 Okay, so thank you for staying so long I'll try so okay, and of course I was wondering whether I would do one thing or two things And I think I will do one and a half things Yeah Okay, so Here I'm going to start again. So remember we stopped at the gene teller theorem, but gene teller theorem was a theorem that was saying Near a point an almost minimal set of dimension two is in fact equivalent to A cone and the cone that you would take would be a blow-up limit of the set at the point And we want to do this at the boundary. We expect things to be a little bit more complicated. So it it means that We're only going to look at two-dimensional sets in our free say It works two-dimensional sets in our end part of a program works as well and Again near a point of a boundary and we try to essentially Classify the singularities of a soap film at the boundary That's what we try to do. Okay, and I'm supposed to remind you a little definitions so the At some point of time you have to tell me where I have to step Okay, I try here. So the definitions Sliding almost minimal sets. So they're almost the same as for a minimal set Host of measure of a competitor should never be much less than host of measure of a set itself Here is the set F is the competition Do you hear all those things are I suppose even better than I right, okay Okay so Right and That's not so bad That's bad So Yes, so someone is playing games on me. So What was I saying? I was here and Again, we're interested at the regularity at the point and sliding competitors is the same thing as we had before for plain Minimal sets except that there is a deformation, which is a one-parameter family and the important Properties of those things is that a point stays in the boundary when it starts in the boundary. Okay right, so All the theorems that I claimed I proved This week are still true at the boundary either boundary is nice and in our case the boundary is going to be either a smooth surface or a smooth curve and rapidly either a plane or A line, okay in particular those the sets are uniformly rectifiable In the case we're talking about at least rectifiable alphas regular limiting theorems are true and The Various monotonicity for the density for balls that are centered on the boundary Right So this is what I call a dream of a Jin teller, right? We are picking a point. We try to Say what does the set look like at near this point? The first thing we do is we look at blow up limits. They are supposed to be minimal cones We study these minimal cones depending on the minimal cone We expect to be able to prove some more regularity for the set okay, and When these equal to two we can hope to find a list of minimal cones because at least There is this description of the intersection with a sphere as a union of arcs of circles. Okay Let me so in fact the program works reasonably well in the Setting that I describe here, which is you start from the following thing. So you're in R3 You're given surface and let's say for instance the surface could be You know something like this, okay Okay, so the surface is the revolution thing that you can imagine, okay the set has to be contained in One of the Connected components of a complement of a surface. So for instance inside the tube. Okay, and you would try to minimize The surface measure of a set E which is supposed to have a boundary along that thing So the sliding boundary is this thing here The surface you are forced to stay on one side of a surface for some reason The main reason being that it's easier to do the proof and the other reason which is that so films apparently do this You consider that E? contains Gamma, okay Which is a way of saying that the the boundary is also Wet okay apparently it happens so it's not much a problem and under all those constraints You try to minimize area or something like this, okay, so that's I think the The thing that I have up there Again pick a point For a minimizer Or an almost minimizer at the boundary and you the first thing you try to do is look at lower limits of this situation and you get something that looks like The wall is something like a plane And it's strange because of course I expected to draw the plane horizontal and I'm forced to take its vertical, okay? and Then you have to look at what is the boundary of E that can happen and you know it has to be a minimal cone Okay, the minimal cone has to stay on one side of a plane and There is essentially two one is a plane Perpendicular to this plane this is minimal okay union the plane because the set always contains the plane, okay? and the other new one is a set of type y which I'm not going to be able to draw correctly But imagine a set of type y which is perpendicular to that plane and you stop here That's the least of two new Minimal cones so that's what I have here except that I forgot to remove half of the sets In my description, okay, so the minimal cones are known There is another good news that the minimal cones they are far away from each other and they don't they are not tangent They don't have a tangent piece compared to the boundary And so in this case what happens that you get in fact you get a description of a minimal cones here It is so here I'm describing the work of a young sin fun, okay So first a list of minimal cones not too shocking you have to prove it But it's not shocking and once you know this you can actually prove some sort of a Reifenberg theorem for the sets of that type and you get that the That in fact the minimal set that you were talking about Here has to be a C1 version of a cone which is there Okay, that's in principle what I have to do first holder That's a first paper and a longer paper to make it C1. Okay, and I claim that this was not Written yet once, you know this it's much easier to prove existence results This is supposed to be the second part of the talk, but we'll probably be cut You know half an hour from now, okay, so that's that's supposed to be the nice situation. Finally Near boundary you get a C1 description of a set as being locally one of those two cones plus small errors and so a typical picture of this is the Picture here where in this case the surface was the boundary of a tube You had to lie at the exterior of a tube. Okay, and locally you can look like this And okay, so you get here a set of