 bounded below away from zero and there are two possible situations where either the curvature stays bounded along the floor, either the curvature is unbounded. I want to discuss today the two situations and what you can say. So the first situation I want to discuss is the fact. So they collapse with bounded curvature and already yesterday I told you that this can happen only when a single curve, isolated curve, collapse down because actually if you have a collapse with bounded curvature you can imagine that you cannot have a collapse of a region because there is some curvature where a region can collapse only if it has less than six edges, five edges and if you consider a region with five edges, we have five at most, five angles of 120 degrees and if you consider a straight segment between these angles, these corners, actually there is not enough curvature to close. So with only five angles of 120 degrees you need some curvature in your region in order to close. With six you can do it, you can have a straight hexagon but less than six. So your region in order to collapse has to have at most five edges so there is some curvature around and you can imagine then when a region collapse the curvature must go to plus infinity, it cannot stay bounded. This is absolutely a non-elementary lemma in general and actually in general we don't have a proof of this outside the word of moving networks. If you find the proof of this, a bound from below on the area for regions with five angles and of 120 degrees and the curvature bounded above, I would be very interested to know it, a proof of this but well in the case of networks you are in a very special situation, you can prove that. So if a region is collapsing the curvature is going to plus infinity so it's the other case that I'm going to discuss. So in this case a region cannot collapse and by the argument that I showed you yesterday that means that only an isolated single curve of your network is collapsing down not other cross curves which doesn't mean that only a single curve it could be that at same time in different points in different areas of your network some curves are collapsing at the same time but locally only one single curve and so the analysis is understanding what you get when you send your time t to big t and what you get is a limit some kind of limit network in order to reapply possibly one of the theorems or possibly one of the short time existence theorems that we saw at the very first lecture in particular the theorem the very general theorem of Ilman and Neves and Schulze in order to restart your flow. Okay now what happens is that actually possibly parametrizing all the curves what is it is possible to show proportional to arc length actually it's possible to show that the bound on the curve which implies some convergence in double two infinity and in c one to some limit network s big t as t goes to big t let me and the very important thing is that this convergence gives you it's not up to a sequence all the sequence of networks converge to a unique limit network actually and moreover even if the I'm not saying well sorry this is great so sorry I made a mistake here so the w two infinity convergence is weak while the w the c one converges is strong so the convergence is strong in c one and weak in w two infinity to this limit network so you only have that well you still have the the curves are of class w two infinity so not necessarily c two for sight but at least they have bounded curvature in a distribution from a distribution point of view. Moreover what happens is actually that s t is non-regular since a single curve is vanishing you can imagine that the situation is that two triple junctions are going to collide along this triple curve this this vanishing curve and what you get in the end since only this curve is vanishing by the argument of yesterday what you get at the end is a four point before showing some something related to this let me try to convince you that give you another idea of estimates that you need different by the previous ones actually several of the things that I said easily hide a lot of estimates behind that actually just give you a taste of the another estimate in order to get this so as you can remember we all we always working with this which means that lambda is equal to and this explicit expression of a tangential part of the velocity of the motion actually allows to write down an evolution equation for lambda which is the following this lambda evolves like t derivative is equal to lambda s s minus lambda s minus two k k s plus lambda k square and let me recall also the evolution equation for k which is k s s plus k s lambda plus k q so I want to convince you that actually you have a unique limit so the what follows by this guy but these two equations that if you consider this guy here and call it v it's the velocity at every point of the motion I want to write down an equation for the square of the velocity actually the velocity since these two guys are or to normal it's the square of the velocity it's simply the square of k plus the square of lambda and now I use these two when I derive this I get two lambda times all this stuff let me write this explicitly lambda s s minus two lambda square lambda s minus two lambda k k s plus lambda square k square then I derive k square sorry here is a four and here is two now I derive k square so what I have is plus two k k s s plus two k k s lambda plus k to the fourth power two and now you see that there are similar terms around this one is like this one so I can throw away this and put minus two here and now I want to rewrite all this guy in terms of v actually so what I can I deal with these two together these are equal to lambda square plus k square s s minus two times lambda lambda s square minus two times k s square I simply taking derivatives and okay now