 Okay, so welcome back to this morning section. So unfortunately I have to say it's always my duty to wake you up. Which I mean it's not that I like very much, so I apologize. And that's why we always try to do this kind of as smooth as we can. So let me always remind you which was the point we had yesterday. So now we passed to study the singularities of these surfaces which are parametrized by by graph or multiple value function. And the idea of the study is a blow up analysis. So we take a singular point, say x0. And then we try to focus on small scales around this singular point. So we consider rb, then 0, and br, x0. And then we apply an amount which sends br, x0 to b1. And what we would like to understand is if in this limiting process we may somehow simplify our objects in such a way to deduce something on the, on the sides of this singular set. So for example, if here we had singularities accumulating to x0, we'd like to understand if the same happens somehow here. And the hope is that the limiting profile of this blow up is kind of simpler than the function itself. So let me do a kind of a parenthesis here. So this kind of analysis of idea of blowing up and starting singularities via blow up is typical in geometrical analysis. And it's also the way singularities are studied in co-dimensional one. So when you have, in the case of hyper surface, you perform the same transformation and via the monotonicity formula, so by, so the classical monotonicity formula, we get that the limit is a cone. In the case of co-dimension one, we start from something which is kind of singular, then we blow up this picture and we get a cone. And here you see we have gained one dimension of symmetry. So the radial direction here is kind of prescribed. And via the Allah's theorem that Camillo showed you in the first week, we know that this cone is not flat, so it's not a plane. Because all the points where the excess was small, so where we were converging to a cone were actually regular points. So we have such a picture and starting from this consideration, you may try, and that's what is done. To analyze and understand the behavior of this singular set. So now what we are trying to do, it's kind of really the same. But the situation is a bit more complicated. So this classical monotonicity formula, which is the ratio between the mass of the current and the mass of the disk with the same values, is not giving a meaningful information in this case. Because here, just blowing up in that direction, what we get is twice a plane. Because here we have seen in our model singularity, which was that complex variety that we had a continuous junction plane. And in the limit, we were getting a double plane there. So performing this same analysis is not giving us the right information. And indeed, what we started to do yesterday was to cook up a new monotonicity formula, which was the monotonicity of the frequency. So instead of looking in the usual ratio between the mass of the card and the disk, we look at this function, which was called my uncle and frequency function, which I remind you was the ratio between the energy of our parameterization divided by the L2 norm at the boundary. This quantity we showed yesterday was monotone. And now this one is the quantity we will use to somehow perform such a blow-up analysis. So let's first understand, so the first hour of today, which I mean will be kind of aimed to conclude this blow-up analysis, will play somehow everything will be based on the monotonicity of this formula, so to say. Let's understand which kind of rescaling we need to do. So in this case, of course, I mentioned one, we were somehow rescaling isotropically in all directions. And then we were getting the code, which was singular, so we were looking at this angle here somehow. In this case, I mean, if we rescale isotropically, we get a plane. So not any information on the limit. The way here we have to scale is kind of an isotropically, so maintaining the structure of the parameterization. So we will rescale kind of vertically and horizontally in a different way. And the scale we will choose is. So now I write you are for the blow-up on the ball be R in Y. So this one will be. So horizontally, the function will be scaled as R. But then we have to normalize vertically this function in a different way. And the way, I mean, to gain compactness for the parameterization, you have various different way. I mean, one way could be to keep one of the norm fixed. And we will keep the energy fixed. So we normalize here by the Dirichlet energy in VR of X0. And then you will see you have a scaling factor here, which is R to the m minus 2 divided by 2, if I'm not wrong. So what we are doing is horizontally, we look at scale R. So it's true this picture here that we are focusing on a ball be R. But vertically, we need to rescale our function in order to maintain a norm. And doing like this, you see that the integrally B1 of DUR is equal to 1. So what we are maintaining is the energy. Now the first proposition we would like to see now is that actually we have compactness for this for this rescaling. So as for the limiting cons, we were able to say that this rescaling we're converging as a current to something which is a cone. Now we would like to prove that these rescales do converge as function. So L2, but actually they will converge with W12 to a limiting profile, which has some extra symmetries. And that's what we prove first or we almost prove first. So the first result we are going to see of the domain or of the current if you want. And L is our coordination. OK, so the theorem, I mean it's a theorem is too much for such a let's call proposition. So we assume that you in W12, let's say B1, AQ is is the minimizing. And then we assume some extra hypothesis. Let's say that we focus on a point X0, which is a multiplicity two points. So you have X0, but I mean we assume zero is two times zero. And then let's assume also that this rescaling is always meaningful. I mean, of course, if at certain point the energy is zero, this means that you is constant and that's the only minimizer. It's it's regular, so we will not look at the issue there. So we may assume that the energy in the R is bigger than zero for every R. Bigger than zero. OK, the conclusion, a subsequence Rk, say, and a function U, which is the limit in W12, B1 or let's say RM, AQ, such that the following code. And then we have, so the first conclusion is that U is still the minimizing and in B1 as energy one, so it's not zero. And the other conclusion is that actually U is a is a now simpler than a generic diminishing is what is called alpha homogenous. That is U of X is equal X to the alpha U of X over modulus of X, where alpha is a number bigger than zero. And that's correspond to the limit as R goes to zero of the frequency of U. So it's it's it's W, W12 log. So let me I forgot to put there. So Uk, U Rk converge to U in W12 log. So so what does it mean for multiple value function converging strongly? So this one has to, has a mouth to say. So this one means that we have convergence L2. So the distance between U Rk, U L2 is going to zero. And we have convergence of the energies. So the integral br of the B of the U Rk squared converge to the integral of the U in BL squared, whatever it is. So we have convergence of the functions and convergence of the energy and that's very bad. Yes, so let's call this is in the notes is G. So but what is important here is that in the end we have a mini measures G, which is non trivial energy is is one. And we simplified our, our geometry of such surface. So for the minimum cone, it was linear. So it was here with alpha equal one. Now we cannot claim that alpha is equal one, but you have still something which is homogeneous. Of the real world also is exactly the, the, the frequency of our original function. So