 Theorem except that the rest of the limiting theorem as a long and complicated proof, which I don't give okay But at least the story about how to measure is fun. It's fine. And also the amusing thing is that yeah the rectifiability of a limit is very useful here uniform rectifiability it is true in many cases, but Was not able to use it for good, right? Okay, why does this thing? Okay Lower some continuity. Oh Just okay Monetize the density so for this part I think I will also try to go a little bit faster than I intended to So again, we stop a second now we can take limits of sets and We can we can take limits in host of measures There's another estimate which is upper semi continuity of a host of measure which actually is true for let's say Almost minimal sets or minimizing sequences which has a proof which is similar to this one and I but I just forget about it. Okay Next I'm trying to look at an almost minimal set a return to the story about proving Some regularity property of those and then as you've been told many times a very important tool is Monotonicity of density or monotonicity of whatever you can write which is monotone, right? It's important. I mean if you find a beautiful quantity and it's not zero and it's monotone You should use it. Okay, but of course, there's not so many proper Quattities like this and density is just your good luck in this case and the thing is the same for minimal surfaces and many minimal things Okay, and I'll talk a little bit about that. So here. I decided to give you a statement directly. Sorry directly for almost minimal sets for also for minimal sets the Statement is density is non decreasing. Okay, so density is this thing that you've seen before, right? host of measure of a measure of a set in the ball and then you normalize and Then the first statement is what I said The you fix a point and I will it's going to be enough for me to fix a point in the set and then think this way things are easier Then the this is a function of our and it's not decreasing in this case And otherwise it's almost non decreasing or nearly non decreasing in the sense that you have this function here You multiply it by a function which is a little bit increasing just enough to compensate. Okay, and then you get something which is Non-decreasing again, okay right, so the function by itself might be Oscillating a little bit, but if you multiply it by the server function You get something like this. So the point here is that the gauge function you take it small enough so that the integral converges So for instance when r tends to zero this guy has a limit so The product of the two because it's monotone has a limit and you immediately get that the density itself by dividing has a limit but those other things Okay, we're good. Okay, and the condition on the gauge function That is needed for making this work is just a Dini condition, but okay Right towards a lot of a proof The and maybe don't even read that too carefully The main point okay, so you can follow I mean depending on whether you like people moving hands better than writing down and reading Things are the other way around okay You have this function you want to show that it's monotone Let's imagine everything is smooth enough So in this case smooth enough is obtained because the measure of a set inside of all is an increasing function So in fact it has a derivative almost everywhere And it's it is more than the integral of its derivative almost everywhere and that's what you use to justify the rest So if you want to know that this quantity is monotone you take its derivative and you find out that it should satisfy some differential Inequality which is well, I'm sorry. I don't have to do it to Look at it here. Okay, and the differential inequality in fact turns out so the second thing that I should say is that the derivative of the mass inside is Larger than the one less dimensional So let's say suppose you are in dimension two. It's it's larger than the one dimensional measure of intersection with a sphere Okay, that's also not so hard The best way is to imagine now your set is rectifiable So imagine the set is a countable union or even a single C1 piece And you just do the computation of a C1 surface and you find out that this is what happens Okay, so sorry. So this is what I said here Okay, and in fact when you just compute what you need to know you need to know that the host of measure inside the ball is less than some ratio times the host of measure of one less dimension of the intersection and It just turns out that this here is exactly the measure of a cone over this intersection So in fact what you have to prove is that the measure of the set Is less than the measure probably I have it up there is less than the measure Of the cone over the intersection with a sphere That's what you have to do. Okay The set is supposed to be minimal in this case because I decided to do the proof in the minimal case which is easier Okay, so the proof would be finished if you knew that the cone is a competitor for the set Maybe it's not true But I'm saying that the cone is a limit of competitors for that set and that's enough when you check the things Okay, and here is the picture to convince you that the cone over this intersection is Limits of competitors so here I draw what I so I draw the set I Decided that the competitor is obtained by taking these very thin annulus here and making it a very large Mechanicals here concentrating everything near the center when you look at this then the sorry So this small tiny piece here becomes something like this and it looks more and more like to the cone There is the rest of a picture inside that essentially goes inside here But you know at the end it's going to disappear anyway, so don't don't look at it too carefully So this is the radial mapping that corresponds to it and essentially You know once once you agree with this picture, you know that the cone is a limit of competitors, okay? All right, and then you just I told you about minimal sets for almost minimal sets You do a proof you follow the estimates and you get what I what I said Okay Near constant density. Let me just say again two words about it as in many theorems, it's good to know that the that the That's of the function is monotone, but it's also good to know what happens when the monotone function is constant and What happens here is that the again if the density so you have a minimal set Suppose the density is constant Then there is a proof that the set is a corner It's not too shocking again It takes more time than I would have expected, but it is true It's essentially it starts by looking at all the inequalities and finding out that the set has to have tangent Plains that go through the center and then it's