type y somewhere. Okay That's and so his theorem is essentially saying there's nothing worse than that No, right that's the good case Here we're now going to try to do the less easy case Which is when gamma is a line? Okay, and again, we start the same program and I first have to tell you what I think is the list of Minimal cones of dimension two with a boundary which is given by a curve There is a first observation is that all the cones that were plain minimal cones They stay minimal in this setting because if you remember the definition of a minimal set all the allowed deformations Have some property here. There is less allowed deformations because you also add the sliding boundary condition therefore At the end it is easier to be minimal in the plane category than in the sliding category Okay So all the cones that were minimal before planes sets of type y and t they stay minimal here No problem, and they don't even need the boundary and then you have two new ones The half plane bounded by the line and you're not surprised. Okay, and the second one is a set of type V bounded By the line so two half planes that make a large angle with each other and again, you're not surprised and Remember we did in dimension one. I said this was one of the options Here is exactly the same thing except that you take the product by line. Okay, so I guess it's clear What a set of type V is again two half planes that look like this No small angle because you could do better by pinching large angles, okay, and I still mentioned below the fact that we have two special cases of plane minimal cones Which is the line that contains? The boundary sorry the plane that contains the line. Okay, which has a special status because Because we'll see and the same thing. There is this set of type y Which is based on the line and it also something special should happen there. Okay Right, so this is the cones We are sure about then there is another one which was suggested by Xiang Yu Liang Which is the cone over the edges of a cube? On the previous lecture, I showed to you that it's not plain minimal because you can pinch the center and Now I'm going to prove you By image that it has a chance of being minimized Minimal with a sliding boundary condition, so that's the next slide and then this will be the list of cones I sort of know okay, I suspect this one is minimal that we cannot prove it and Then there might be other ones in higher dimensions there. It's almost sure there are other ones But let's just talk about dimension three So my bet is that there is no other one, but I'm not sure about that. So this is the proof that Xiang Yu is right So this is you know, I mean you recognize the cone I was talking about and so apparently Things that it's also a minimal set with a sliding boundary condition. Okay, that's my proof Some puzzles so the sliding boundary is the diagonal Which it is strange because I don't recognize my picture What happened to my picture? So in principle there is a wire which is The great diagonal and for some strange reason it disappeared from the picture and I cannot tell you why but anyway So one of the great diagonals. Oh, no, okay. It's the other way. It's the one that is up there So I mean I thought it was a little strange that the pictures would just erase. So this is the Siding boundary, okay, and it goes along one of the singularities of a set and because of it You know, if I try to pinch the way I used to I would have to Near the center here. I would have to Tear people away from the diagonal, which I'm not allowed to and anyway, so that's the reason why this is Yeah I mean this Like this I think I Mean essentially the largest that I was able to construct and plunge Into a limited quantity of soap, right? So that I could show afterwards I mean you need a bath. I mean you see what I'm Saying but the you know the construction is not impressive Whatever I've been doing with soap experiment For the thing Whatever I would get was fairly stable Which means that you know if I had been you know, even my cube had been even worse. I would probably be seeing exactly the same thing and And my red wire is just because yeah, I had some electric wire that I could use But I've been trying other materials like you know without the coating works the same. So, you know Okay, oh No, and actually you can have an idea of a size this used to I think contain ice cream Does it make? It gives you exactly the size. Okay Right, so, you know a true proof would be better, but we don't have that okay Okay, so now well, I'll try to do so of course now we expect that This will be easier for simple cones and more complicated for more complicated cones So I start with the good news The good news is the simplest cone, which is the half plane and I have a fear I'm here that looks a lot like the Geneteller theorem So in principle, I'll not try to read it It says that it's even slightly better than the statement that I gave if you have a minimal Or almost minimal set with sliding boundary a line and it would work also with a curve Provided it's smooth enough if in the In the unit ball or twice the unit ball the set is close enough to half plane So it really looks like a half plane when I look at it Okay, and this will happen if you have a blow up limit, which is like this at some small scales Then in fact in the ball it is exactly one of a half You learn no strange topology going on no little holes In the set Okay, a nice description of a set I cannot hope for better. So that's When you're close to half plane Again, the main story is this I didn't say What are the ingredients for the geneteller theorem? And the same ingredients here Are used for this sort of theorems. It's always sort of the same Uh, there is just one thing The geneteller theorem is essentially based On the monotonicity formula The way it works extremely vaguely Speaking is the following You want to have a differential inequality that says that one of the set is not so close to a cone Then what happens is that its density decays at some speed And then if you control the density Uh, things will get better at the end. Okay, how do you prove this at least how do you start? You say, okay, the density is constant when the set is a cone Uh, if you find a competitor, which is better than the cone Over the intersection