this one plays exactly the role of, of the blow up for the one case. So let's see a sketch of the proof of this, of this proposition. So let me see what, what we can skip. So that's a roof, but won't be complete. So what we, what we need to observe. So first of all is that. So first of all, we would like to say that we have compactness this UK. So let's check which properties as this function. So we, we start noticing that. So that's, I mean, we have said already before the energy of this you are K. And then we know that we were blowing up a point of multiplicity Q. Which in this process is fixed. So all these functions at zero as values Q zero. So now by the continuity theorem, we knew that all these functions, which are diminishing on the ball be one with bounded energy, they are uniformly all that continues. Because the older norm was bounded by constant, which was, which was proportional to the energy to the square root of the energy. So by the older continuity result, you are K uniformly continuous. So you are, you're still to take a sub sequence. Then up to taking a sub sequence, we can find a converging sequence. Say so in the uniform norm, but let's say in L2, it's enough somehow. Okay, this one seems quite, quite standard. But I mean, let me say that here a few details are missing in this, in this argument. So what is missing is that we have just verified that the energy in B1 was bounded. And the older continuity result is true in interior. So in principle, we should say that this convergence is in interior. But if you look carefully at the monotheistic formula of the frequency, you see that you have a bound of the energies at every scale, one with the other. Which is a corollary that I should have put in the notes somewhere. It's a very simple consequence of this, of the computation we did for the frequency. So actually, not just the energy B1 are bounded, but the energy in all the balls. So here I could put BN and here a constant and this one remains still true. So now I can apply some of these compactness in any ball smaller than BL. So meaning that up to some sequence, I have convergence locally in the red. So this one, this argument is missing somehow this, this point here. But you'll find this in the, in the notes. And now a point which is missing is a, which I won't prove here. But again, I put it in the notes. Is that, and that's also is kind of very, a very standard argument in calculus of variation. Which has to do with the concept which is called gamma convergence. Is that when you have a sequence of minimizers and, and some properties, of course, you converge to a minimizers. So the claim is that our limit G. So converging subsequence in L2 towards G. So the claim is that our limit G is the minimizing. So it's still a minimizers of our function and I have the convergence of the energy. Which is what I claimed here and the convergence of the energy holds. So you may see this as a consequence of very general principle, but actually the proof again in the notes itself contained is quite simple. So you know, you may assume that one of these two conclusions is not true. So this means that in the limit G may miss either G decrease the energy of your sequence or G being not minimizers. You may find another function which decreased the energy with respect to G and the boundary value of G. But then since you have convergence in L2, you may select a good slice where you have the convergence of the slice. And this means that the competitive function for G, which was improving the energy for G, it's also improving the energy for the sequence. And this one cannot be because our sequence where the sequence of minimizer. So this one I won't prove you find all the details in the notes. But it's a very standard. It's kind of the same conclusion is for the theory in co-dimensional one that when you have a sequence of minimal currents, they converge to a sequence of minimal, minimal currents and the mass is continued. So the only thing now I would like to verify and what is important for us is this point two, so the structure of the limit. Why we can conclude that we limited this extra symmetry. And for doing this, we look at the frequency of the limit. So now the limit is a minimizer. So all the conclusion about the frequencies we had yesterday still holds for G. And now we see what is the frequency for G. So compute this frequency in zero. So this one we say this R, the energy B, R of DG squared divided by the L to norm of G squared. And now, first of all, we know that we have uniform continuity for the U, R, R to G, and then we have continuity of the energy. So we can write this in terms of the limit with respect to the U, R, B, R, D, U, R, K squared divided by the L to norm of the boundary of U, R, K squared. And now U, R was just a discrete function with respect to U. So if you perform here the change of variable back, you obtain this formula. So R, R, K integral in B, R, R, K of D, U. Now, squared divided by the L to norm in this most of U squared. But this one is nothing else than the frequency of U at the radius R, R, K. R going to zero of E of U at R, R, K. Which means we know this quantity was monotone and was convention to the limit which we called F. And now you see we have that for G and for a given R, so this one is a limit R, K. For G and for a given R, the frequency is always half. So the frequency of G is constant. And we argued yesterday that this one can happen if and only if G is homogenous. OK, and now the last conclusion. So why, why alpha is bigger than zero? Now comes from here. This was a question also left open yesterday. Why alpha now is bigger than zero? Assume alpha is zero. We would have that G now is zero homogenous. So meaning that it's constant on the radii, but it's continuous and in the center G as value zero. So this would be the only possibility for alpha being zero is that G was zero. But G had an energy which was, which was not trivial because it was one in B1. So this cannot happen. And we conclude also that that actually alpha was positive. OK, so what we, so here it's kind of hidden what we have, what I told you yesterday. But basically using this monotonicity of the, of the frequency we deduce that in the limit via this, this suitable rescaling which maintains the norm, the energy of, of the function. We obtain something which is not trivial because the energy of G is still, is still one. And which is, which is radially symmetric. Not really as a code with the power one, but as a power of the, of the radius. And now we start from here and, and we see how this first blow up has kind of simplified our, our, our situation. Because, because now if in some sense we are able to, to say that in, in this first blow up limit we don't lose track of the singularities, then somehow we may directly argue with something which is already, already radially symmetric. So, and that's what we do next. So now we come really to the core of the matter and we, we start the proof of the estimate on the singular set for such function. So let me see if I forgot on the frequency. Yeah, here for example. All right. Mm-hm. So for the energy you normalize planning out to do? Yes, which, I mean by the way, it's not, it's not the only thing you may do. So for example, for the proof in the, in the case of currents we, we normalize for the L2 norm of the function. And still I mean since the frequencies thank you that these quantities are all comparable you, you may conclude. So it's not the only thing you may do. Now in this case of, of functions, it's very elementary doing like this, but for example, already for, for currents we do something different. Mm-hm. Okay. Okay. So let me now, now proceed with the, with the main part of the, of the proof of the argument. So, so now we come to the proof of the main theorem, which was I give you a very minimizing