unpleasant, but it's true Okay, and here is a variant of this obtained by what I just said plus a small compactness argument and the compactness arguments are allowed because we have theorems about limits Okay, which says that now if you have a set and let's say it's almost minimal, but we have a gauge function Which is extremely small It has a density which is almost constant between two radii so the density of a large radius is Extremely close to the density of a small radius Okay, then The set itself was very close to a corner Again constant density and minimal implies cone and I'm saying by shuffling things around Almost constant constant density and almost minimal with small enough constants implies as close to a cone as you want And this is very useful. I think I mean I said that I would not use it in the rest of Okay, but we'll see. Okay, so this sorry and this is the almost constant density Property I'll see whether I can leave it Okay Right blow up limits. We're finally ready to do blow up limits So blow up limits is what you've seen before you take the set you look at a point and you just expand Little balls near the set I mean near the center into large size and you try to take limits of the sets that you get by expanding so again The blow up limit is you take a sequence of radii if it's at the point X you first put X back to the origin you Dilate the set so that it becomes size one and then you take any limit of the sets And you call that a blow up limit Okay, those things exist because again we took the notion of convergence sufficiently weak To be sure that we would have blow up limits. Okay, and they are not in principle They're not unique. It's part of us stories that you have to ask given an Is it true that? At a point there is a unique blow up limit or not in principle. We don't know there might be two or three For instance if a set is a spiral At the center of a spiral even if a spiral is beautiful you could have that the blow up limits are any line With any angle or more complicated things. Okay, right? And the reason why I only mentioned blow up limits now is that before that? We didn't have all the material to say what I'm about to say here if you take a blow up limit You have a convergence result about host of measure Which says that the density of a blow up limit is the limit of the densities of what happens in the balls the densities of what happens in the ball tend exactly to the limit that I was talking about which is the density at the point so some number, okay, and the result is that the Measure the density of a blow up limit is a constant function of a radius Therefore the blow up limits at least you know that there are cones because of what I said before. Okay, all right The limiting theorem saying that if you had taken sequence of almost minimal sets Then the limit has to be an almost minimal set with the same With the same gauge functions when you take blow up limits the gauge functions actually get better and better because of scaling So the result is that the limit is a minimal set and not only an almost minimal set So blow up limits are minimal sets and there are cones Okay Good I'm going slowly to a Regularity result of gene teller of a reason so will soon come to low dimensions The reason being that in high dimensions. We don't even know the list of cones. Anyway, the idea now is the following It's very easy to understand what is Minimal set of dimension one part list of minimal corner dimension one minimal set of dimension one We have an idea of what it is. We talked about that before The next step is to look at a minimal cone of dimension two because when you look at a minimal cone of dimension two you might take the intersection with a sphere and It's you know, you have to draw a picture on the sphere and It's It's a set which is not exactly minimal, but it's sort of minimal on the sphere So you could try to understand how it happens by your knowledge of one mean of one dimensional Minimal or almost minimal sets get a description of the intersection with your sphere and continue Okay, so what I what I started saying here is the beginning of a long program, but we'll never end Which is minimal sets are complicated But they have blow up limits and at the blow up limit the minimal set becomes a minimal cone a Minimal cone should be simpler because you look at the intersection with a sphere You get something that looks like a minimal set of dimension one less and by induction You finish up. Okay, but the induction will close at dimension two Anyway For you you also can forget what I just said and say, okay, we know that blow up limits are minimal cones It's interesting to study minimal cones. Let's try to see if we can make a list of minimal cones Of course, this will happen in low dimensions and of afterwards will not be able to okay so This slide is supposed to list minimal cones and I minimal cones of dimension one we talked about this I Continue never to mention the empty set and then there is only two options the lines and the sets of type y Which is three half lines. Okay, that's Minimal cones and dimension one of us as you guess minimal as cones of dimension two are more interesting and What happens in dimension three is the following. There is a full list the full list was finished. I think by Jean Taylor And the list is the following you're not surprised the planes there's cones of type T Why you've seen a picture before cones of type T coming from a tetrahedron. You've seen picture twice also and that's all the lists of Minimal cones in dimension three, okay, nothing more and again, there is the Story about this before you could guess the list reasonably easy and then you have to prove that there is no other one All the minimal cones should be obtained by looking at the sphere drawing curves on the sphere the curves have to be arcs of Geodesics because you make a local study here a Compute even you could even compute the mean curvature or do something and you find out That it has to be Arcs of geodesics and then you have to glue arcs of geodesics on the sphere you find out that maybe there is something like 20 possible clients and then You Say that all these clients are not really minimal cones By finding better competitors one after the other and you're left With those three here Okay This is it's easy to prove that it's minimal this it's also reasonably easy to prove that it's minimal and this one Typically, there is some arguments involving calibrations, which is integration by parts of the domains bounded by the set against the right vectors okay, and Finally, I have a reference here for a site where you can see pictures of all the clients that didn't work and Better case and I'll just give you one or two So this was the simplest