I'm looking at I use this guy here minus two and this guy here two together if I collect lambda this is equal to minus two lambda lambda lambda s plus k k s which is equal to minus two well minus lambda lambda square plus k square s and finally if I take one and two I collect the k square that is plus two times k square lambda square plus k square now you see that our v v square is equal to lambda square plus k square appears here minus lambda v square s plus two k square v square minus squares so throwing away these two I can write smaller or equal than this guy here so I wrote down a partial differential inequality for v square that actually works very well for maximum principle if you can use maximum principle because at the maximum point of v square this guy is non-positive this guy is zero if the maximum of v square is in the interior and what you find out that simply the time derivative of the maximum of v square is smaller than two times k square v square max and in our case the curvature is bounded so this guy here is a constant it can be bounded by a constant so this is bounded by a constant times v square max which means that if you can use maximum principle on this equation v square can only have exponential growth in particular infinite time cannot go to plus infinity you can bound the v square at some time at every positive time by its value at time zero and if you bound v square you are actually able to bound lambda square so a bound from k on k gives you a bound on lambda also in particular on the velocity of the motion all of these if you are if you can use maximum principle on this equation which means if the maximum of v square is in the interior not at the triple junction okay but at what happens at the triple junction at the triple junction we know from very beginning computation computation at the beginning that the sum of the curvature square are equal are exactly equal to the sum of the lambda square at every triple junction and now you know again that your curvature is bounded which means that at the triple junction all also the tangential velocity tangential part of velocity is bounded so at the triple junction you have your estimate that under the assumption that the curvature is bounded also the tangential part of velocity is bounded by some constant which is more or less the same that bounds the curvature so either during the flow inside the curve the tangential part of the velocity is still bounded by this constant the same constant which bounds the tangential velocity of the boundary either if it is larger than that then the maximum must get must fall in the inside and then you can use maximum principle because when you have this you also have that v square at the triple junction is bounded by two times the maximum of the curve well by some constant the maximum of the curvature square which is actually bounded by another constant by hypothesis so this either v square is bounded but its value at the boundary of the curve which is already bound uniformly bounded otherwise you use maximum principle in the interior on this equation to say that it cannot grow too much and so you get a uniform estimate on v and then also on lambda so bounds on the curvature implies bounds also on the tangential velocity which is very good let me underline only for this flow for this special flow that actually it's we can always put our self in this situation so we have this bound on the velocity and then I look at these guys I take one curve and the curve one point on this curve gamma i of x and t and I look at the difference between gamma i of x and s so I look at the square different times this curve the this point is mapped by the curve and this difference is actually equal to the integral between s and t let's take t larger than s s and t of delta gamma delta t and x xy this is simply deriving integrating but we know that this guy here is exactly the velocity of the motion it's exactly our v by the equation over there so this is equal let me do carry the modulus inside of the integral so this is equal then the integral between s and t of the modulus of the integrand which is actually the modulus of v at x c the xy but we we just derived the bound on v depending on the curve but sure so this is smaller than a constant times t minus s and the constant is independent of time actually we did it for the generic x and x doesn't appear here so actually this is you can also write that the uniform norm between the same curve at two different times is bounded by a constant times the difference in time so then it's easy to see that this implies the curve the map gamma i indexed by the time for time going to big t it's a Cauchy sequence in the c0 of zero one as t goes to t so it's a so you have a limit a unique limit actually of the curve gamma i that we can can call gamma big t from zero one to r2 which is our candidate limit network unique there's only one limit because of this fact that it's a Cauchy sequence in time indexed by time but this is only continuous map but then the parameterizing arc length and using the bound and the curvature what you conclude that possibly not excited this map but the what you see in the limit the support of the network in the limit it's actually well it's actually something which is parameterized in c1 with bounded with bounded curvature and it can be unique the limit is unique there is only one limit network in doing in doing this this convergence so we have a limit network at big t uniquely network it's curve this emotionally bounded curvature actually with some effort you can prove that actually far from from the possible multiple junction the curves are not only w2 infinity they are actually c infinity and far from the triple junction the convergence is also c infinity that's kind of regularity t or m that follows by several estimates for instance by interior