function. And I would like to see that the singular points are of dimension m minus 2. So now we try to prove that the dimension, the out of dimension of the singular set of u is less or equal to m, m minus 2. And they claim that somehow the core of the matter is this, is, is, is this lemma here. So lemma, which is, I mean, it's sort of equivalent to the, to the theorem, but so the core of the matter is this, is to give an estimate on, on the set of points of higher multiplicity. So I, I said, let's say sigma q, the point x in our domain, such that u of x is equal to q times the point p for some p in, in our n. And q is the maximum multiplicity because we assume that we are a q cauldron of our domain omega always. And the claim is that then either this set is the entire omega, which means that u actually was q times a, a regular harmonic function, in which case we have no singular sets or the, the out of dimension, so the dimension of this set is at most minus 2. And this is, it's countable or even discrete when m is equal to 2. That's all. So this one is the, the analog of this theorem here. In the case where we just look at the points of multiplicity q of higher multiplicity. And once you can prove this, so you may understand the singularities of the higher multiplicity points, then you may argue by induction to prove the theorem. So the proof of the theorem is kind of by induction on the number of values q. So when q is equal to one, of course, there is nothing to prove. We have just, so the basic case, everything is regular. So the singular set is empty. And that's because u is a harmonic. And now we assume that we know the result for every q less than a q star. So, so assume the proposition q less than q star. And then we want to prove for q star, p there. So p is a point in Rn. So it's equal to this for some p in Rn. So there are points where all the leaves of the function are collapsed into one, one point. This, this one, no, this one is one value. So when you have one value, nothing to be discussed. Because one exactly over omega. So what we have is that omega is our domain and here we have two values. So when the value is one, nothing to discuss because it's a maximum one. It's a, it's harmonic. And now we assume to know this result for any q less than a given one. And then we prove for this new one. Now what we, we take is, we consider our set sigma q. And now two things may happen according to this proposition. Either sigma q is the entire domain. In that case, I mean there is no singular set. Everything is a q copy of, of an harmonic functions or the sigma q as dimension M M. If this one is the entire set, then the singularity of you are empty and we are empty. So we can assume that the dimension of this sigma q is less than M minus two. And now it's, it's very simple to see. So the function is continuous. This set is defined by an equation. So this set is closed. So now we can consider the complementary set. So since sigma q is close, consider the open set omega minus sigma q. And now what happened in omega minus sigma q? That's locally, so we have no maximum multiplicity. Our function is continuous. This means that locally we can split our function over this set. So let me make here a picture. So case of two or even three, three points. So here sigma q is just this point here. So now q here is equal to three. And this one is the only point of multiplicity three that I have outside of this point. I am always able to split my current in the graph of my function in different packets, so to say. So for example, here I have actually three, three different leaves. But I may have also a situation like this. So around this point here, I'm just able to split the current in two blocks, where one is a one-valued map and the other one is a two-valued map. But anyhow, what I know is that locally, so for every x naught outside of this maximum multiplicity set, there exists r bigger than zero such that u in the ball, b, r, x naught is equal to the sum as measure of a function u one and u two, which are diminishing and are q i value functions with q i less than two star and q one plus q two equal to star. So at least I find two blocks. So for example, a two-valued blocks and one block where the multiplicity is strictly less than the maximum one because I'm away from the maximum multiplicity point. They are still minimized because this would be continuous. They are just isolated on a very small scale, which is arbitrary here. And the sum of these values, of course, is is the total value of the covering. So above each ball here, we have exactly the superposition of two functions, which falls in this case here. They are diminishing and q value for some q that is less than q star q star q star because now we are trying to prove this for just us. Sorry, sorry, sorry, sorry. The maximum multiplicity here is q star. That's the content of our proposition that if I take the maximum multiplicity, then the set with the maximum multiplicity satisfy this conclusion here. So now for this one, yes, this one is this, this conclusion here, the conclusion of the proposition, the theorem. Yes, sorry, that's the proof of the theorem assuming the proposition. So let me repeat this. I mean, now it's completely unclear. So let me repeat this and fix the notation. So the proposition says that when I take the maximum multiplicity, then this set is either the entire domain or as I mentioned, M minus two and in this in this proof, the higher multiplicity is q star. So let's put also the q star here. So there is no confusion about this. So the proposition claims this second line here or this first two lines that either we are exactly a superposition of the same. Function for q star times the same function or we are very small in measure. So that's what is claiming by the proposition. Assuming this improving the theorem by induction. So assuming that we are now in this situation here outside of the set. So now this one is q star. I'm able locally to decompose in simpler in simpler functions. And once I decompose in simpler function, I apply the inductive hypothesis. So now what I know is that the singular set of you one and the singular set of of you two as dimension. And now you may argue via a simple covering arguments. So this one is an open set of this of this bounded domain. So for each of the points in this open set, you have a ball where the singular set is the right dimension. Then you cover with counter remaining of them and you have that all the dimension of the singular set, which is the one for each package in each ball plus this one as dimension at most and minus two. So now by simple covering include that all the singular set of you as dimension at most and minus two. So the proof of the theorem is really what I'm saying is that via this very simple, induction argument, the theorem is actually reduced to look this property for the higher multiplicity points. That's really what matters because the rest may be treated by by induction because we reduce the complexity of our problem. So all the effort should be done somehow to prove this theorem here. So that was just very quick. But I mean, again, in the notes you find all the details to say that the core of the proof is estimating the higher multiplicity points. That's all. And now let's come to this point of the multiplicity where we meet another of the main analytical issues of this theory, which as for the others, I would like to explain you via an example. And now this issue may be called kind of sentry. So do you repeat this so that the singular set of of in this ball. We are of X naught is equal to the union of the two singular sets. So it's the singular set of you one union, the singular set of of you two. This means that you as not singular set on that ball. No, I'm saying you in the ball be are of X naught because you one and