 Znamo se razvršenje, da je dobro vse dobro odličenje klas. Vsih zelo je več nekaj zelo za vse. Znamo, da se pričoče... Zato, če je in izvaj, in nekaj, nekaj ne zelo. V tudi, tudi, tudi lektor. Če je nekaj nekaj klas, pa je stavnja klaso, da je zelo, da je Kaminu, da je Leldi, če je zelo. Zelo, da se vse, dont we are going to present, this is something of the regularity of minimal services. And actually of what Camilo introduced, this are area minimizing, OK? Camilo sho you, in theory of internal currents and some of the issues that you see in the regularity for Chodimation 1 currents. So, especially he gave a proof of a partial case of the Georgian decay lemma, which is at the base of the study of the regularity for collimation 1 current. In this week we will see something related to higher collimation, so this one will be the focus of my course. So, understand some of the issues which are involved in the study of regular points and singular points in higher conimation, minimal surfaces. And the plan of the lecture is, in some sense, to focus on some of these issues, which are kind of basic, in my opinion, but in this theory, but in the simplest case we can admit. So now this one will be clear in a while, so we will try to avoid all the technicalities to focus on the geometrical and analytical problems which are behind. And maybe the best things to do is to start with an example and always to keep an example in mind, which I saw Camillo already introduce, and that's the example. So that's kind of introduction via examples. Camillo already introduced a class of minimal currents in higher conimation, which are the complex varieties in CN or in a scalar manifold. So since it's just an example, let me just introduce the most basic one, which is a complex curve in C2. So the one which Camillo introduced, I think, was ZW in C2, such that Z2 is equal to W3. So let's understand a bit the geometry of this complex curve. So my favorite picture is try to write this as a graph, which is clearly impossible, but gives kind of the ease to the geometry of these objects. And if we write kind of naively, we have that Z amount is one of the two square roots of W to the cube. So let me write that's not correct, but it's plus or minus these two square roots. In some sense, if here I put a two-dimensional plane, so Z in C, which is a real-dimensional plane, and here I put W in C. So what I see is plus or minus, so ZT is plus or minus W to the 3 half. So something which looks like this. So this one is V. Now what is V actually? If you see here, it seems kind of disconnected, but it's just a two-dimensional projection or of a four-dimensional space. So this V is living in C2, which is a real-dimensional form, which has four real dimensions. So what is actually V? V, if you see, it's a very nice topological disc. It's the image of the flat disk in R2 via a smooth parametrization. So V is equal to U of B1 for B1, the unit bowl in R2, and U goes from B1 to R4. In the following, I will always identify this introduction, and see with R2. I mean, we can also give a parametrization for such map. So in polar coordinates, a parametrization is the following. So rho cos theta, rho sin theta, rho 3 half cos 3 theta, rho 3 half sin 3 theta. So it's the image under a continuous map of a disk. So actually it's a very nice surface, much more regular than a general current. And if you see, this parametrization is actually very smooth. It's an embedding everywhere except at the origin, where it is just an image. So this one is just to put all this information together. It is an example of a minimal current in R4. So it's a two-dimensional minimal current in R4, which is regular everywhere except in the origin, where it is just immersed. And there's a kind of double point there. So what we will try to analyze in this week is the behavior of these singular points here, which are called bridge points. So in some sense the aim of this, of what I'm going to present, is to find some analytical way to understand in the presence of this such behavior. And this surface here is a very clear example of why the techniques and the tools in codemation 1 cannot work in higher codemation. Because, as Camino showed you, the main parameters in codemation 1 for the regularity was the excess, which is an integral measure of the oscillation of the tangent plane. And here the tangent plane is continuous there. So remember, this one is an immersion. It's a C1 immersion. So in some sense looking at the excess in this case would not detect the regular point. So that's just a remark. So the excess of this current in Br, around the origin is going to zero. So at a certain point becomes less than the constant in Allah's theorem, but nevertheless the surface is not regular. So just looking at the excess is not enough to understand the regular, the singular points. And indeed what we are going to see is something which is completely different because I'm out from codemation 1. A different even in the results. So maybe I give you immediately the final results, which is something I mean, we won't see in this course. It will be too long. But it gives, in some sense, the idea of why this theory is so different from the one in codemation 1. And the results, I don't know if Camillo mentioned to you. In higher codemation, so in codemation 1, so let's say, so which is the case of hypersurface, so we have that every minimizing current is regular. So regular means that it lives on a smooth manifold and is given on integration by integration over this smooth manifold. It's regular in the interior except for a set of at most m minus 7 dimension, so out of dimension, where m is the dimension of the current. And this one is the most basic result in codemation 1. Actually much more is known, but just to point the difference, we just talk about out of dimension. Which I mean, for those of you who do not know what is out of dimension, just keep dimension. It's a very reasonable notion of dimension that you may introduce in this problem in geometric analysis. This one is a result of a very long analysis and effort, which starts with the Georgian and then was completed just in the 70s by a great federal, other than many other