 Starting more or less where I left which is we're interested in almost minimal sets Almost minimal sets are the guys that are defined above so you said the set is always going to be called E and A competitor would be F. F would be a competitor in a ball and there is this main Property here that the host of measure of a set in the ball cannot be much Larger than the company the host of measure of any competitor, okay And there is something that I meant to say that I usually always say and then surprisingly not yesterday When you're doing competitors you deform and deforming does not mean that you cannot Pinch and in fact one of the main things that you do when you try to find better competitor is pinching so for instance if You know your you had a set like this Okay This should not be very good if this should not be almost minimal Because we've a picture that I have here. There is a better competitor, which is just Union of two discs one here one there and a wire in the middle Okay, and the way you see that the red thing is a competitor for the white thing is you essentially pinch everything here and then Project on the disc along this direction right or if you prefer when I pinch here I can do it all the way up to here, but I will have To include the two discs here So if a measure of a two disc is smaller than the measure of this thing here you get a Better competitor and that's a way of saying that my initial white thing was not almost minimal At least not with a small function H at that time. Okay, right If the discs were larger there would be a better competitor, but which would be The one we've seen before This would also sorry so you have to imagine what I'm doing here But you know some of the parties hidden here so a middle disc full here and then Two sides here. This would could be shorter to this one. We've seen okay All right, so that's four definitions So what do so I have a certain number of remarks that I just And then just reminding you of things the first thing is that it's important to know how you do the accounting of post-off measure But the most important thing usually in these sort of problems is to make sure you understand What is the set of competitors for a given? Problem so that's number one. I suddenly have again the fact that the definitions that I'm giving are Close to the definitions that were given by Armgren and Taylor for more or less the same reasons There are lots of different ways of organizing definitions So what I do what I did is I tried to find something which was reasonably simple But okay many many many sets of definitions are essentially equivalent. So let's just not not worry. Okay, I Don't know yeah, I should probably have said that before but the pharmacist minimizers of the Reifenberg homology problem Satisfy this condition. So it's not only potential solutions of my preferred plato problem that would be almost minimal sets almost minimal sets also Contained more people than that Same thing for the supports of size minimizers as soon as you prove that they exist. Okay, so that's not the bad and Again, I claim that This is supposed to give the best description. I know of soap films To be taken with a grain of salt Okay Right and I'm supposed to start giving you simple examples of almost minimal sets and I'll try to read So maybe I'll put myself and I'll read on my screen. So When the dimension is one in fact the minimal sets of dimension one they're just this I'll call this a line Imagine three equal angles Here so this I will call a Y and I will just put it like this right and this is minimal If you try to deform this and get something better You cannot and the angle again here comes from a fact that if you try to distort by moving the central point you have to have Point where a derivative is zero and this force is the 120 degree angle and in fact These are exactly the only non-empty minimal sets in our end that can exist. Okay Again, this is not minimal and the reason why it's not minimal is For instance because you know just do a competitor in this ball Try to do something reasonable and I guess Typical thing that you would do. I'm sort of lost so for instance you can pinch here I'm surprised because I thought something like this Should work and if I did my job more correctly, you would see 120 degree angles Anyway, I don't do anything Really clever. I just pinch here and then notice that once it's pinched. I can try to improve the situation, okay What did I have? So this is I think everything for one dimensions for two dimensions planes. You're not surprised What I call Y set with a bold Y is The product of a set Y here by a vertical line coming this way or the union of free half Spaces and that's fine There is a set T and I there will be a picture in a second. So it's the cone of our something about tetrahedrons, but You know, this is the Y and this is the T okay Right, so where am I my description? Why is T's and Then afterwards I have other types of examples this this what I gave you is in fact the total list of Sets for which we know they are minimal in the whole space And it's suddenly the list of minimal cones and cones are important in this business. Okay now The next ones are more local. So catenoids. So this is not really a catenoid But imagine something like this, right and probably there will be a picture next so or any Other things people call minimal surfaces are going to be almost minimal sets But usually it's locally that it happens. Okay, so for instance a catenoid would look like this In fact it more looks like a fatter Version So let's say this will be my picture of a catenoid Okay, and I'm saying it's essentially not minimal because At this scale this will do better. Okay But for small radius r. So if you take just h of r is equal to I Don't know zero for r less than one over ten and Then h of r is equal to plus infinity everywhere else. It's almost minimal. We'll be set the gauge function Okay, and then I have the expected thing which is that I hope it's a good description of soap films soap bubbles and so on and so far Okay, that's Okay, the sets so this is a catenoid. Okay, and again this one is probably already starting not to be minimal With the given boundaries Okay, right This is last precaution that I am taking so in fact I mean already you've seen I mean