 In principle should go after I complete the thing that I cannot complete which is you take a wire, a thick wire, you look at the minimal set that you get by plunging the wire into soap and lifting it up. Each of these sets satisfies the fung conditions so we control the sets for the thick wires and we just ask what happens when the thickness of a wire goes to zero does the minimal set that you get tend to a minimal set we bounded by just a line if it does how does it happen exactly and things like this. And I think the best way to solve this is to solve the other problem to know exactly what the limit should be and then afterwards you can try to see whether the limit is reached normally or in a weird way. Okay so that's but I think it's an interesting question to see you know when you have a thick wire how does the picture change okay so this is a picture of that this is a reasonably thick wire and you can think that the wire will go down what is the picture that you will see in principle at the limit you should get a minimal cone for the line there is no minimal cone for the line where there's something a plane which is like this so what is expected is a picture like this when the flat thing becomes more and more vertical and tends to a plane that contains the line and then things have to arrange to themselves but for instance this sort of picture I cannot justify it's something that I hope is true okay don't feel bad about asking questions because this will just slow me down and then you'll suffer more you'll suffer less for the last part okay no problem okay so I hesitated and finally I've been rushing the previous part so that I can talk a little bit about this part and I will not be able to complete this part but let me tell you first a little bit why I want to talk about it part of this is the I believe that if you want to prove existence results for I think this sliding problem that I've been talking about and probably also for the existence of size minimizing currents there will be no hope if you do not control a little bit the regularity of the potential minimizers okay so in other words at school we were told the way to solve things is first you prove existence then you prove regularity I believe it's exactly the other way around right you first prove regularity of minimizers and then and then each minimizers are regular you have a chance of proving existence okay which is bad news for me because it means that in dimensions larger than two I don't hope to prove existence results because the situation is more complicated but anyway let's talk about dimension two so I will not be that precise but I will try to describe to you a scheme which starts from Vincent Fabry's thesis which I think is neat in itself which people usually don't like so I'll try to sell it one more time okay and it's a general scheme for proving existence and here I claim the scheme works in a very special simple case which is the case where we are working on a manifold but the manifold is a very simple manifold is the manifold that you take by picking a finite number of cubes and identifying some of the faces of the cubes so that you get something in which you can continue to draw dyadic cubes okay and which has no boundary and if it has no boundary it's easy because I don't have to decide which sort of plato problem I'm looking at okay and which has some topology because this way I know that minimal surfaces can exist okay so that's the that's the setting but in some sense the setting is not so important I want to talk about the scheme and again we want to prove existence of minimizers let's say in the sliding category okay oh sorry or in this case it's the plane category because there is no boundary okay right there is only one way that I can prove existence myself is by taking minimizing sequences making them tend to a limit and hopefully the limit will be the minimizer that I expect that's what I want to do okay uh so again let me see if I read if I said everything that I wanted to say we have this simple situation where you have this manifold without boundary which is composed of cubes so that I can talk about dyadic cubes in there I need non-trivial topology so that the problem is not trivial I start from a given set E0 and I try to minimize cost of measure among all the deformations in that manifold of a set E0 okay and okay right and the general scheme is the following and then what I have here is the advantage of what I took as decisions to make the problem simpler there is this flatness about dyadic cubes so that I can draw dyadic cubes in my manifold without thinking about it too much otherwise you would have to be I mean otherwise you would have to play with charts it's okay it's painful uh Vincent is currently doing it if it didn't finish it but it's more work no boundaries so that I don't have to worry about the boundary but in this case I know for at least one example proved by Feng that you can deal with boundaries it's just this way we don't have to mention this and unfortunately there is this last advantage which is I'll stay in dimension two because this is the dimension where there is some regularity result for the minimal set okay right okay so we start uh we look at either class that we're interested in which is all the deformations on my initial set and we try to minimize uh host of measure in this class that's what we call MV infimum then as I said there is only one thing that I can do I'll take a minimizing sequence for which the host of measure tends to minimum and I try to extract the subsequence that converges and show that the limit is okay okay uh so so far we all do like this except that from time to time the object of interest is not a set but for instance is the case of Antonio a measure in the case of many other people are current and then the difficulties appear in a different place each time okay all right so the general strategy I repeated three times so it's okay now what are the obstacles there is obstacle number one which is not the bad one and I'll mention that a little bit later which is there is only one way to make sure that there is a limit when I take a subsequence which is to use host of