authors, Simon's and so on. But what is striking is the final result. If now I would like to write the final result in either codemation, you see it changes completely. So in either codemation, the singular set may exist. So I mean, the question is where this magic number is coming out. It doesn't mean for me it's I think as magic as for most of the people. So if you see the computation, it comes from an analysis of the stability of cones and then you see or geometrically, you see as a possibility to foliate a space with a foliation, which is singular in one of the leaves. But why this comes first in this dimension I cannot say. Honestly. But I didn't say what is my degree of optimism because now I knew before the deal. So in either codemation, instead, the singular set may have at most, so is at most of dimension. So I was warned not to write down here, so just change. Report is at most minus two, where m is always the dimension of our current. So you see what changed is not the techniques and the way now I show you how to approach the problem, but the result itself is different. And this result is optimal. So we have shown already a curve, which is a two dimensional object, so m that is equal to two, where we have a singular point and the singular point is dimension zero. So this means that it can happen to have a minimal current with a singular set, m minus two as dimension. So it's not that we cannot prove more than this, but it's that this result is sharp. And this result, this theorem is due to angle, so in the x. So now what we will do in this class is in some sense to understand some aspects of this theorem. Revisit it with some modern terminology and modern concepts. So this one is somehow the starting point for our analysis. And again I think the best way to understand where to start is looking at the examples I gave you. So the examples was the one of the complex curve B, which is roughly speaking z equal plus or minus w3r. But somehow correctly is the one given by the parametrization. I wrote you before. So, and why I would like to start from this picture, because it gives somehow the idea of what we are going to do. And the point is in starting minimal surface, also in this very weak context of minimal currents, all the effort is done somehow to reduce to the case to a nonparametric problem. So the case where we can use an equation, the minimal surface equation or the linearized equation which is the Laplace equation which is what Camilo showed you for Allard's theorem that once you reduce to a graph, to ellipsis graph, then you may prove the decay of the excess and then you transport this to the case of minimal currents. Now the same would like to do here. So we would like to write a nonparametric problem for these objects. But now you see, so from the picture it's not so clear, but I mean you may realize by yourself that you can write this surface as a graph of a function. So morally the tangent plane with respect to which we should parameterize this one is the tangent plane of the parametrization of the immersion which is this horizontal plane here. So somehow the first claim is that this horizontal plane is the preferred one to write a parametrization. But around this point there is no way to write as a graph of a function. And the reason is because of this branch point. So if you start from say now let me write this at the plane W here. The W plane. So if you start from a determination of the square root at E1 say and then you follow a circle around the origin, you will end up somehow on the other determination of the square root. So there is somehow no way to write this as a graph as the union of two continuous graphs over the W plane. And that's because of the nature of the branch points which is exactly defined by these properties that when you follow a continuous path around the point you will end up with another determination of your leaves of your coverings. And that's what prevents to have a representation via a graph. Unless I mean we are allowed for a singularity here, but we would like to solve an equation which gives regularity, so we cannot have something which is discontinuous. So somehow starting from the beginning it's kind of clear that we cannot use a very classical non parametric theory for treating this problem. And here comes the first main idea of Andre, which is okay. This one is not the graph of a function, but above every point in the W space, we have two values of the z. And always two, also in the origin because this one is a single point but it's kind of approximated by by two points. And if you see the density in terms of currents this one is a double point. So the density there is two. We cannot treat this as a graph of a single value function. Let's treat it as a graph of a multiple value function. So in this special case the function is so we can write kind of formally. So to each W we associate two values which are in so let's write, I mean kind of informally, but it's clear which are W to the three half and minus W to the three half. So I have a map from a unit disk, say in R2, taking two values in R2. So which is this set here which is a space that in a while I will introduce and we call atomic measure when mass two of R2. And now what is one of the main points here? One of the main issue in this parameterization that when I give you a determination of the square root of a complex number I cannot tell you in a consistent way which square root I am taking. Because as I showed you once you follow any continuous path on the plane in the branch point, we change this determination of the square root. So in this representation what is important is that here I do not use this bracket so it's not an order pair. I'm not saying what is the first value what is the second value. I just give you the set of values and that's as a whole, it's my value of my function. So this space here is not R2 squared. But it's kind of new space of points which is the space of unordered points. So that's kind of the beginning of the analysis. So let's start from introducing this space. Both times the same but now it's clear from the definition and this one is the space of in this example R2 we will always consider Q point which was