I'm just addressing a problem that we've talked about this morning Which is if you take a nice set and you add a countable set to it or some really horrible even close set But of measure zero you still get Minimal set if you started from a minimal set But we want a nice description of minimal sets. So we want to get rid of this extra set that you could add Okay, so that's what I'm doing in this slide. I'm saying I start from a set I Look at what I call its core which is just the support of host of measure on that set so for instance on this red set here the core would be just the union of the two discs and This guy here Does not support host of measure of dimension two so it disappears Okay, and what I'm saying in this slide is that I might as well just forget about it So again for the descriptions for since if you if you're thinking about trying to solve a plateau problem The red part here is important because it tells you something about the topology What I'm saying in this slide is that if this set here is minimal Then when I remove the red part, I still get a minimal set It's not in the same category in topology as you had before but first I will study the regularity of Of a reduced part. I mean the coral part is star I'll prove some estimates and then later on If it's needed, I will return to the extra piece here to talk about topology But in fact the later now the later on will not happen today, right or the this week So again, this is just some way so that I can say if I have an almost minimal set I just look at it. I Say I look at its Core I say it's also also almost minimal with the same constants and I study the regularity of that guy and then afterwards We'll see what happens. Okay So from now on all my almost minimal sets will be reduced or coral which means I don't care about the extra piece Yeah, sorry, I mean e-minus is time I I Really hope it's the only Error in my spin was slides, but okay, you know, you're right. Yeah, so again Yeah, the the little red part here might have some topological Interest, but I just throw it out okay, and So for instance here, I will say that the union of those two discs is alphos regular satisfy some other properties And I will not say anything but the wire in fact the wire here I drew the best most beautiful one that I could find but of course, you know, if you just draw a set of Dimension one like this. It's going to have exactly the same measure. It's going to be equally I mean, I don't control the red part. I mean the green part, okay so So again from now on I'm only looking at almost minimal sets that are reduced and what I'll try to do is Explain a certain number of results of regularity low regularity for those things Most of them are true including at the boundary Which was my initial goal, but the proofs are easier. So are easier when you're far away from a boundary So I will just now consider myself far away from the boundary Okay, and I'll try to to give proofs of various things and the list the initial list is here I'll first regularity rectifiability uniformly rectified uniform rectifiability big projections and then afterwards we'll talk about limits. Okay Let's start so our first regularity is this thing here so again, you take a point of a support and On a ball centered on the support the host of measure of a ball is comparable to r to the d with some constants that in our case will not will depend only on the dimension and Will be satisfied as long as the gauge function h at r is not too large Okay So yeah, I Tend to forget I'll do as if I was working on the whole space and in my slides I'm apparently doing it on an open set and If I'm working on an open set, I will just take balls that are contained in the open set okay Right, and I'm saying there is an easy part and the hard part and actually even not the easy and hard is Not so clear, but it's in co-dimension one. That's what's happening Let's say the logical thing is that the host of measure should never be too large Because you would imagine that this is going against being almost minimal or minimal This part here you'll have to say that either said is too thin. It was essentially useless. That's the idea Okay, or you could contract it even more Okay, right So I hesitated and in fact, I'll give you more statements first and then more proofs later as opposed to Just little by little so the second thing that I'm going to talk about is uniform rectifiability And I don't want to insist so much on uniform rectifiability itself So it's a quantified version of rectifiability. So for me rectifiable will really mean We are always having locally finite host of measure. So it's not Counter ball anything so we're looking at a Set with locally finite host of measure and it's rectifiable if it can't be covered by a countable number of Either leapsheets images or C1 images or C1 surfaces of dimension D Plus a set of measures you know, okay, right What else? Yeah, and uniform rectifiability is something that looks like this but more Uniform so instead of asking for instance for many pieces to cover of a set we ask for a control number of pieces or just one and Instead of having leapsheets mappings we asked for leapsheets mappings with a given bound and that's One of the definitions of uniform rectifiability. So I always be looking at alphos regular sets because my sets will be alphos regular and I'd say that it's uniformly rectifiable when there is some constant so that whenever you pick a ball Center of a set you can find a leapsheets mapping from the Unit ball let's say of RD or let's say the ball of radius Where do I have it? It oh, yeah, okay It's essentially the best is to define it on the ball of center zero and radius are the same radius Leapsheets mapping and leapsheets and you want the intersection of a set with the image of that ball to be large Okay, larger than some number times are to be okay Just away. I mean I essentially took the definition up there And you'll ask for a single piece of not so small size and here this thing says that Our set E contains big pieces of leapsheets images. There are other definitions I don't want to insist so much. Okay Right, but maybe there is one thing that I should say because otherwise I will forget Alphos regular doesn't seem to be so much of a property Yet when you don't have