estimates of hacker whisker so actually what you conclude that you get some unique limit big s big t one and w2 infinity weak to some s big t which is actually a c2 because if you are c2 with bounded second derivative outside the point then you are c2 up to the point actually and what's happening here that s of t from the geometric consideration is something like this with this curve is going to collapse so when t goes to big t what you see these other curves are not collapsing they are all converging c1 actually that you find the four point and actually you can expect that could happen something like this for instance 40 and 80 120 since the c1 convergence forget this part these two curves must converge to something that forms an angle of 120 degrees like here and also the other one so other two here this a priori could be possible but actually you can exclude this possibility where these two angles are differently by 60 degrees because in a way in order to do this there must be some curvature here on this curve that can is in a way concentrating and concentrating at this point making in a way some kind of turn in the in the collapsing but concentration of curvature means that the curvature is concentrating in a small length of this curve but going to plus infinity but here it is not a situation here we assume that curvature is bounded so actually what can be proved is that when you when this curve collapse what you see at the end are two curves like that with 120 degrees between them and exactly 60 degrees in the other angles so a single situation so all this stuff force to have a limit s t network forming a four point here actually not any kind of four point exactly with these angles 120 120 60 which is exactly what we saw in the simulation possibly i didn't underline too much the very first day that i show you the simulation but when there is a collapse of a single curve the angles behave like this actually we can re-see the simulation maybe tomorrow at the end to show you again the the the video so this is a theorem under multiplicity one conjecture that we use several times what we get in s t are it's a a network with triple junction triple junction where nothing happens stays triple junction regular triple junction with 120 degrees and some for every isolated curve like this which collapse you get one of these four points with this angle condition and as i said it's exactly what we saw in the simulation this is a better drawing i'm not very good at 10th but this is better thanks to alessandro plude okay just a remark this analysis can be extended also if you one boundary curve boundary arriving at an end point of your network if this curve is collapsing actually instead of an inner curve what you get in the limit it's two curves arriving at the point p forming between them an angle again of 120 degrees which is exactly what we saw when i show you the example of the spoon and the curve connecting the spoon to the boundary collapses and you remember if you remember in the end we got something like this with these two angles all 120 degrees with this angle of 120 degrees this is exactly an example of this situation of the boundary and actually if you get this depending on the what kind of physical or problem you are trying to modelize you have to ask yourself what you want to do with this guy when a curve on the boundary vanish actually how to restart these guys well actually all the theorems that we developed worked only as a underlying only for regular networks with triple junction so if you get a four-point you absolutely need the short time existence of ilman and neves and schultz in order to which is a more geometric major theory based okay now i want to consider very special networks three like so there are no loops around that's simply graph so you don't have any closest loop and what can be shown is that if you work if you consider net which is a tree and the one of the simplest one like this four point on the boundary and two triple junction is one of the simplest well actually what what can be proved is that if you look at this guy well it can happen that this curve we are collapsed and we are in the same situation here and this is in a way some kind of converse of the previous theorem if you are a tree well whatever the curvature cannot go cannot be unbounded actually the curvature is always bounded during the flow must then if you know that if your curvature is bounded then you are in the first situation so the only possibility of singularity is a single isolated curve collapsing down and doing again exactly this phenomenon here so it's a kind of if and only if if you are dealing with it if if you are dealing with a tree actually this is the only possible singularity is that you can that you can you can see during the flow okay before discussing some consequences just to give you a hint why this works well actually the idea is similar to what I just mentioned yesterday about the evolution of a single single tree actually that in a way if you control the curvature far not so far but the triple junction actually you are able to control not the curvature but the growth of the curvature close to the triple junction sometimes you control so good if a tree of this straight enough flat enough that they cannot go to plus infinity fast enough in order to produce a singularity actually the proof involves several estimates but the idea is that if you consider this guy a tree and you take as usual choose a point x naught as before and take a blow up with the standard rescaling dynamic whisking and dynamical rescaling take a blow up and get the limit shrinker as before of a tree well taking a blow up is like taking a limit and you can think you can guess that if you take a limit of something which is a tree you still have a tree because you cannot produce new regions you cannot produce loops also