traditionally introduced by and let's put a sign here that means that these Q points are not different Q points so in the way to formalize this concept is to take so there will be the Q points in Rn and I consider this as the atomic measures of mass Q in Rn. So positive atomic measure of mass Q in Rn. So I write the sum e from 1 to 2 of the Dirac delta in Pi so this one is a Dirac delta such that Pi are points of Rn. And then you see these points are not necessarily different so for example a bridge point could be two times zero. So our values will be always a measure which is atomic so leaving on delta and mass Q so positive with mass Q. So this one belongs to aQ of Rn. And just to understand a bit this space that's just a very nice formalism because we can think a bit about summing measures or stuff like this but another point of view to see this space is actually the combinatorics one I was kind of suggesting here. It's true that it's a collection of Q points in Rn. So this space you may also see as Q copies of Rn. But then you have to take care of the order of the points. So you have to take a quotient where the equivalence relation of permutation. So P1 P2 is the same as P1. And this one gives you a bit the idea of the geometry of this space. So it looks like a nuclear space because it's actually R to the NQ but then it's you took the portion with this group of permutation. So you create some corners and some angles in this space. So the space in the end will turn out to be not a smooth space naifli. So let's say the first structure we can give in this space is the one of a metric space. So let me introduce the metric for this space. So here we have a lot of freedom to design which metric but it will be clear in a while why this one is kind of the correct one. Traditionally it's written like G for the metric so this one is really the distance between two Q points T1, T2 and imagine T1 is the sum of the data in PI and T2 is the sum of the data in SI. So this one is the minimum among all the permutation of the square root of the sum always 1 to Q of the Euclidean distance I minus S sigma I squared. So just to understand this with the picture we have two sets of Q points so let's take the case of exactly I don't know two points so T1 and T2 and we consider the pairing of one point with the others. We have once this and then we have another one. No, this is the only one I shouldn't know it's this and this. Then we see which one realize the minimum of this distance here and this one will define the optimal pairing between these points and last week you had a week of optimal transportation so I think for you it's kind of now very clear that it's actually the vastest time to distance in the space of measures. In this very special case of atomic measures so that's nothing else than the objects we have already seen I think already in this course and now it's very simple but this one is I will leave you as an exercise that this space of Q points with this metric and by the way I posed some notes for the class so some of the material you find here and also some of the exercise with the ins also to solution are in the notes if you need to consult but not this one this one is very simple OK, so now we have our space of points let's define what is a function to this space of points so nothing easier than this space and we just consider functions between two metric spaces so let me I should never erase the picture of the complex variety that should stay there with us so now which kind of function we would like to look in this problem so our function from a flat space a is this so these are two valued functions which are just map U from say an open domain so open bounded and let's say also smooth that's not the point in a flat space and M is always the dimension of my object of my current to this space of two points in Rn so which may be one or greater than one in important cases is when n is greater than one so in this picture it's like that here we have Rm as the horizontal space and the vertical space will be our Rn and we consider two points in this Rn so that's the splitting of our and now this one is of course a metric space here I introduce a metric and I can talk about continuous function measurable function the same as application between metric spaces so this one is kind of clear the next step is to try to define a derivative for such a function because remember our aim in the end will be in some sense to analyze the minimizers of the area of the mass of these currents in this graph it's expressed by the integration of a first derivative so that's so the area of the graph of u is the integral of the Jacobian of u a Jacobian is nothing else than the square root of the determinant of du transpose du so somehow we need to have a notion of of derivatives there and now this space is perfectly nice it's Rm but this one is a metric space so the point is trying to define for this class of functions a notion of derivatives there or or weak derivative so what we introduce now are generalized sobolev functions ok let me say there are many different ways to introduce sobolev functions between metric spaces and here actually it's also easier because this one is a nuclear space and but the way I find more convenient is a way which is going back to to Ambrosio in the 90s which is looking at composition with the Lipschitz functions so I say that the function is sobolev and the Lipschitz function is a sobolev function so let me give a formal definition so we say that u from omega into this space of two points and when I don't indicate the dimension is kind of always a medend there is let's say w1p if there exists a function phi in lp of omega and this one now is the usual lp space so this one is a function from omega into r and actually I mean we can take positive that there exists phi in lp such that the following points and we need two conditions here so the first one is that every composition is a Lipschitz so instead of taking generic Lipschitz functions I will take composition with a distance function so I say for every t in my space of two points the map which now it's a real valued map so it goes from omega into r and each x I associate the distance of u