implies this. OK? Notice that you have this. It can be useful. It's a hint for your proof. The integer, sorry, quite tight, is a hint for your proof. OK, what we are interested in, we are interested in networks, can imagine this is related to shrinking regions. So now I consider, instead of this, this is a theorem. Instead, I have a conjecture. I suppose we have a region, curved C2, with five curves or less. But if you can do it with five, you can do it also with less, with angles 120. Curved polygon with angles 120. Five edges. And I want to conclude the same. I want to conclude that if k is smaller than 1 on this curve, then there exists epsilon, such that positive is such that the area of omega is larger than epsilon. This is false if I have six corners. Take a regular hexagon. Curvature is 0, so it's bounded by 1. But you can make it smaller as you prefer. This is false if you have too many corners. Suggestion to connect the two things. In this case, integral of k is equal to pi, 6, pi. Yes, should be 6. For hexagon, integral of k is 0. If you have a curved hexagon with angles 120 degrees, the integral of k is 0 always. So this guy is positive. So I think this can suggest you how to generalize the result if you are able to prove this. Because, let me tell you, it's open. And it will be very interesting because it will simplify some arguments in our job. Actually, we don't need in our work because this question, we ask it for moving regions, so they have other special properties, so the conclusion is true for the special region, moving regions, that we use in the network flow. But the question can be put in general. Can we ask in general? And you can see this, which is my opinion, in particular, the first one is independent interest for people in geometric analysis. Because actually, it's a kind of isoparametric inequality, reverse isoparametric inequality, without the perimeter entering, only curvature and a contended area. I can also tell you that the niger dimension is false. There are counter examples. So try to find a counter example for surfaces, for instance. But actually, the consequence of this is that k squared max times the area for a closed curve is always larger than part. And the consequence of this, if we are able to show it, would be that k square max of a larger than epsilon for some epsilon, open for curvilinear polygonals with these properties here. OK, I think my course finished here. Let me, before closing, thank you, thanks to the organizer for the very nice week here in Trieste. Let me also thank Alessandro Pluda that left today morning. Since several of my slides were based on other slides that she made for another occasion, in particular, all the figures was done by Alessandro. And let me also thank you for your attention and for following the course till the end.