because of the multiplicity one conjecture forbids the possibility that you the network touch itself and produce produce a closed loop so limit of trees is a tree is a tree so the limit shrinker that comes out from a blow up in the in the situation of a tree is a tree and then you can work out a lemma that tells you that the shrinker which is a tree must be actually only it's an absolutely non-elementary lemma that if a shrinker is a tree then it must be composed only alfines that actually as we said to yesterday since an alfines has unbounded length it must be passing must be must be passing for the origin so composed by alfine for the origin only possibilities are guys like this for the origin but actually we are dealing locally dealing with a situation like this because also in this situation isolated curves there are no regions that can collapse so we are in the other situation only curves can collapse and they must be isolated otherwise multiplicity one conjecture is false so locally you have a collapse of an isolated curve so locally you are in this situation always so if you local you are in this situation and you take a blow up of things like this well you have very few possibilities actually you can you can have at most four alfines so at most you find something which is composed by four alfines moreover the fact well the fact that the the convergence is in c1 tells you that you must have 120 degrees 120 degrees here and here and again there is another lemma tells you that at the end again you can only have this situation 120 120 60 and 60 so the only possibility when you have a tree are the old ones so single line empty set single line triple infinite flat triode more this guy extra so four a tree without assumption on the curvature you only have an extra guy because of the topological structure that forces the shrinker to have this structure you only have an extra case to deal with in this in the argument that I used to yesterday so again you put yourself in a in a you take this guy is the this curve here is is vanishing you put yourself close and close to the point of vanishing what you see is that your guy is getting close to this so which means that even enlarging you are getting close to straight lines here and again you use a similar argument to the one we used for the trio in order to say that at some point you are too flat the other is not enough curvature in order that the curvature which is inside here goes to plus infinity fast enough and you use a very similar argument the one that I roughly presented yesterday that okay the curvature inside here can make well can go to plus infinity but it took too much time at the end you find out an estimated tells you that the coverage is going to plus infinity after some interval but this interval is larger than the interval given by big t the time of singularity so you conclude that the curvature at the big t is not going to plus infinity and again as yesterday as a heuristic principle yesterday I said when you find out the blow up which is flat there is no singularity the curvature is bound and also in this case we are adding to the the old list another possible blow up which is actually flat so it's not about to expect and actually it is like that the your curvature cannot go to plus infinity and in fact in this situation by this argument at the end you conclude that the curvature is bounded in the situation of a tree where some curve is collapsing down it is more or less it's very rough but otherwise I will spend a lot of time in showing the the technicalities of this but this is more or less the idea the more ideas that if you get the flat blow up curvature is bound without the consequence of this well a very first consequence is that what yesterday called type zero singularities actually exist singularities without with the bounded curvature that actually you can also imagine that possibly they cannot exist because if you consider a very long rectangle but the minimal connection between the four points here it's not so difficult to show that it's given by this guy 120 here 120 and a long line in the middle so now suppose change color that instead our initial network is done like this let's make it regular so angles are here 120 120 and run the evolution for this network as I said yesterday if the evolution doesn't develop any singularities you expect and that through that you converge asymptotically to the minimal connection between the fixed end points so if you don't have any singularity of this pink network you should get you should converge to this guy here but that's clearly impossible because they have a different apology in a way different structure they are not equivalent if instead so at some point a singularity must happen and when it happens since this guy is a graph by this theorem must be or this way the curvature must be bounded so there must be one curve collapsing and so and if you make it symmetric actually this curve must be a collapsing one because if you make it symmetric on one side and the other side it's not possible this curve collapses because otherwise also these other one collapse so the only collapsing must happen on this curve forming a four point like this with bounded curvature because of the theorem and there is starting by means of Ilman and Neverschultz's theorem produce something which will be something like this asymptotically you get the minimal connection but the point is that this must happen and this is a type zero singularity so you rigorously prove after this theorem that type zero singularity must really happens in some situations let me also say that all this can be localized which means that the theorem can be localized so a easy corollary is that if the network is locally a tree maybe over here there is some regions but locally a tree means that uniformly in time in that neighborhood