of x which is now a number so this map here it's now actually a real valued map and I ask that for every t this map is sublev w1p so meaning lp and the generalize generalized gradient but then what is important is that all this composition needs to have a common dominant for the gradient so then which is this function phi here I ask that the norm of the width derivative of this composition so for every t the norm of this width derivative is point wise dominate by phi a point wise means of course the width derivative almost everywhere in omega no that's point wise so for almost every x and it's a finite value and has to be controlled by this function phi and also phi is defined almost everywhere because it's just an lp function so it's really point wise almost every x say in omega that's what we ask now this one is an exercise which I put in the notes so that's somehow the most in this space but it's like asking this property for every Lipschitz composition and that's what is important exercise belongs to w1p omega aq if and only if composition u belongs w1p omega and the derivative f composition u is less or equal than the Lipschitz constant of f times phi almost everywhere and that needs to be true for every f Lipschitz from aq so it's to remember it's a metric space so every Lipschitz function from aq to r that's an exercise I've been asking that property just for the distance function of the Lipschitz function a Lipschitz function it's a way to know a metric space somehow so that's why this kind of notion it's so reasonable question on this because this one is one of the key points of what I'm going to present and next one is the energy so trying to define a the area function for this kind of multiple values so in some sense we need to define so the next part of the hour we will spend to understand how to define the area function or a variant of this now we will see and the first step we do here is in some sense to immediately to simplify our object which is also what is done in the theory instead of looking at the area function we look at the linearized energy so as I wrote you so the area of the graph of a function u is given by the integral of the Jacobian and now kind of linearized this function in the first order you get actually this one is the integral of one plus du squared over two plus a reminder which is a big of du to the fourth so in some sense a very cheap way to define this energy would be just to define the modules of du and that's what we are going to do most of this hour will be to understand this object so this formula don't take too seriously that should be another u which is the one which is also the first components of the identity but it remains true that this one is the expression so now try to define du and looking at this definition here this is a very natural guess which is ok, I take the minimum phi realizing these properties and this one is my du because every gradient of this one leap sheets map is dominated by phi so this one is very reasonable but it's the wrong energy and actually the energy will be defined and will be clear at the end of this hour excited by this property here of the area function so it really needs to be the directly energy and now if you look at this energy here for vector value map this one is not the directly energy this one is the operator norm of the gradient so it's not the invert smith no, so we need to do something else and for this the expression of the energy we introduce is a sum on the partial derivative in which case the two norm coincide now we have a map like this and it's a different u you have x, x u of x so this one is I should put here say big u x u of x so that's the map and this one the derivative of this gives the one in the expansion ok so what about du so we define so before defining the modulus of du we define the partial derivative the modulus of the partial derivative of du so we take dj of u and look I write like this but this one is kind of an independent object so it's not the modulus of a derivative up to now but it's just a quantity positive quantity which will be an LP function and it's given by the optimal phi there for the partial derivative in j which I mean it's not difficult to regularize that this one is the supremum over a dense a countable dense set of of points in this space of two points of the modulus of the partial derivative of this composition so I select a countable dense set of points there I look all of this composition all the partial derivative and I take the supremum point wise almost everywhere for this quantity and then the right quantity to look du squared is exactly the sum on the partial on the on the different direction from 1 to m of this partial derivative so this one now we will see in a while why it's the right quantity for the energy and note that in general when we have a map with the values into a metric space it's very simple there are many different definitions for defining the modulus of a gradient but not for defining a point wise gradient this one is something which is general much more subtle in this case I mean we can define also a point wise gradient and now we will spend the rest of today lectures for doing this in order to understand a bit better this class of maps here and we will see that this point wise gradient will match this energy we have introduced here so up to now all the theory up to now it's possible to be done for generic metric space here so complete metric space here you may define this energy so breath class like this and from now on we exploit the structure of two points actually a bit more in details so now we try so let me see if I forgot something to give a notion of point wise differentiation so let me first say something on the representation of this multiple values function so given a function u from omega to this space of two points it's always possible to find q in terms of number different measurable functions would sum as measure gives this function u and I mean huge unique measurable so u1 uq from omega into Rn such that u of x is exactly the sum of Dirac deltas in ui of x so this one is a very a very useful way to name the values of this function there so in some sense I'm not claim anything