it things are much more unpleasant. Okay with alphos regularity for instance This is just a simple thing rectifiable sets are known to have I Proximate tangent planes almost everywhere I mean principle if I wanted to do to be very nice I would tell you what is an approximate tangent plane. In fact, if a set is alphos regular Approximate tangent planes are all automatically True tangent planes so that I wouldn't have to worry about little fuzzy sets and densities going around Okay, but many little things like this happen when as soon as you know that you're alphos regular So it's very convenient to work with anyway, even if again it does not Look so much like, you know strong regularity property, okay right And yeah, the theory that goes with it is So I'm going to prove regularity anyway couldn't prove uniform rectified it because uniform rectified it because it didn't exist at the time and essentially I'm saying that the Sets that we're talking about our uniformity rectifiable. Okay, I will probably comment more on that This one will not be able to prove It looks very nice We were very happy with it because we like uniform or activity It seems to me now that it is not as important as we thought it would be and I'll discuss that a little later Okay, and there is a little bit more which is written on the bottom of a slide But I'm not sure it is so important not only the sets are uniformly rectifiable, but they also have big pieces of leapschitz graphs Which is a little stronger property Don't necessarily pay too much attention to it So again instead of leapschitz images, it would be leapschitz graphs that would have big intersections And this is a little stronger than the other one and the difference is having big projections And we'll have to talk about big projections for some other reason Okay, but the difficult part of a uniform rectified it will not do but I claim it doesn't matter so much Okay, so that's all the definitions the fear is we want to prove then let's try to start doing little proofs, okay? So let me be put myself in Codimension one because things are easier and try to prove the upper bound for alphos regularity Okay, so you give yourself a ball and you want to estimate The size of a set in the ball and here I'm saying that we have this estimate which will actually be easy Which is cost of measure of a set in the ball is less than the so omega n minus one is the measure of a unit disc in Rd in this case d is equal to n minus one times r to the n minus one plus the error term coming from almost minimal and Okay, I mean I can try to hide what's going to happen, but it doesn't matter so much so, you know suppose the set is really You know lots of mass here You're in this ball This is clearly not Almost minimal you can do better than that for instance. What can you do which is better than that you'll keep? you know anyway you'll have to Find the competitor in this ball, okay, and here is a competitor which is clearly as good So a competitor in this ball means that whatever is outside you keep the same and Here all this mess here Essentially you replace it by yes, so I'll try to continue in red although it doesn't work so well And I'm cheating in this case. I think it's like this. Maybe it's going to be a subset of a sphere Okay, so if I can prove that the red thing here is a competitor for my initial guy deformation of the initial guy I win because The measure of this thing here is exactly Omega to the n minus one R to the n minus one Okay Now why is it a competitor that's not so hard so this set still is as finite measure locally So it's easy to find of dimension and minus one So it's fairly easy to find a point here So if I find a point here Well, I mean I will prove my point of the time before and what I will do is I will essentially use a radial projection Okay, and I change colors to so This was the right for voice for the estimate and I essentially Projects all the thing that I had in here on the boundary So the projection is going to be a deformation because I could go I couldn't interpolate between the two and what I will get Rather than what I said is this part Right and maybe a little bit of that part here Okay, even a subset so it's even better. Okay, so that's my first proof of the day Very impressive All right Okay So in fact in this case I can use any arbitrary point which is not on the set So that the thing is defined the point that I chose was more clever than another point which has you know I hesitated I hesitated a little bit I could have taken this point here and I would have I mean it would be having a little bit more stupid because I would Have obtained the whole sphere But not so much, but any point will do For the argument So that's fine And it's working well because you're in co-dimension one and the host of measure of co-dimension one of a sphere is finite And of course if you are in co-dimension larger than one and you try to do this Estimate it does not work right So this is so yeah, so this is the bottom of a transparency So what should we do now and the answer is that there is a well-known technique That was introduced a long time ago, which is the feather flaming projection That was probably introduced by feather and flaming on their paper In their paper about You know, it's a paper about currents What I will do is I will take exactly the same construction as they have Make it make it for sets. I hope it becomes a little simpler and use it many times So today will be a lecture about Feather or flaming projection Okay, and this is the right way to solve this problem about you know projecting on the sphere and then You have you have a set you don't know how to estimate it and so on and so forth. Okay, I feel light so I think I'll go here again so Again feather or flaming projections and I decided to cut the construction into many little pieces. I think three of them Because it sounds easier Okay, yeah You could try to do All those things at the same time, but let me try This way, so I will do a feather flaming in some given face Face means face of a cube and the face would be k dimensional. I think At least it will have a dimension which is strictly larger than d the dimension of our set Okay, so I give myself a cube Which I suppose is called q I give my set a set f usually F is the intersection