you two are just defined on this local ball. So when I have two values, I'm not looking at these points here because this one is a point of higher multiplicity. So this one is my sigma q, in this case, sigma two, and I'm looking at the point here and here actually my you, like my you one and you two are completely regular. On the other hand, if I have three values, so this one will be sigma three and I look at this point here. This one is a singular point for you. And it's a single point for one of the two functions, say you want. So the singularities of you are exactly the union of the object. This one you verify easily from the definition of singular points. So let's leave this here for a while and let me start in this this kind of last last chapter of this of this story. So now we reduce to consider the higher multiplicity points, but still we have a problem with what I told you about the persistence of singularities. So we would like to say that when we perform our blow up, if I have a single point, I remain with the single point, which will for current is not that easy and there is no this this correspondence for functions a bit simpler. But there is kind of a very key analytical issue behind this. So this one, I mean, we can say it's a problem on the on the persistence of singularities. And one of the issues was this frequency that I mean we don't have to collapse to zero because otherwise we lose all the singularities. But there is another issue, which is somehow connected with the centering of the lease of our functions, which says that I mean, even if we do not collapse to zero, we may still lose all the singularities in the limit that's may still happen. And for understanding this, as usual, let's consider an example. So I mean, I mean, take the same here in the notes. So I take an example of a complex variety. As usual, it's a both an example of a minimum current and of a diminishing function and this complex variety. I called the first one being this one we call W. We take the point that W C2 such that let me maintain the same notation here. W minus that square square equal W to the final. So in the notes, I gave all the formal way to express this formula here, the break, but let's do like informally and informally. I'm saying that so this one is a Z informally. I'm saying that W is equal to Z to plus or minus the two square root of that to the fifth. So the two determination of the square root. So that's if I if I solve this equation in W, roughly speaking. So when I try now to to draw a picture, what I get is this that at the first order, I'm looking at the solution of W equal Z squared, which is this function. So that's W Z. So at the first order, I have this picture here, which is a very smooth embedded disk. So this one is a very regular surface, but then I'm I'm branching on this surface. So the real the real calligraphic W, which is the yellow one. Now it's it's a branching on this. So this one is it's a complex car, which is not smooth in the origin because I have the same phenomenon of a branch point, but I'm not branching around the z plane like in the first one. But I'm branching around this smooth disk, which is this W equals X square. So that's the picture that that I have for this complex curve here. And now you may try to do the blow up by yourself for this function when you do a blow up here with the formulas I gave you. And I mean, actually, I invite you to do this, this exercise. You verify that you converge to this function here. So you converge as a blow up to the smooth car that square equal W. There is no way. Just take your formulas. We scale center here. You should do all the computation explicitly passing to the limit. You obtain this. As what as a two dimensional is a two value function. So this one will have multiplicity to this. So you compare it to a double copy of this of this disk, which is an embedded disk. So we started from a single point here. We perform our block. We don't have any more singularities. So we can understand anything about from the blow up for such function. So what is here the problem? The problem is that these these minima surfaces may have a regular part, which we have some amount to mod out in this blow up limit. So somehow here the best would be not to see this as a function over the z plane, because what we see in the limit is just the first order part, which may be kind of perfectly regular. But the best would be to see this W as a function over a kind of centered manifold. So to change coordinate, find what is called after I'm going to center manifold, which in this case is exactly the smooth a this smooth disk. So it's actually very smooth manifold. And then look at W as a parametrization on the normal bundles of this center. So the main idea here would be. OK, just performing a blow up analysis. We we may not be sure that the first order of our surface is a singular order because something like this may happen. This one is an example of a minimizing surface. What we should do is somehow mode out this first order, which is regular kind of finding this this first regular part of the surface and then just looking at the singularities over this this manifold, which is a very nice and smooth method. So the idea here would be. So in this picture now I change a bit. So I don't look this on the normal bundle for me for function. It's not necessary, but I may look this kind of basically parameterized on this. I mean, for me, my current you would have to go to the normal bundle, but here it's not necessary. So the best here is OK. Instead of taking that squared plus or minus the square root, I just subtract my my center bay. So I don't take this, but I look at a new map which says that not to so not this that to that square plus or minus. That's five over two, but I just subtract this middle sheet. So I look at the at the function, which is F, which sends that into plus or minus that five over two. So what I do is I modify my my minimizer, parametrizing it, not with respect to that, but with respect to this middle sheet here. So that's I mean for function is what is going to work. Technically, it's very simple, but geometrically is a key point of the of the story. So in order to have this kind of of persistence of singularities, we have to subtract the average of of the values. So what we consider is not you, but we have to consider. The function of X, which is the sum on all the values of what? Of the values of you minus their average, which we call it a composition of you. So we eat a composition of X, which is just the function given by some of UI of X divided by Q. And this one now it's a function from omega to a red. So it's a classical function. So what we do, it's exactly what I have. Picture in this, whatever I've drawn this picture is at the state. So here, for example, the two values of you are that plus the square root that minus the square root. So in this case, the average of you is exactly that square. So this one is the exact average. And then we take as a function the values of you minus that square, which are exactly these two. So this general formula is what they did for this example here. No, the other one was not a smooth one. That one, the isotropic limit. But if you do the scaling I introduced for the blow up, you remain with the same function in that case. Yes. So let me say in the other example, which was as I remember now, which they say that three equal w two. And in which case, I mean, you see something like this. So let's put here, W here that and then you have this multiple values function here. So which which now corresponds to this. So you send that to the two square root of that tree. So when you do the blow up of this, you are constantly equal to you. So when you do you are with the formulas I gave you, so which was now it's two dimensional. So it's just the dilation divided