about this ui so I'm not saying for example that when we start from a continuous function the ui are continuous that's not possible or when we start from a sublet function the ui are sublet I'm just saying that we can give a label in a consistent way which is very non unique superposition superposition of measurable functions ui but in general this one would be just measurable and the proof of this statement is by induction I gave like an exercise in the notes and there are hints for the proof of this and it's not so deep I mean it may seem strange somehow because I said we cannot order the value but when I said we cannot order in a consistent way in such a way that I mean when we have a continuous function our values are ordered in a continuous way that's impossible but in a measurable way so just giving a label which is measurable we can do and that's I mean it's always possible and we will use this writing here just as a way to write our function so nothing special just having a way so that's the only use we will make of this proposition which I mean by the way the proof is very simple by induction and it's given in the notes so you're measurable, yes so that's kind of the minimum regularity I always assume OK, what is wrong is when you substitute this measurable word with any other words then so if you have leap sheet and then you find Q leap sheet function it's impossible so obolev that's the only wording in this kind of statement so but this one is useful because I mean it allows us to use this notation and it's to introduce this notion of point wise differential so definition so we say that U is differentiable X naught in omega if L1 and I mean we see as linear map Rm to Rn so linear map and now we give the two condition for the differentiability there so the first one is that now we set a notation for the first order differential of a map so T at X0 view that's kind of the tangent map to U at X0 which is let's use now Y as a variable so this one is the sum of these deltas at Li Y minus X0 plus Ui of X0 and then here you see the first use it's just a way to label these values so this one is a sum of Q different linear maps based this affine map at Ui of X0 so once we set this then we have that the distance between U of Y and this linear map here as usual for a differential is a little of it's an infinitesim map of the distance so that's so I'm telling you what is the linear the differential of this map and I ask for the usual differentiability properties that it has to be infinitesimal with the distance to the form so this one is quite it's quite natural but there is an extra condition which makes this notion a bit more precise and an extra condition is that every time here a point with ion multiplicity also the linear map should coincide should be the same so Li is equal to Lj if Ui of X0 is equal to Uj of X0 so now let's explain this this definition with a picture so basically what we say is that a point is so let's do the first case where we have exactly the superposition of two different smooth functions so let's say Q equal to smooth function which are kind of disjoint so we have here our two functions and our two values function is the superposition of this very very smooth two function then at a point zero in the domain we may take the differential of these two so this is a fine map here and our differential for the map will be just the superposition of these two differentials and this one will be a point of differentiability for our map another case is when the function are not disjoint so always Q equal to but we have a situation like this so like a branch point for example so this and this so the map comes together and this one is our point X0 then we say that X0 is a point of differentiability if we can find a double linear map which approximate this function there why double? the linear function has to coincide so here we need to find the common tangent plane to the two which approximate to the infinitesimal of the distance our two values function so this one will be taken with a multiplicity 2 and this one is still a point of differentiability for us what is not a point of differentiability and it's kind of questionable but this just make this definition a bit more restrictive is when we have a two value function which behaves like this in principle now it's linear let's do less linear in principle here we have two very nice differential function and the superposition of these two is a good approximation for our function u but nevertheless we don't consider this point the point of differentiability because the point is the same but the linear map are different so that's what we exclude from our definition of differential point and I mean the reason why this case is excluded now it's a bit vague I cannot explain kind of completely but it comes from the fact that in some sense all the all the core of the theory is to look at the space of q points as a union of stratified space so the two the multiplicity of two points would like to see as independent from the multiplicity of one point so to do this means that they needs to have their own differential which is a multiplicity of two differential so the definition may seems a bit weird because that picture is perfectly nice in terms of surface that one is kind of an image with a self intersection of surface for the theory is more convenient to take and actually suffice to take this more restrictive definition but it's mostly used later this so at this point it's not completely justified why we consider this second condition here ok, and now we make a little break of ten minutes and we assume at 9.40 and for the second hour I mean we will see that actually this kind of point wise differential exist almost everywhere for sovereign functions and we will give another way to write the energy which will be kind of more convenient for certain computations ok, so we assume in ten minutes here ok, let's start again very very slowly ok, so now we have a notion of differentiabilities for our maps which is this one which is shown in these pictures it will be completely unuseful if we cannot say that our function is differentiable so the first thing we have to do is to prove a sort of rather