the network is always a tree then you can use the theorem that says that whatever happens inside here inside this your network even if you find a singularity the curvature is still bound so where the network is a tree in this area even if it is not a tree globally the curvature is there in that neighborhood is bound and actually this is more than a mark if no region is collapsing actually the network is a tree because you can always shrink your neighborhood and without collapsing you only see a tree and as a consequence if no region is collapsing locally locally no region is collapsing locally the curvature is bounded and this is more this is actually the the converse of what we saw before that if a region is collapsing curvature cannot be bounded in this case if the region is not collapsing the curvature is bounded that tells you that bounded singularity with bounded curvature coincide exactly with this kind of vanish and curvature with a bounded curvature coincide with the collapsing of a region and now also the scroller can be clearly localized and now we deal with the collapse of a region the other family of singularities in this situation we assume so these two guys are equivalent after this argument so we have collapse of a region and unbounded curvature one implies the other so curvature is unbounded and even is the one multiplicity one is true again you perform a procedure what you find out then now you are no more it's so nice situation that you can only find lines infinite flat triads these guys composed by two lines crossing at 120 degrees but now you find a lot of things because every shrinker is a flow by mean curvature think of these guys suppose you have this guy is flowing by mean curvature is shrinking to a point and vanishing if you do a blow up at the shrinking point you find itself so you really find these guys for instance this one and this one the or the blackest blue bucket spoon or all these other possibly you can imagine that if you fix this guy at the free end points on it on a circle and let him evolve you simply look see him getting down to a triple to a triad to a flat flat irregular tree so there are a lot and classification in order to understand the singularity is important because of this because of this reason and clearly the more your original network is easier it's from a structural point of view the more you can find a small number of these possible network because as I said taking a limit to top the structure of the top of the complexity of your network simplifies because maybe there are taking a limit a blow up limit means that you are enlarging things so maybe some parts with higher complexity you are sending them to infinity so your complexity is getting is getting lower actually in the case then the region collapsing you have you see macroscopically your regions collapsing then you enlarge and what you actually what actually happens is that in a way the two things compensate and your region you after the rescaling the dynamical procedure of whiskey and your region is always there past the limit you find it in the limit and actually in fact what you find in the limit it's kind of it's the memory of the region which is collapsing down that you don't see without the rescaling procedure because if you look at think of things only without rescaling you see this region going and you if you're able to take a limit in the limit the decision has gone you have no idea how was the region if it has a five or four ages you only see a point where the region has disappeared instead the in the in the shrink in the block shrinker limit shrinker actually when you see this region it's what actually collapsed down and also if you find if for instance find something like like this supposedly this three for a shrink at these three lines are unbounded these three lines are it gives you the limit tangents of the non-vanishing curves which are the curves arriving at the region disappear suppose for instance we are something like this and now this region is disappearing what happens here that these three curves are going to be something here here the original hay collapsed inside this point this is time t and this is time at the limit time but you take take the proceed the block procedure on this guy what you see here is that you get something like this as a block shrinker and the direction associated these three alfines are exactly equal to the direction you find in the limit so the the the unbounded curves and mounted alfines here give the direction of the limit tangent of the curves which are not vanishing and arrive on the region which is actually collapsing down this is more or less what I said unfortunately in this case with unbounded curvature unbounded curve bounded curve is very good because it gives you immediately compactness in c1 here you have no control on the curvature so you don't have seen one compactness and moreover at the moment we are not able it's a contracture to show that when you take the limit of this network to to have something when t goes to big t you get something but actually we are not able to to show that it's a unique guy like instead I was able to to do before because before because of the argument about the Cauchy sequence bound on the velocity in the case of bounded curvature so this is we don't know and in fact at the moment we let it we conjecture it through that actually there is only a single guy that you can find out as a limit but it's a apparently hard open problem assuming such uniqueness actually what can we prove that possibly after reparametization there is a vanishing part of the evolving network in this case these three curves bound in this region that collapse to a point like this in this limit network s of t and a non-vanishing part that actually converge in c1 actually two curves that arrive that concourse actually at this multi-point