by its energy on the R square root. So this one is just a multiple of you. So it's it's just a constant times you. That's another computation you may you may do. Because it's not an isotropic blow up in which case you are right. You get a flat plane with multiplicity two, but this this blow up is getting horizontally vertical in different ways. And if you verify this one is just maintaining the branch points of you up to up to a multiplicative constant, which is the one which is making you with energy one. In this case, you as not energy exactly one you have to normalize by its energy. So this constant is is exactly equal to to the energy of you in B1. So that's what you see in this club forever here. So actually you get something which is fixed there. So this phenomenon is kind of of genuinely new. So you may still converge to something which is not trivial with energy one, but all the singularities may collapse. So that's that's something we have to deal with. OK, and the general solution in this special case of parametrizations is this. Just just subtract the average and why this works because the average is a very, very smooth manifold that's a that's due to the linearized in behavior of this of this function. Because what you can prove very easily and I leave to you as an exercise is that if you is diminishing them, the average of you is harmonic. So that's first the first thing. So you were solving this generalized Laplace equation, but the Laplace equation is linear. So when we have all the values, we get a classical function, but the function, this one is harmonic and the other thing you may prove is that. If you is still minimizing and we subtract of some an harmonic, then we get something which is still minimizing. Then F of X. So that's formula there, which are the values of you minus its average. This one is still the minimizing. That's a very simple integration by part computation. So not just the average is harmonic. So this middle manifold is sent and manifold is an analytic method for this very smooth, not just this, but when we perform this operation of translating our values, we end up with something which is still a solution of our problem. I mean, nothing to say this one maintains all the singularities of of you. So now that's just a simple observation that the singular set of F is equal to the singular set of you. But the Q multiplicity point for F now are just the points where F is equal to zero. So within this operation, we simplified a bit more our set of higher multiplicity points and not just the one with higher multiplicity, but exactly the one taking multiplicity Q zero. So they are all so now with this with this translation, the singular points are all on the horizontal lines. So in some sense when it's now very simple to understand that when we perform our blow up, having that everything collapsed into a sheet means that everything should collapse to zero because the only high multiplicity leaves is the one at zero level. But we cannot collapse to zero because of the estimate on the frequency. So somehow subtracting this averaging take a into account this problem of collapsing everything into a single list because the only possibility would be zero, but zero is not allowed by the frequency. So let's so I'm a bit late. So let's do just five minutes break, meaning that we assume in a let's say eight minutes at a quarter to 10. And then we conclude this this proof. So now we really come to the final final argument of the proof. So something on the blackboard will be missing, but you have in the notes and now and and now we will do the proof of the proposition, which is let's write in its simpler form the dimension of the multiplicity Q points. Let's say for the M minus two and we will do this proof in a simplified case. So the case so we consider the case M equal to the case. I'm a beginning to it's just a bit more technical, but if you know sufficiently some machinery, geometric measure theory, it's anyhow very standard. So it's very similar to the case M and M equal to and in this case, we prove slightly something slightly more, not just the dimension is zero, but we prove that that Sigma Q is made of isolated points. That's what we are going to prove or to give the sketch of the proof in the measure two. And the proof now it's very it's kind of very standard scheme in geometric measure theory. So what you have, you have a singularities. You assume that that the singular set is very big in this case. For example, we have an accumulation of other singular points towards it and then we blow up it. So let me just just recall. So this one is the set of X such that f of X is equal to times zero. Did I assume Q equal to or not? I think I think no. So this one is this set here and without losses or generality, we may assume that zero is a point of this set. So we zero is a singular point and we try to understand if we can have accumulation towards zero. The proof is by contradiction. So by contradiction. We assume that there exists X K in Sigma Q. Accumulating to to zero on this part of the blackboard. Let me do a picture so we have zero, which is a singular point and then we have this X K, which may accumulate towards zero and they are all singular. And that's what we would like to exclude. So we assume this and the operation we do is the following. So we blow up at the radius, which is the distance of X K to zero. So this one will be our balls for the block. And so what? So we set R K equal the norm of X K. So let's be very funny. So and now we consider our our escape function. Let's just call for simplicity of K. Of why so to change variable that's F of R K Y divided by its energy, which I mean, let me write in a concise way yesterday. We write D for the energy. So D R K one. That's the direct energy in the ball. The R K. OK, and look now at these functions here. So we know by all the previous staff that up to subsequence, we are converging to a de minimizing G, which has not one average zero and so on. It's not it's not trivial, but let's look now the point wise behavior of this G. So what we are doing is we take these points, which is points Q times zero and we sent in two into radius one. So so this ball here. It's sent to be one and I have here the corresponding. Radius, which is X K divided by its norm. So it's a point in S one. Where the function as value two times zero. So here. F K of this point is equal to times zero. And we know already that everything is converging in the in the uniform. No, because everything is is uniformly all that continues because the energies are all one. So we have here a limit of points where the value is zero. So if if we consider now as one is compact, if if if you consider up to sub sequences W, the limit of X K divided by its norm, which are points in S one. Then. By uniform. Convergence, the limit. So G is converging FK is converging uniformly. The points are converging. So the limit of this value is exactly two times zero. So we have now let me. This one was the extra conclusions I wanted to to deduce. So collecting all the conclusions. From B one to a Q is the minimizing and it's not trivial. So the energy of G is one. And since they were all function with the average zero, it's very simple to see that also they are converging uniformly. So also the average of G is zero. So G is made of values which sum up to three. We have that G is alpha homogeneous. That was one of the conclusion of our blow up analysis. And finally we know that we have a point of the boundary of S one where G is zero. So G of zero is equal to G of W is equal to Q zero. For some. W in S one. And now I show that starting from these four conclusions, we have the contradiction. And for doing this we will again something which is very classical in. In geometric measure theory, which is called cylindrical blow up. So the first plot is