marker theorems so that at least our differential almost everywhere for our functions so this one will be next step which is a generalized for the markers theorem so which at the beginning I mean we will state just for for Lipschitz function so the theorem will be we take f from omega to aq the Lipschitz and I recall you the notion of Lipschitz function between two metric spaces it's the standard one the distance is not expanded up to a factor L then so then is differentiable almost in omega so this means that for almost every x0 in omega we find such a correlation of linear maps as it is I wrote before OK and now we try to give a proof of this result so won't be a complete proof but almost complete let's say because all the ideas of the proof will be inside here so proof with more sketch and what we do to illustrate this phenomenon of the proof is to consider the special case of two values function so q equal to I don't know if I started a bit in advance so and the point is the following as I told you the idea what we to treat this multiple value function is somehow to go to go down and distinguish between multiplicity one, multiplicity two and so on in the activity with the formalism now which is completely independent from these combinatorics because we have this notion of metric space but nevertheless this one is kind of behind this argument so when q is equal to what we can do is distinguish between points of multiplicity one and multiplicity two so we call omega tilde let's say the points x in omega with the multiplicity two which are the one where the two values of our function are equal and then we look separately omega tilde and omega minus omega tilde and notice that omega tilde is a closed set because f is it's a it's continuous and this one is defined by inequality so it's very simple to see that it is a closed set ok, what happened here so outside of omega tilde we have two values which are different so let's argue with the picture so for every x naught in omega minus omega tilde f1 of x naught is different from f2 of x naught and so we see basically so that's our x naught in this case we see basically two different values for such a function now the function is continuous so now I'm not going to prove this but it's a leap sheet it's a leap sheet continuous the two values are different this implies that these two leaves of the function cannot meet in a neighborhood of this point so there exist we see two different two different leaves of my functions and now the function was globally leap sheet it's very simple to see that f1 and f2 are also leap sheets in a neighborhood of x naught so in some sense in this neighborhood we can apply the classical or the macro theorem and find differential for both functions almost everywhere I mean we cover this set so here and now it's a very simple to verify via the definition so by the very definition at the beginning this map here so i1 to 2 of dfi x naught y minus x naught plus fi x naught this one is a differential for our two values function so I say a differential but you see from the definition that when exist the differential is unique so it's the differential or the derivative so when we have these two values which are split then we argue as if we had normal normal or classical function so the only thing we need to understand is what happened in omega tilt is it clear this with this classical or the macro theorem which is said that when you have a Lipschitz function it is differentiable almost everywhere with respect to the back mesh so all the things we need now is to understand this multiplicity 2 set in omega tilt f1 is equal to f2 and now omega tilt is not the entire space but we can use theorem for Lipschitz function so for example the Kitz Brown theorem and say that f1, f2 coincide with the restriction omega tilt of a Lipschitz function ok so on this set the two values coincide the function there was was Lipschitz so in particular considering these two values kind of separately they are Lipschitz they are the same value and they coincide so with and extend them and look an extension of this Lipschitz function and now this tree I use for defining which points I look for the differentability of of this and I say that I consider points x0 in omega tilt with these two properties so the first one is that they need to be points of density 1 for omega tilt which I recall you so does it mean it means that if I look at the measure of omega tilt in smaller ball around x0 it has to cover almost of the ball so the limit are going to 0 of Br x0 intersected omega tilt renormalized by the measure of Br is going to 1 so it's covering everything around this point x0 and then since G is Lipschitz itself I look at the point of differentability for G so x0 of differentability for G and now what I claim is that for all these points x0 with these two properties actually my function f was differentiable and notice that by the lebek theorems almost every point in omega tilt there is a density in one point and this condition which is satisfied almost everywhere and now here I apply again my Rademacher theorem in the classical case and I know that G is differentiable almost everywhere so these two conditions are satisfied almost everywhere so would suffice to prove the differentability for points with these two properties it's a full measure set so that's actually a theorem which is not a trivial theorem which is based on the basic of which covering theorems which says that if you take as a measure for example the lebek measure restricted to the set omega tilda you can find a differential almost everywhere with respect to the lebek measure and this one will give you the density of the measure which is 1 on omega tilda 0 outside so you know that almost everywhere is coincide with the density which is 1 inside and 0 outside but this theorem I mean it's not trivial at all it's a very deep theorem in measure theory so now we are we are left with this task so which is to prove the differentability at this point there so let me fix some notation so the claim claim is so f is differentiable at m so let's make this claim a bit more precise the linear function to f at x0 is exactly twice the linear function associated