convergence if I use host of convergence uh a priori I don't know exactly whether for instance host of measure will be lower semi continuous will whether it will go to the limit or not it's much better if host of measure goes to the limit okay but anyway that's one of the problems here uh I mentioned in the slide that there are other problems about parameterizations let's try to parameterize at the last moment because otherwise parameterizations will go to infinity and that's not good okay so we'll take the limit as a limit of sets in host of convergence and we'll have to make sure that the host of measure goes to the limit and I tell you how it's going to work we'll have to take a special subsequence which satisfies some quasi-minimality property so that one of the theorems I talked about about lower semi-continuity or host of measure applies and we we we get out of trouble because of this so this is host of convergence the problem number one problem number two is hairs so what do I mean by hairs there is this beautiful surface here that is the minimizer and the minimizing sequence in taking this one plus extremely thin little additions like this just think about tiny forms okay our hairs okay it costs almost no measure so this minimal the minimizing sequence could be like this something like this and of course if you if I don't cut the hairs what will happen that the minimizing sequence will converge to the whole space for instance or anything that I want so that's not good so we'll have to cut hairs sometime so that I have an acceptable minimizing sequence and after that I can try to think about limits okay this is also what Reifenberg did I think I mean cutting very some session of cutting hairs in Reifenberg's initial paper about the homology problem okay so that's number one okay and again the reason the way we are going to cut the hair is to do a feather flaming projection on a network of cubes so that it becomes quasi-minimal okay was it really okay right so lower semi-continuities I sort of mixed host of measure and lower semi-continuity but this is this is linked with the fact that we have to take host of measures so it's this thing here that we want at the end and in order to have it at the end it's enough to take a minimizing sequence which is uniformly quasi-minimal I have here a definition of quasi-minimal it's just a little bit more general than the almost minimal sets that I have and the properties that I mentioned before like rectifiability alphas regularity and so on and so forth and the lower semi-continuity of host of measure are true for quasi-minimal sets okay so we'll try to take a sequence of uniformly quasi-minimal competitors and then we'll save for this part okay all right okay so if we can do this we'll be we'll solve half of the problem the other half and this is the part which creates trouble all the time we get this sequence of this minimizing sequence we take a host of limit of this sequence how do we know that the limit is a competitor right so in the case of deformations you know you have deformations for instance by lip sheets mapping each of the competitor is a lip sheets image of a set the lip sheets functions get worse and worse and the limiting set maybe is not the lip sheets image of the initial guy because the you know there was some jump or something like this so there you have always to include a last step where you verify that the limit that you get is still a good competitor okay so for instance in the Reifenberg homological problem it's okay because you choose a homology such that when you take a limit of guys which are competitors automatically you can go to the limit and the limit also satisfies the same condition there are linking conditions by Harrison and Pugh and again the limiting conditions sorry they connect the in linking conditions that they have they are nice because when you take a limit of linking sets it still links so this part works fine okay and there is the typical bad problem which is size minimizing current you have this set of size minimizing currents you look at the support the multiplicity in the current do what they like maybe they tend to infinity you get this limiting set how do you find a multiplicity on the limiting set so that it is a size minimizing current but might be tough okay and the way it's going to be less tough is by saying oh this set is beautiful so by hand I can put multiplicities okay that's in the good case right all this so this is issues then okay so again okay so I said now I was wondering whether I would say it now or later so in the case of a sliding competitor what what's happening for this last part this is what we're going to say we have this minimizing sequence of quasi-minimal sets they are uniformly quasi-minimal so you can apply the limiting theorems and the limiting theorem says that the limit is a minimal set you don't know if it's in the class we started with but at least it's a minimal set and we're in dimension two so in dimension two we can control the regularity of minimal sets because there is the gene teller theorem remember we have no boundary here so things are easier so we get this description of the limit and the limit is just a bunch of faces that make nice angles with each other which may be a finite number of singularities something nice okay now we know this thing is nice so it's very easy to build a retraction from a neighborhood of this limiting set to the set so a mapping that sends a neighborhood of a set to the set which is lip sheets okay we build this retraction and now why is the set a competitor that's easy because you have a sequence that tends to the set those guys are lip sheets images I'll just draw a picture so you had the initial set let's zero so you have the set in infinity so it could be more complicated than that but that's okay I said infinity is nice and smooth and there is this retraction here from a neighborhood of zero to the set is zero zero is the limit of sets ek here is a ek for k large enough it's close to infinity we have actually a green map from easy row to ek which is a lip sheet mapping and I'm saying I compose this with a retraction and I get a lip sheet mapping from easy