client So this could have been a minimal cone So I don't know if you see two triangles one on top of the other when you draw the line then you get a Prism and then you take the cone of the edges of a prism and you get this picture and This is this has the right angles and so on and so forth But it's not minimal because you can do this you can pinch and it's better Okay, some people didn't see the previous picture Here so you take the you pinch at the center and then you had the multiplicity cities that go up But you don't count multiplicities so that's good and you get a simpler. You get a thinner guy number two the Edges I mean the cone over the edges of a cube this also looks like a reasonable client So here I'm reasonable client means that you know this Comes from an arc of great circle on the sphere or if you want its faces and the free angles here are equal at that point That's the typical conditions that we need, but it's not minimal because you can for instance Pinch here you could also have pinched there or the other way around, but anyway you do better, okay? And I think that's all and this is yeah this is proof that I Don't know either so fumes follow the Follow the math or that we didn't make a horrible mistake in the computation. Okay Gene feelers Tell us fear M. I warn you don't read the thing in the middle because it doesn't help This I mean I'm also trying to write down the notes for a thing And I do realize that this was not helping so here is the typical statement that we would like to have all the time But we only have In this case and a few other ones So we have an almost minimal set of dimension. Let's say in this case two in some domain But it's a local fear M. So it's near a point you pick So we've a small enough gauge function. I decided to take a power in fact You can do much better than bad, but don't worry Pick a point of this set Then the statement says that near this point so in other words there is a small ball near this point where the set the almost minimal set is exactly the same as In fact, it's blow a blow up limit at that point, which is a minimal cone And when I'm saying almost the same I mean through Defeomorphism of space that essentially maps the cone to the set in the small ball. Okay, so in the small ball the set is just a Copy of the cone with C1 plus alpha distortion okay, and So again, don't read the middle and the end The main ingredient in the main story here is not so much the deformation itself It's just the structure in terms of faces. It essentially says the set has the same Structure in terms of faces as the cone not so complicated like for instance three faces that make that meet Okay, and the angles are the same also because because if you look at blow of limits at a point of a singular set Angles you're gonna find so anyway, the set is just composed of the same sort of faces they're all C1 and they meet with the right angles and In fact, I claim this is the best description. The rest is just to impress people Okay, or to use a Reifenberg theorem when you don't need one Okay, I mean I'm guilty of that too, but okay Right, so this is again a claim you cannot do much better in principle I have one or two pictures saying that the Ginter theorem is true So This is so again, so bubbles with large bubbles. They are almost minimal here You see that this set is composed of Points where is there is a tangent which is the smooth part then you can see here Places where the set as three Files meeting with 120 degree angles and then you have isolated points like here and the other one up there Where you have a singularity of type T? And so in principle Which would I should say is you can do as many sub films and bubbles as you want You will never see something more complicated than that Okay, and it's always going to be sort of C1 Here so this is probably the same picture. I mean our picture same story here So here you have two T points one here one here, sorry and two more there There's a bubble in the middle of the rest of the faces here are sort of flat and Okay Again, yeah, okay, right? Yeah, and I do last comments about the Ginter theorem and then we'll see Okay, I go down so that I can read So again the list of singularities we know The the situation is more complicated than minimal surfaces, but that's the thing we like because we know sub films can have singularities There is something a little bit unpleasant which is that the radius at which the theorem holds You cannot guess so much in advance. It's not unfortunately We don't know yet the following thing Suppose the set looks like a T in a ball of radius one and it looks really much like a T Is it a C1 version of a T in half a ball? It's probably true, but we don't know how to prove this So there is a small point and in fact the main question is that even said he's really looking like a T In this ball is there a point of type T somewhere in the middle or not and this is what we are not able to prove Okay, for why is it is still true and so on so that's the drawback But otherwise the situation I claim is as beautiful as it can be I said C1 plus alpha, but if you are minimal instead of being almost minimal You can improve on that and it's easier because you already have some structure with faces So yeah, you have some some way to start doing PDE's for instance There is an extension, okay Yeah, so this is what I said here, right But I don't want to insist so much on it For two-dimensional sets in larger dimensions What happens is the following Part of a theorem is still true But there are two defects first the least of minimal cones in dimension four is not known We know a few but we don't know we know them all that's a little bit of a drawback and Second for all some of those cones that we don't know it could be that they don't satisfy a property which I call Full length which is a technical property and if they don't satisfy this then I don't have a C1 plus alpha result I just have a byholder result The byholder coming from Reifenberg as you might have Imagine okay, but otherwise the situation can extend a little bit Okay, and I think so about the proof Yeah, well and Okay, and let me just say why it makes sort of sense to stop here so next time I should try to talk about the boundary again and The typical I mean the best result that we would hope to prove out of the boundary would be the Gentler result Here we would just say at every point of the boundary there is a blow up You know there is a blow up limit which is a minimal cone Try to list all the minimal cones and try to say when you're close to a minimal cone You are actually a C1 plus alpha version of a minimal cone nearby That's what we would dream to do That's what we'll not do, but this will be tomorrow. Yeah, okay. Thanks