with g so twice dg x0 y-x0 plus g of x0 of x0 so this one is my claim so let's prove this so we set r equal we consider a point y in omega and set r equal the distance between y and x0 so now always with the picture so this one is our point of our candidate point of differentiability we have our set omega tilde which is a full set around x0 so something may be missing from omega tilde but always few in few once we approach x0 and now we try to verify the definition of differentiability at this point so we consider another point y and we try to prove that the distance between the linear part and the function is infinitesimum with the distance of these two points and what we do is so we consider the bolt which is twice this b2rx0 and let's take the closed bolt and now what we do is so in some sense when y is a point in omega tilde we know what is the function there because in omega tilde g was coincided with our with our multiple values function but when y is not in omega tilde we don't know what we are doing so the best is first of all to find the projection of y onto this set omega tilde and we call this projection y star and y star is a point in our boot set omega tilde in so much that the distance between y and y star realize the one between y and the set omega tilde inside this closed bolt and now we will always verify our condition passing through this point y star so as amount here what we use is the fact that everything here is Lipchitz continuous and once our Lipchitz continuous here it's too low let me go up here so then in order to estimate the distance between f of y and the first order differential at y we can put in between our comparison point which is this y star so using the triangular inequality here we can put f of y the distance between f of y and f of y star then here we put the distance between y star and the first order differential at y star and finally the distance between the two first order differential now everything is Lipchitz there so we use the fact that f was a Lipchitz function so here we can put Lipchitz of f y minus y star so here we know that in y star it's actually our good set so in y star our functions the two values were coincided with the common Lipchitz function so this one is actually a differentiable function there and I get a little o of y star minus y actually because this one is the same as the distance between so twice the distance between g of y star minus the linear part of g at y star which now is the usual differential for one function so this one is by the second condition that g was differential of the distance and here we use that again this one is a Lipchitz function it's very simple to see that the Lipchitz constant of the linear part is the same of f so here we have again the Lipchitz constant of f times the distance and we would like to see that all of these so that's our claim is an infinitesimum of the distance between y and x naught so that's what we would like to prove so that's the claim OK, let's see first first this one here so this point y star is a point inside our ball b to r and r was chosen exactly to be the distance between y and x so I have here an extra factor 2 but let's term there first this 2 this term here is at most 2 times r which is exactly y minus x naught so this one is fine now the only thing we need to care the Lipchitz constant is finite of f we need to care what is the distance between y and y star which is the only thing we do not control so this distance here we would like to see that it is infinitesimum with the radius of the ball so why this? here we use the first condition for our point and we use that actually this y could have been outside our set but the space I have outside omega tilde is very small because it was a point of density 1 so here I take a ball around y with radius y minus y star so this is b y minus y star center at y and you see that this one was the point realizing the distance with omega tilde so this one is contained in omega minus omega tilde intersected with the ball b to r so in some sense I have an indirect measure for this distance because I know that the volume of this set is very small so it is contained in the complement of this set here so how can I estimate the distance there? so this one up to a dimensional constant which depends just on the space dimension is exactly like the measure of this ball here to the 1 over m I renormalize for the ball b r so I put here r and here the ratio between this measure the measure of this ball and b to r and here I am changing this constant there is a factor to an extra dimensional factor there but I use now this information here that actually this ball was contained in the dimensional set so that volume all to the power 1 over m so this volume here is less or equal than a constant times r but remember what was r r is the distance between y and x0 so that's r and then I have this ratio here which is b r b to r so we have connected omega minus omega tilde divided by b to r to the power 1 over m and now we are done so omega tilde at x0 density 1 so this means that the complement of omega tilde has density 0 at x0 so this ratio here this ratio here is going to 0 as r goes to 0 that's by property a so all together this one is y minus x0 times in infinitesimal so little o of y minus x0 which is what we wanted to prove because now this one is already a little o of the distance this one as well and this claim is proven that we are summarizing this proof so what we have done was looking first at the points where the function were splitted and then we know that we can apply our classical or edemacher theorem finds two differentiable functions which approximate our function there and then we looked at the point with the multiplicity 2 and we select kind of good points which were points of density 1 and points of differentiability which is the definition living on that set which is what we call g the central point now for that point there with this computation we verify that we have a differential and the differential is exactly twice the linear part of g which I mean turns out to be good with our definition because for a double point we need a differential that's the distance between f and twice this