now here you find the triple a triple junction only because the region here has as free edges but see other examples and also in this situation you are not guaranteed that this angle are 120 can be whatever another example suppose you have something like this so you have one two three four and what you get is something maybe something like this with four curves depending on the number of curves arriving from on the region which is collapsing down far from the triple junction as before because of local regularity or a car we can estimate for mean curvature flow what you get is that the the convergence of this curve if you are a little far from the triple junction here it's actually infinity and the curves are infinity but the best you can say about the curvature is that if you measure the distance of some point x here let me call it p here to the triple junction in arc length well the curvature here it's a small o of the distance of one over the distance so getting close to the triple junction to the to the multiple junction that you get in the limit the curvature can go to plus infinity but not so much smaller than one over the distance to the to the multiple junction this is a qualitative a quantitative very important estimate in this situation and this is an example of what can happen this is a nomothetic situation when you know very very special situation which is homothetic and why I was underlining this this estimate because actually the idea is that when you assuming the uniqueness conjecture when you find out something like this in the previous situation the guy in the situation with bounded curve the guy that we found as a limit time big d was with bounded curvature in this case the curvature is not bound and if you look at the at the theorem of in my neighbor's truth well actually I didn't write it clearly actually the the in the in the version which is up now published or at least on archive actually you ask that your curve have bounded curves apparently you are able to restart the flow in the situation of the of a tree but not in this situation where your curvature is unbounded then in a personal communication and without a written proof but I'm personally convinced and the argument that should be there where in this theorem you can have actually put as an initial network something under this only this condition so weakening a little bit the condition that are present in the paper on archive where they ask for bounded curvature or all the curve in order to have this bracket flow bracket flow that possibly up to now you saw the definition instead you can push the the proof in order to deal also with a case when the curvature has this kind of growth so actually it's an unpublished an unpublished proof of this of an extension of this theorem you can actually restart the flow after this after this limit s of big t under the uniqueness condition also for in the second situation of collapse of a region okay so let me get back and let me conclude maybe a little bit early with the a big picture of what we've done up to now that possibly resume everything that we've done kind of lien scheme okay like this okay on this here I put hypothesis on the singularity possible blow up network shrinkers and their conclusions okay first case that with yesterday the first case that we discuss it when when on the singularity we assume that all the length of the curve were bounded by some epsilon larger than zero uniformly and we assume it also in order to get a contradiction that the the supremum of the curvature well the curvature was bounded by some constant bounded a plus infinity what we work it out that the possible blow up must be either empty or actually at the point where there is a singularity embedded infinite length and regular hold only triple junction and 120 degrees only three points 120 degrees all of this under let me write in red multiplicity one condition under m1 multiplicity one condition it is one condition entered here embeddedness of the limit after you got this what are the possible shrinker with all these qualities all these properties actually a line for the origin only okay or empty set line for the origin or infinite flat only these two conclusion that we we got at the end that no singularity at all was a contradiction conclusion singularity by means of white theorem or the extension that we did with the manovaga and mani or the extension the analogous extension of ilman and nevus sheuls k in this situation is bounded so there is no singularity k stays bounded which actually was the idea to get this to get this proof you assume k was not bounded you get a contradiction at the end second situation let me write it like this inf of some curve is to zero so some curve the length some curve goes to zero and curvature bounded that actually I call type zero singularity singularity with bounded curve type zero single something that possibly I didn't say that in this situation of bounded curvature all the length of the curves of the network converged to something so not only the inf of some curve is zero but actually some curve the length of at least one curve must go to zero but really is going to vanish so there is at least one curve such that li was to zero what we discover we discover what are the possible blow up must be the empty set as usual embedded again no curvature because since we are doing a rescaling bounded curvature implies that the curvature in the limit is going to zero which also means segments because there is no curvature they are what we call the generate well our shrinkers and no regions because the argument that if you're bounded curvature no region can collapse so locally you are dealing with a tree without region collapse so in the limit you cannot have regions because the limit it's easier just less complexity than the approximating so no regions the generate