of a cube all my cubes will be closed by the way So usually is the intersection of a cube by some with some set I'll just say that the set is contained in the cube. It could have a part on the boundary. So this is my set f Okay So and I want to project the set on the boundary of a cube And the idea of a feather flaming projection that once this is done. I will do it again, but let's do just one step So we pick one point So to make things to make the geometry A little bit easier. I will always take my center Near the middle of a cube Because this way I control the geometry of projections. I pick a point psi Here And again the point psi I try to take it outside of a set f Okay For now, it's easy because my among my assumptions. I think you have the fact that the host of measure of a set Why Okay, well anyway Usually my set is as dimension d and the face is a larger dimension. There will be an exception Later and that's the reason why my slide doesn't say what I expected it to say But nevertheless, there is always room to point to point to find the point psi Outside of a set f and that's where you project Okay Usually it's because this set has dimension strictly less than the dimension of the face And from time to time it's because the host of measure of this set has measure less than one tenth of a measure of that Half square and you project so you project Radially in the way You would imagine okay start from xi and any Point for instance this point here Will be projected to that point there, which is the Okay the point of the boundary of a cube which is on the same line So for instance The new set that I get here is going to be of course the picture is easy If I take a set which is large enough Okay, something like this All right That's step number one But I want to control the house of measure of the images because that's what I'd like to do And there are two ways to control this. There is a simple way Which doesn't always work the simple way is to say okay, look I pick this point psi and I look Near sorry Near here, and I'm trying to find what is the lip sheets Uh constant for the mapping that sends a point here to a point there Okay, and you see that if a point is close enough to a boundary of mapping is essentially going to be one lip sheets Maybe 10 lip sheets because there is a corner here Or because the center of the center xi was here and I'm sort of projecting Sideways, but I don't lose more than a constant And the second bad thing that can happen is that the point here was very close to xi Like for instance for this point and near this point the mapping is not something like 10 lip sheets Maybe it's 10 Multiplied by the inverse of the distance here Normalized by the distance here. Okay, so that's what I have on my thing here. The first estimate is again You know the projection near a point is diameter of q divided by the distance Lip sheets times the constant and so the obvious estimate would be that the host of measure of a projection by xi Is going to be the host of measure of a set that I started with multiplied by the lip sheets constant to the power d Which is what I have up there. Okay That's usually not the best estimate Okay Okay, but it works when the set is under control so for instance as soon as we know that the Our set is alphos regular we'll be able to find points that are Here that are far from the set And we'll be able to do that but the first is at the beginning of the arguments. We don't know that Again, it could be that the set is Sort of epsilon dance like before And if the set is epsilon dance, you're not going to be able to use this estimate the same way Yet on average when you take xi an average point Uh things sort of average out and what you will have anyway is that the host of measure of a projection is less than c times the Host of measure of the initial set Okay, by a fumini argument that I will do in one second. Okay So again, uh, the map is not going to be lip sheets with uniform constants But uh, when you average, uh, it will act as if it was lip sheets. That's what's going to happen And I have uh the argument for the proof. So let's go for it very slowly So I want to prove this uh globally So what happens is that I will just average on xi In the half cube and uh, I forget about the set f. Okay But it's an average host of measure of the measure of a projection So I repeat the average here and then I write down the host of measure of the projection And I write it's the integral on f of The lip sheets constant local lip sheets can't hold them up to the power d Integrated against measure Okay, and the dixie measure is just the average that I was talking about. Okay, right So then when you have a double interval and you don't know what to do you use fumini. So let's do that Uh So this is the same thing except that now I'm integrated. I'm integrated in xi for a given x The same story here and what happens that this Distance of diameter don't pay attention to it. It's just a normalization. You could have decided it was one In the way, so you have one of our x minus side to the power d And the face is at least d plus one dimensional. That's the way I required it. Okay The result is that this integral converges The normalization says that the interval makes some of the diameter disappear And what you get is c times the diameter to some power and then host of measure Of you know this set Okay, the measure of this set and again Things work out fine, okay I hope I'm not Maybe this this sounds fishy to me and I'm not able to Read correctly. But what I'm saying is that the important part here Was that the integral inside was converging no matter what the point x was To something which was bounded And then afterwards what you you're supposed to recuperate is the measure of a set f that you started with No, I think it's fine. Okay Right, so again on average Things will be acting as if it was c lip sheets with some constant and that's what we want okay And maybe some you know some pieces of a set so let's say here of course this this looks This one looks small But of course it will end up being large, but it doesn't matter, right? It's on average. Okay That was uh Step one of feather flaming Step two is you do it again Okay, let's try and I have a long story here about repeating the thing. So I still