around the singular points. And now we blow up off of the line of the single point. So the picture there is the following. So in the limit we have a function G living here. And at a certain point W, which is the accumulation point of this of this boundary value, we have value zero. But now G was was alpha homogeneous. So all this line, all this line here, it's a line of points of zero. And that's what I've been saying since a while that G has simplified our picture. So starting just from an accumulation of of singular points in this limit, we get because of this homogeneity, we get all these lines of zero. And now we make a blow up at a point here offline. So we just consider the second blow up, which is called cylindrical blow up. So by the way, all of this is done also in co-dimensional one, when instead of this block maps, we have the detention codes. So this one is very, very standard. Somehow in there knowing a bit of machinery in geometric measure theory. So we make this, this, this cylindrical blow up around, say W over two. So we take, let's give another name, R S of Y equal G W over two plus S Y divided by the square root of the energy around S W over two D G squared one half. And now here comes a technical lemma I don't want to show at the at the at the blackboard, but I can give you an intuition for this name. So every time you have something which is, which is kind of radially homogenous, so it's, it's invariant somehow on the radii and you blow up not in the origin. So at the middle of one of this, of this, of this radii, you get something which has, which has a symmetry, which is cylindrical and that's why it's called cylindrical blow up. And you have to think like this. So this radii close by to the yellow one, as you get closer and closer, are kind of parallel and much more parallel. So when I, I blow up here, the picture is, is basically making all these radii straight. So when now from here I do this second blow up, I get such a picture here and this middle one is, it's always the yellow one. So here there is really nothing deep. Just take the formula, write the limit of these functions, remember that G was, was already radially symmetric and, and you see that what you get is to have an, an invariant direction. So these, these functions here up to sub sequences are converging to a function H, which is a function of one variable. So, so if, so without loss of generality, put coordinates, this point W was actually in E1. What you get in the limit is that H of X is just a function of X2. So you have the invariance in the direction H1, but this one is still a blow up. So as all the properties of the blow up, so in particular we have that now H. So let's, let's talk about H, which is the entire function, but it's just a function of one variable, but H is, is diminishing its energy. It's, it's not real. It's energy is, is one in the right ball as still the average, which is zero and is a, okay, is, is homogeneous. But this one, I mean, we don't know. We may forget. So what is important is that we have, we have taken the blow up at the point where the function was zero. So in the origin H is still zero. So now let's make a picture of this. And now we look just this section because this function is just described by H bar. And I make the picture here. So now we have a function which is a, so I can draw this picture with a function which is invariant in one dimension. So let's draw just the picture of, of, of H bar. Maybe it's better. So for each bar, we have, this one is the e to the dimension in zero, we are zero. And then we know that it's not trivial. So it's not constantly zero and it's average. Sorry, it's zero. So, so I have basically a number of values. Which, which balance to zero, which sum up zero. So if Q was equal to two H was looking like this. So each value was plus or minus. So H bar of X one is equal to minus H bar two of X. If the values are two or otherwise they really sum to zero if they are more than two. But now this picture is quite, is quite a simple picture to analyze. So we have function which is zero in the origin and it's one dimension. And that's to minimize the Dirichlet energy. That's a very attractive exercise that the only one dimensional minimizers are the linear function. It's kind for, for the minimum surfaces. The only minimizers are segments in a row, but, but not just segments. You want the segment, but as for minimum surfaces, you know that if you have a minimizers, a one dimensional minimizer, they cannot be to intersecting segments. So they are not intersecting. That's kind of the minimal connection between points in a ramp. That one is done for, for the area, but it's the same for the energy. And I invite you to, to verify this as an exercise. So when you have to connect a few points and let's say we have a boundary given by, by disorder here. So this, in this, you have to connect with this and this. You know that this one is not a minimizer for the length, but the only minimizer is this because you reduce the length of the two segments. And the same is true for the energy. So now we have at the boundary a number of points here. And the only way to minimize such an energy with this given boundary condition is to beat segments which are not intersect one with that, but they should all pass through zero. And this one gives the contradiction. So since, but they all pass through zero to the origin, we have the contradiction. Okay, so this one is, is the end of the truth. So let me repeat somehow this scheme was a bit fast, but I wanted to save some, some minutes for discussion. So then the proof is like this. So we do our first blow up in the hypothesis when we have accumulation of singular points. This first blow up is putting us in a very far ball situation where we have still a minimizer. So a solution of our problem. And now we have an entire line of singularities. What we do now we do an offline blow up and then we, we get something which is cylindrical. That's really where we see the reduction of the, the complexity. So we had this radial, this radial kind of homogeneity, which is already a reduction of complexity because it induce this cylindrical, the composition. So we lost one dimension and we have now to discuss a problem, which is a one dimensional problem. But the one dimensional problem is completely characterized. The only minimizers are non intersecting segment, join the points at the boundary. But in all these conditions, zero was a singular point. So zero remains a point in all of these segments. The only possibility is that the segments are all the same, which is not possible because having all the segment the same and being balanced to zero means to be zero. So that's how we get the contradiction. And now in a dimension is kind of the same. You blow up up to the point that you reduce to the one dimensional case, but you have some out to work out with the arguing with the dimension of this set in terms of house of mesh house of dimension and so on. So these are complete. In some sense, what I wanted to present to you in terms of, of details of this analysis of all these singular points. So which is, I mean, it's, it's deeply different already in the geometry, in the, in the result from the dimension one, because you don't see for each singular point, you don't see a singular code, but you see something which has a continuous tension plate in order to analyze this continuous tension plane. A the great achievement of one and was to introduce all this machinery of multiple values functions and to discover this new monotonistic formula. From which then everything was was derived. So first of all, let me ask you if you have a question about this, this part here. I mean, of course, I will stay here also next week. So we will have also time to discuss when you have digested a bit more all these. All this