means here that take a regular shrinker and possibly collapse some parts some straight parts actually in this situation because we are dealing with straight segments okay if now classify if you forget about every part forget about the part before if you classify regular shrinkers with no we're composed only by segments straight with no regions and with this idea that if they hide something which is that they are something that can vanish yeah you have the same as before this guy like we saw before so there is only one more in that situation and the conclusion is that actually you always have k bounded by an argument similar to these ones and you can describe the exactly what happens so it's standard transition this guy becomes two curves forming angles 120 and moreover also are isolated so it collapse the lapse of isolated curves with bounded curvature so this is s t and this is s big t final case again if of length of some length 50 and k infinity unbounded so which means that here we are dealing all together with type one and type two singularities I didn't separate the two cases actually we all have a conjecture and when I say whole let's say at least me, Matteo Novago, Alexander Pluda, Felice Schulze, Tommy Mann and that type two singularities for the motion of embedded networks are not there they don't they cannot be developed like the same for the motion of a single closed curve embedded type two singularities for curves in the train can only appear if you consider the motion of immersed curve possibly a curve like this very possibly when this loop will shrink will develop a type two singularities but this is only possible if your curve is actually immersed not embedded so if we look at this guy look at this situation type one and two because up to now it's a conjecture only conjecture type two singularities is the last case we discussed again non-empty or embedded curvature non-zero regions around again they generate so possibly there are some hidden parts and there can be a lot and actually as I said and you can find on the survey that I uploaded on the as a material for the first lecture there are several conjecture about many of them coming from Tommy Mann and about the what is what can be in this class actually the strongest conjecture that you can imagine is that this class is actually topologically finite so the possible structure are actually finite but anyway there are a lot and they are for sure they contain regions number a finite number of regions actually and again also here we have as usual M1 otherwise the conclusion is not there and again conclusions which is the last conclusion that I showed you under under the multiplicity the uniqueness assumption uniqueness of the limit again let me also underline that possible blow ups here at the moment non-unique again you can conjecture that the blow up is unique but even in the smooth case of mean curvature flow in several cases the problem of uniqueness of the blow up in the rescaling after rescaling procedure it's still open also in this case we conjecture that actually the possible blow ups could be unique but at the moment we are not able to prove it could be non-unique here instead we are asking if you remember uniqueness for the limit network when t goes to big t and the region is collapsing in this situation you can have a lot of things like a lot of things you can find here so stuff like this that actually globally are only c1 only c1 c infinity far from from far from the triple the multiple junction and with curvature or small o o1 over d also let me mention another conjecture here that actually we don't believe too much this estimate in the sense that we actually hope also in this situation k is bounded this is not absolutely necessary for restarting the flow like I will try to do tomorrow because actually the theorem of mean man and emissary should that could be extended to this situation so we can also in this situation this case is clearly in the hypothesis of their theorem this case apparently is not in the published version in the public version on archive but in an unpublished version you should be able to use the theorem also in this situation here if you have this estimate on the growth of the corvator getting close to the multiple junction okay just to conclude one minute I hope that apart with the analysis of the network for flow I hope I convince you of the power of the power of these techniques of using blow up in getting information on what happens but the blow up tells you about something microscopically close to the singularity but then what you really want to have information macroscopically on your flow and you see that at every step from information here you get information on the blow up then here you're trying to use this information to classify and hopefully restrict strongly restrict the possibility of the blow up that you find out hoping that you are good enough in using this information to classify what appears here and then once you classify you use the microscopic information that gave you knowing what you can find to get some conclusion on the macroscopic behavior of your flow this it's a very good scheme in general outside the network flow for mean curvature flow rich flow network flow or other kind of flows you can have in mind possibly in other situation we have different way to do blow up different way to do classification that usually this part it's usually where the geometry enters you use the geometry in order to classify the object that has these properties and write them fill the the holes here and hope that they are not too many when they are too many like here you see it's quite clear the last things you find here the more you are precise in the conclusion here this is more or less the philosophical message behind this scheme of work okay I think I can conclude here and see you tomorrow