of the set we think Uh, then if you put it back in the estimate you will get that the density is decaying at some speed So in fact the game consists for these theorems in proving That if the set is not looking enough like a cone Then the density decays and the density decays because you find a better competitor than the cone And usually the idea behind finding a better competitor for the cone is that uh A harmonic graph is often better than the graph of a cone That's okay, and and then you have some technicalities, but uh, yeah No, you're not going to get the technicalities. So here, yeah Oh, yeah, yeah, sorry. So I usually say c1 and then I try to make efforts to say c1 alpha In most of the slides and then what happens is that yeah, so c1 is equal to c1 alpha for the rest of the lecture. Okay Okay And there is an additional thing which is a little bit unpleasant and will become more unpleasant later The you need to control also what happens on balls that are not centered on the boundary on gamma And for this there is a difference. I think there is a formula here. There is a different function that is monotone And we use that so instead of using the monotonicity only for ball centered at the boundary We have to use some monotonicity of some function But we devise on purpose near the boundary and it will show up later Otherwise, it's more or less like the proof of gender here. And so this is the good case Uh case two the simplest case afterwards is the case when it looks like a cone of type v And again, we would like the set to look like the cone of type v And I think okay, so let's forget about this comment here. So this is the picture that we expect Okay, and this is a picture that we get most of the time and there is one case where you could expect something else And here is the something else And I'll tell you right away Instead of pretending I pretend I I don't know where I'm going Exactly what's the general case So there is two types. I mean there are three types of sets of type v There is the plane which is a set of type v, but very special, right There is uh, let's say something like I'll I'll just take a picture. Uh, there's uh This one I will call it sharp And there is uh, all the other ones which have angles between one other than a 30 degree angle and pi Sorry and 180 And I will call that generic Okay, here is gamma Here is gamma and here I move the picture in a different way. Okay So let me back Up and go to this picture. Here is uh, so sorry l is the same as gamma The set is this thing here Here you have a generic angle And then when you have a generic angle you have you get exactly the same sort of picture What I had before and of course as you expect the angle can depend on the point and very Slowly but it still can vary So maybe the angle here is a little larger than the angle there and so on and so forth And then eventually you get a sharp angle of two pi over three at this point Okay, and then if the angle either thing is sharp You are allowed to do something that you were not allowed to do Which is sort of leave the set In this way, so the main two folds go up Along the set There is a small thin face here, which will say is vertical in the middle And that's one of the things that can happen, right? You're sort of detaching the two things Okay, and essentially I claim a theorem is that you can only do this When the angle is sharp Before you first get sharp and then you can lift Okay So we have something like this and maybe it's coming back here with a sharp angle here Maybe again, there is a small piece here where the thing is generic And then again, it's going to lift Like this, okay, and when it lifts there is a singularity set of type y along here Okay, where you have three Faults right the three faces this one the other one here and the small vertical one here, okay And that's what I claim is happening In other words, this is really this gives you the singularities in this case With a c1 plus epsilon Regularity, okay, so that's that's what's happening and again the proof is more or less like gene-terror theorem So I'm not Going to see too much. So I think all this description is In case I forgot how it worked, but I then I could read it Okay, there is still one case. Sorry, so Okay, where am I, okay, right Generic case is the picture that I was talking about Sharp case then some curve can leave In a plane, I have one last comment to make And oh, sorry and the comment goes with a picture. So here's the picture So I said earlier that Plane minimum sets stay minimal whatever happens. Okay. So in other words, if you take a set of type y or a minimal set locally and you add A boundary, I mean sort of a needle through it Okay, if you do it carefully the needle can go through the thing and it's not going to change anything In other words, the set will still exist the way it was and it's just that there is a needle that goes across That's one of the options. Okay And there is another option that can happen in the case of a plane which is tangent to the line Which is that most of the time The thing looks like a minimal surface. Okay, and then along This piece here where this is the intersection This is contained in the intersection of the set and the line Okay, and again the line is called l because I picked I mean I took a picture from some other place So along this thing here, you can have a v set which is very flat It's the same story as before The v set was infinitely flat at I mean it was a plane here. It's a plane here in the middle. It can make a tiny crease Okay, and again, this is the c1 description Near a point Where the tangent is a plane Going through the line. Okay. So these were the examples that we know how to tweet There are a few other ones, but okay, let's discuss it. So If you are slightly disturbed by this picture, but in principle it makes sense You can also do a nice change of variables Very nice change of variables should not change so much the notion of almost minimal set Get to This picture Okay, so this is the same picture after flattening the set. So now the set is a plane The curve gamma becomes a curve which is slightly different And what can happen is that the curve goes along the set from time to time and Maybe leaves the set a little bit and then goes away and then returns Okay, and I'm