material you can have quicker assumption. So indeed, the revision of the monotonistic we never used the minority property of the of the function, but just the first variation. So if you have a function which satisfies the first variation, then the quantities are small, but all these theories not working just for stationary maps. Because in a couple of points, we heavily used the the minimizing property of you. So getting this strong convergence, we always have to argue via via the minority and also this last argument is an argument which use the minimizing property of. Because two crossing lines are stationary. As very for this map, I mean, you may you may show by yourself. So also here we several points we use kind of every diminishing the minimizing property, not not for the monotonistic of the frequency. That it's not it's not not. Or let's say I mean it in it depends. I mean, what you consider as a as a statement somehow mean that's so so when you have two intersecting planes, so you have a one singular set, so it is a stationary surface. So if you just ask for stationality depends on the notion of stationality, you consider you have probably to allow for bigger singular sets. That's a but you may kind of refine a bit this notion, but it's a it's not very natural. Okay, if you don't have another question about this, the last 20 minutes, I would like just to discuss with you a bit what happened for the for the linear problem. So we did everything now for for this linearized equation. So we consider the Dirichlet energy and everything was working perfectly nice. So we had this exact monotonicity. Everything was already parameterized as a function. So we could have used all these PD's argument in bandings more estimates and so on. Now, of course, without giving any details, which actually at the beginning was part of the plan. So you find in the notes the last chapter with some details for the linear equation. Let me now skipping this details just telling you what is the end of the story. So this one is proving the analogous of this partial regularity result by arms and this very special case of of parametrization, minimizing a Dirichlet energy, which corresponds amount to the to the linearized equation. But then from here to the end of the story, it's a it's a quite long road and what we've done is is more or less one fourth of of all the process to get to the last to the last result. And the first point is that in general, I guess this also Camino mentioned to you last week, we know no way to claim that a minimal car and minimize current is a priori a graph. So there is no graphicality in the representation of of of the minimizing current. So one of the key points. So now it's K. So general theory area minimizing integral current. So the first point, which I guess already Camino mentioned to you is that I mean we know no way to say a priori. If I minimize in an area minimizing time, it is a graph of the function. So no a priori estimates to to claim graphicality if these words exist in English. So to claim that we have we have a graph in India. No, we don't know. And for example, you could I mention one what what Camino showed you is that once you have the excess, which is small, you are a graph. But before getting to this conclusion that we are a graph, you have to argue via an approximation and then find the decay of the S and they get all the regularity basically before seeing that you are a graph. So I mean, for example, in condition one is true, minimizing current is locally a graph where it is regular. So except this a single point, but in general, there is no way to say this a priori. This one would simplify the theory a lot. And now in coordination, in our condom mentioned, it's not known if actually they are they are graph or not some, but we don't have counter except. So this point here, which is, I mean, it may seem a kind of technical point. It's it's it's very important in all this theory of regularity and what plays the role of this graphic is an approximation result. So behind passing from this non parametric case to the to the real one, we have always approximation result approximation theorems, which kind of did destroy all of the things apparently that we have done up to now. Because when we find the function you from omega into the space of two points was graph approximate our minimizing currency, this function you is not solving any equation. It's not the minimizer of any function, but it's just an approximate. Minimize. And now let me say that this one is one of the key points which changing from one dimension one to a co-dimension in a co-dimension. At least for what concern the study of the dimension of the singular sets, approximate minimizers are enough and they join the same regularity properties in a co-dimension is not true. So you cannot just say I'm solving a perturbation of my problem and then I get the same regularity. But we have always in some sense to refer in a kind of continuous. I mean today to the fact that you are actually an approximation of a real minimizer of a linear problem. So this one is kind of technical, but it's it's the link. The kind of the link between what we have seen in these cars in this class and the case of of minimizing cars. Another point which is very critical in my opinion is the starting point of this of this nonlinear theory. Even if I mean to discuss this property, you pass first via a first approximation result is what I called you the problem of the centering. So deconstruction of the center made which was what I planned to do in the last part of the class, but I didn't have time. So deconstruction of the center leaves, which is somehow doing the having the role of the average of the values. This one is a very is is a very problematic point in the theory. And the point is that I mean you have to match kind of a couple to condition, which goes one against the other. One condition is that the center medical that's really to behave like an average. So so let's call this center manifold M. Then we look at it at parametrization of let's say you let's call and they are normal parametrization from the center manifold to this space of two points such that the graph of and is similar to the to the current T. And what you need to have is that and is almost average zero from one end. So one condition is is that the average of this map should be approximately zero for which I mean you would like to take the exact average of the values, but they are not regular at all in the case of harmonic function. The average was harmonic. It was very it was very smooth. In this case, I mean when you have a current, you don't even know if it is a graph when you approximate with a graph. The graph is just lip sheets. So a priori, this point is just lip sheets, but you need to have the center manifold that's the second condition which enjoys some some regularity. And exactly what we need in the proof is that the regularity of the center manifold is at least C3. So in our case, this average was was analytic, so we had no problem in all the computation with it. In this non-linear case, we have to make to match this regularity here. And now these two conditions are kind of deeply related with the approximation issue. So you have to approximate in such a way to achieve this. So all the standards approximation you find in the liter, we're not enough to achieve these two conditions here. And what I'm going to prove was a new approximation result with better estimates in order to achieve this, these two, two condition, but it's not yet the end of the story. So another point is that in the general theory of minimizing currents, we don't know if in general we have uniqueness of blow up. So we don't know that's maybe the most famous problem in the field, which plays a role also in this in this context here. So because you would like to