row to the limit so this is the way you prove that the limit is a competitor right and you need regularity things are hard in higher dimensions all right okay so so far so good I think the only thing that is left for me to tell you is why on earth can I find a minimizing sequence which is composed of quasi-minimal sets so same I mean they have some radioity with uniform quasi-minimal bounds that's what I had to do right which I call the quasi-minimal haircut so again I will start from any minimizing sequence and I first make it better so that each set is quasi-minimal and so on and so forth okay that's what I need to do okay and again have this comment saying that since no one understands this proof I will repeat it up to the moment when people are so bored that they will say yes it's fine okay all right so here is a definition of quasi-minimal sets and I will skip it for you it's just you know instead of requiring that the set the measure of a set is less than the measure of the competitor plus something small here I allow the measure of a set to become k times larger and I have to measure only among whatever I change wwp is the place where I change things okay I'm saying whatever I change I can maybe divide the measure by k but not more that's the definition of a quasi-minimal set and the main properties that it's fairly easy to obtain and the theorems that I had before are still true that's the two main properties okay right and for instance by Lipschitz's image of a minimal set is a quasi-minimal set because the definitions are like this okay but it so it gives you some stability okay and I had a list of examples but this I will not insist so much again the reason why I like the notion is that it is flexible enough so that I can work with it and we'll see the way we get quasi-minimal sets okay so how do I do a quasi-minimal haircut so I have this set ek it's one of the competitors it's not looking so beautiful how I can how can I make it look more beautiful let me try a first attempt I take a dyadic grid make it very fine it's not going to be more costly to make the grid finer so think about the finely very thin grid of dyadic cubes okay and then I take this set and I just feather a flaming project it on the grid because I don't know how to do anything else so I feather a flaming project okay and I get a new competitor for this guy which is living on the grid okay right and I'm saying okay so this is one competitor okay and then you know I can look at all the other competitors that lie in the same grid and maybe some of them will do better so what I do is I just minimize among people in the grid so unions of faces of dimension D and maybe faces of dimension D minus one that are competitors so that are deformations of my minute and my initial set and that live in that grid okay so I start from ek I projected it on the grid and I say okay maybe this one is not beautiful I'll take the best competitor which lives in the grid okay so it's a minimizer in the grid it exists because the grid is finite okay uh and uh what I claim somewhere here except that I have lots of definitions so I claim that if I do this if I take a competitor which is living in the grid and minimizing in the grid by definition it has to be quasi-minimal okay it's probably on the next slide okay F is quasi-minimal so this is what I call the important lemma F is quasi-minimal with constants that don't depend on the grid and in particular on how small the grid is it's just quasi-minimal with some constant why do you prove it you have this set that lives on the grid you imagine a competitor that maybe lives outside of the grid you project it by by back by feather flaming back on the grid the feather flaming projection is not doing things much worse I mean maybe it multiplies things by a constant c but it's okay it stays quasi-minimal so the initial was minimal in the grid and I'm saying that when you're minimal in the grid you're automatically quasi-minimal in space because you can always you know given a competitor you can always feather flaming project it back to the grid and compare everything okay and again this is you know this is where we like the flexibility because of course we you know a competitor which is just composed this is this could be a piece of competitor living in the grid of course you can do better by replacing the two segments by this but you don't win more than a factor of two or something okay that's what I want right okay almost finished so there is a problem is that I started from a minimizing sequence I projected it on the grid and of course if the grid I mean essentially you know this was my minimizing sequence it just happened to be nice okay and then I decided to feather for project it on the grid which means that I've been essentially replacing by something like this okay and of course I've been losing I mean I started from a minimizing sequence and the new one would not be a minimizing sequence okay so here comes the trouble so this is because I've been taking a dry-dick grid what I have to do instead is given the set of a minimizing sequence construct an adapted grid which is not the dry-dick grid but which is composed which for instance locally here would be more a dry-dick grid like this okay I can do this locally wherever the place is wherever the set is let's say locally rectifiable when I get lots of little grids there is this complicated theorem of Basson which says that when you have local grids like this you can complete this into a global grid which is not composed of squares but of objects that are fine with and again there is an adapted grid to the set ek with uniform bounds on everything you project on this grid and this way you get someone with surface is almost the same as what what you started with okay and then you complete as before so in other words you have to replace the dyadic grid that I was talking about by some adapted grid so that the minimizing sequence stays a minimizing sequence and that's the hard part okay what it can be done okay and the rest I think was under control and I'd have to thank you for being patient for a long time question no question okay question directly to the poor guy later