saying the position below is very clear So this one is clearly minimal because the plane is minimal and of course if if I fancy to take Sliding boundary, which is the curve that is here. It will stay minimal. So that's allowed Okay So in other words the picture below is clearly allowed And I'm saying the general case is the picture above Which is essentially the image of a picture below by a nice mapping That's one way to say it. Okay Right On the previous slide I had mentioned that in some earlier paper But I am not completely sure but we have the same definition of minimality Bracker said that in fact from a plane you could leave the plane directly by going up I claim not But I'm not completely sure that Bracker really meant this And I hope I'm not wrong. Okay But anyway, that's okay This is the same picture about these sets. Okay, so Here is the same picture that we had before more or less And I'm saying also you could see this as a small deformation of this picture Where this is clearly minimal because Well, no in this case, I don't say that it's clearly minimal because it depends on the boundary But where you would have you know a v set and then the curve that is allowed to Go along the boundary or not But I mean in this case, it's not clear that the v is minimal Because v is only minimal because there is a boundary. Okay So this picture is About the proof I decided I'm not saying I'm saying extremely little about the proof It goes like the ginger theorem, which I did not prove again And there is just the slightly different function here That I use which is monotone Because the density is not monotone. Okay, and I just show this to you On one example Let's say that e is a half plane And let's say that you pick a point a center x A little bit outside. Sorry Let me not be A little bit in the plane But not on the boundary and if you look at the usual density On both centered like this you have density pi Up to here Then the density decreases Eventually for extremely large balls the density is pi over two So it's not monotone in the right direction, right? And this functional in this case You just add here The density Of what I call the shade seen from x of the line What is the shade X of this line? It's exactly the other upper half plane Okay, and so in this present case, you know, I just add This increasing quantity, which is just enough to compensate the trouble that I had here So this functional is monotone In this case, it's really doing what I want it to do because there is this case where it's constant Which is the case of a half plane. It works also In the case of sets of type y that are Placed in the right position And afterwards it stops working and that's the reason why we'll have trouble soon. Okay But anyway, so that's and otherwise you follow gene Taylor's theorem Okay I don't know if I should be happy that there is at least one extra monotonicity formula Or if I should be unhappy because I don't have enough of them For completing the job. Okay Now here is the bad case and maybe don't read everything that's here. The bad case is the case Where the tangent cone is a set of type y with its Uh singular set equal to the line. Okay horizontal Okay, and in this case what I have happened is that well, but what happens is that I have a conjecture But I will give you a picture in one second. I cannot prove it. So I'm in trouble And uh, the reason why uh, I cannot prove it is because this monotonicity formula that I introduced there Essentially doesn't work in this case. I mean it's I mean I've been adding too much So for instance in the case of the y that I was talking about The formula is not constant. It gives you this strictly monotone function that I cannot help Then I cannot really use. Okay, so that's so anyway, this is I claim the main case Probably along the slide there is this fact that there are lots of other cases that I didn't mention But this is really the one that causes that causes trouble for me Okay, and the reason why it causes trouble for me is that you expect points of type y or maybe not along the whole boundary for the other Cones you might have an isolated singularity, but nothing so bad. Okay Here is a vague idea of what my conjecture would be that I cannot prove. Okay, so on top is uh Something which is clearly minimal, which is a set of type y because it's plain minimal and then I decided to draw a curve You know in an annoying place, but maybe not completely related So I decided to take a curve that goes along one face here then along the Uh center of a y there then along another face here and then again here and in this place here It even goes away from the set y because it's allowed to okay Right and this so the picture here is a minimal set The picture down here You can believe that it's going to be almost minimal. Okay It's the same picture except that I just straightened the curve and made in made it the line gamma Otherwise I essentially You know pushed things a little bit down and I have a set like this And I'm saying this should be the description Of this type of singularity except that I cannot prove it. Okay Okay, so it sort of makes sense But uh, what happens is that if you try to do a proof Other Possibilities appear and I don't know how to rule them out. Okay, and I think I don't want to say so much more about this So again, the this may look like I mean the picture here may look like a complicated picture Yet you can give a description of it, you know There is a curve along this curve You have a singularity set of type y and then here maybe there is a little hole But it's okay because it's a smooth surface there and and so on and so forth It's a complicated description, but it's a complicated description But it's a description and anyway the best way to understand this description is just to look up here and say You know the set was a y Distorted a little bit. Okay What's hard is to prove the theorem that I would want Okay, so these are pictures of It's the same pictures here. I've been cutting things into slides and I get all possible behaviors and Okay, this would be so the other possible behaviors that would Appear here This is the typical thing that a typical section that I have to rule out But okay, I can't And Other questions are so that I think I have a main other question here