say, I take a singular point of my current, which I mean in principle may not be a graph or whatever, but I take a branch points there, I blow up and I will end up with a diminishing function. So this process is kind of behind on the back of this process. There is the assumption that making the blow up, the usual blow up to this current. So now is a tropic blow up, not the one we have done in our proof, but the real one you get a plane. Maybe with multiplicity, but you get the plane. So this one is not known. There is a theorem always due to Andren, so which says that up to a set of dimensions at most M minus three, there exists a blow up. So that one block, which is a plane, you have the set of points, you throw away set of dimension, which is still okay for your result, because you don't see in your final result that you know that at that point, you may blow up your currents and you know that in the set of blobs, you get the plane, but you don't know if that one is the only plane. So in particular along the sequences, I mean you may have this plane, which is rotating. So all the things we have done actually, so all this theory of the study of the singularity does not solve this problem of uniqueness or the blow up. So you have somehow also to deal with this issue here, that when you parametrize your currents on infinitesimal scale, your reference plane may change. So the only thing that you are able to do is to perform this non-parametric analysis, not on entire balls, but on annual light. In C2, to be minimized, they meet on a point. With kind of the x-atter, you decompose these irreducible components of your currents. And here what you need to do is sometimes to cope with this problem, that you don't know if it's true or false, that you have an extra plane. So what you do is actually you perform your analysis not on the entire ball, but just on an annual light, just here. So here you may parametrize and make this analysis with the frequency functions and so on, over your centered manifold. But then when you change, in order to have this extension plane, which is flat, maybe you have to restrict to a much smaller class, which usually it's not comparable with this one, because otherwise the extension plane will be unique. So then you have to jump to a new site of your scale. And on that scale, I mean in principle, you lost all the information about your currents. So what was happening from here to here? You don't know. And here there is, in some sense, part of the analysis is to deal with this problem. So how can we understand what happens where we don't have a small excess? Because converging to a flat extension plane is saying that basically we look at the scales where the excess is small. So how can we understand the scales where the excess is not small? And here comes another part of the theory, which is mainly related to the way to construct this center manifold. And which says that the only way for the excess not being small is that your currents has to have a big diameter. And this one, I mean, it's something which, I mean, that's at least, I mean, it's the way in the work with Camillo we understood the problem. And it's a phenomenon which goes, which I mean, we call, after the work of Tristan Riviera, we call splitting before tilting. So tilting is a word which refers to the rotation of the tangent plane. So that the tangent plane may rotate. This we cannot exclude because we don't know if we have this kind of uniqueness. But before the tangent plane rotates, so before we pass from this blue scale where we know that the tangent plane is this to this yellow one where maybe the tangent plane has tilt has become this, before doing this, the different leaves of the currents has to separate. And this one is the splitting. So the current splits before being able to tilt the tangent plane. And this one is an a priori estimate, which is, I mean, it's kind of at the basis of understanding what happened passing to the right scale where we have this small excess. And it's kind of necessary because this middle anuli here, it's a black hole for us. So we don't know there what is happening to our current. And this estimate, I mean, analytically, for example, you see in the frequency functional analysis we have done. So I told you the only key information was that the frequency was bounded. And we got a monotonicity formula, which is saying that the frequency in all these interval is bounded by the frequency on the external radii. We would know the same on the interior anuli, but we're saying that the frequencies remains bounded in all of these starting radii, which are in infinite sequence. Why can we claim that the frequencies on these starting points is always bounded by the same constant? And then it's because of this. So this splitting means that the L2 norm, it's so big that the frequency was the energy, which is an excess divided by the L2 norm. And this one is so big to maintain this quantity bounded. So this estimate is sure in some sense that we can start our analysis with the frequency function. There. And then let me mention a last point before concluding. So in the case of the minimizing functions, we had this uniform convergence. So a singular points, once we rule out the problem of collapsing to zero or collapsing to a single leaves, a singular point was remaining a singular point. But here it's not anymore true, maybe. I mean, we don't know, but it's not so clear. And we have to show that in some sense we have the persistence of singularities in all of this process. And what is here, the problem is that the current that may be perfectly singular at this point. But then we use always an approximation of the current. And the approximation is done on a big set, which is missing a set of positive measures, small but positive, but positive measures. So it may perfectly be that our approximation is making this current there. Regular. So there is no way in some sense to claim that in the, in the leapshitz approximation. So at the beginning, when we start our analysis, we may preserve the singularities. So that's not true. What is true is that these singularities somehow are preserved in the limit of this approximation. So in the limit, if the approximation is quite good, then it has somehow to converge to a singular point there. But for this you need another a priori estimate somehow telling this, because point wise is not true. And it's not true even in the limit point wise. But what you may prove, and here there is kind of another argument in analysis that you find in many contexts. It's a capacitary argument. So the capacity, it's what you look, the capacity of the singular set, you may conclude that what is preserved is the dimension of this singular set. But not the point wise behavior of the singularities. So that's a, and this one is just to give you a kind of idea of the difficulties that you have in transferring this theory to the case of minimum currents. So the analytical issues and the geometrical issues are the one I showed you. So you have the point of B force to have a parameterization which has multiple values. You have the problem of the centering. That I mean you have to exclude that everything is getting regular in the limit. And then you have the problem of getting trigger blowups which is this infinite order of contact which is the frequency. Now in the case of this linearized problem everything is very clean. The formulas are clean. You don't have any troubles in the computation and the theory is very elementary. When you try to export this to the case of the linear problem, you have to deal with all these difficulties that I showed you very schematically in these five points. So if you don't have a question, I think my time is exactly over and I will stop here and of course I will remain at your disposal for questions or discussion afterwards.