linear function is actually infinitesimal with the point so I conclude the proof here but just few words how to proceed for further clue that's just by induction again so now look at three values we may split in one values 2, 1 and 3 and the only thing you really have to carry is the multiplicity 3 because the 2, 1 you can treat already and the 1 was the classical arithmetic theory and then you proceed up to your q so this one is just part of the proof but all the ideas are somehow here in general the omega 2 is the closed side and in many cases it has zero value yes but in that case I mean you don't care you just measure zero you don't need to prove anything about that set because the final claim of the proposition is that you are differentiable almost everywhere exactly exactly so in that case the classical Rademacher theorem would suffice ok so this one is is kind of an important theorem because we had a notion of differentiability but we need to know that the notions are differentiable because otherwise the notion was kind of empty and now what we know is that a ellipsid function which is nothing else than a soublev function w1 infinity is differentiable almost everywhere and now this one is not difficult to see in the notes I gave the details but as for classical soublev function the differentiability almost everywhere when the exponent of the of the soublev class is not infinity so this one is now a corollary of this of this theorem but let's state as a proposition so every u in the soublev class w1p omega aq is and now here I have to have a little word approximate differentiable so what does it mean let me just explain in words what does it mean approximate so it's not the point wise notion of differentiability like for ellipsid function we cannot claim for soublev functions but for every point you have a set of full measure at that point where your function turns out to be differentiable so it's really kind of the same notion approximate that's a way to see this in terms of measure theory but what is important is somehow that this one allows us to define the differential almost everywhere in the notes you find the proof what is behind this proposition is a losing type approximation theorem you know that the soublev function coincide with ellipsid function on big sets so where it coincide with ellipsid function we have a differential since the set is not the entire set this one will turn out to be just an approximate differential but the proof is actually just a kind of losing type approximation theorem but you find details in the notes so I don't want to spend much time on this the only thing I would like to fix now is so now we know that given such a soublev function we have a differential so let's fix a bit more notation on this so always I called this linear map Li but actually the second condition which says that so this is just a remark on the notation so since Li was equal to Lj whenever fi of x was equal to fj of x so now we make no confusion in writing Li so we can write Li equals to dfi at x in this case our differential in terms of notation takes the usual form which is the first order Taylor expansion of f at y is the superposition of dfi of x0 y minus x0 plus fi of x0 so but let me remark that this one is just a notation so I'm not claiming that each fi is differentiable I say that I call this linear map dfi and I make no confusion because when two fs coincide also the linear map coincide so there the fi would be the same so this one is just a way to take advantage from the notation the map and introduce has to the next point which is how to rewrite the energy at the beginning we introduce this energy and now what I claim is that proposition again is that for w1p omega aq d of u let's say square is exactly the sum of the q values of these linear maps almost everywhere so at the beginning we gave a metric definition of d squared taking this supremum all the composition taking the partial derivative and summing these values now we have a point wise notion of differentiability and actually the claim of the proposition that these two energies coincide so you can take the sum of this norm squared and this one is the Schmitt norm of these matrices squared and we will reconstruct our energy at the beginning in some sense this proposition now I don't have time to give details of the proof but in the notes you find it it's important because it justifies our energy so because when our u was actually the superposition of smooth functions you know that so u1, u2, u3 you know that the right first order of the area functional is this so now what I'm saying is that at least the energy I introduced at the beginning is giving me the right first order expansion in the smooth case and that's a way to see that this quantity is one way to justify the introduction of this quantity here it's exactly the first non-constant terms in the area function and this one close somehow the first part of the theory so let me just summarize what we have seen up to now so we started from this example which was so our leading example we will always draw this picture on the blackboard so the picture of complex branch points would like somehow to write this branch point as a graph of a multivalent function because there is no way to write as a graph of one function but for this multivalent function we need to develop a first order differential calculus and that's what we did so we have a notion of sublet functions we have a notion of point wise differential we have a notion of of modules of this differential in the sense we can start doing the analysis on these objects and try to find solution to some minimization problem or to some PDEs and so that's of course a no smooth analysis because our space is no smooth it's just a metric space with this combinatorical structure but this one is kind of enough to allow to do all the procedure for the for the usual calculus and now what we will start today and probably we will finish tomorrow is to see that actually we can find solution to minimization problem in this context now of course we should look at minimization of the area in principle but that one is a nonlinear problem which is technically more difficult than looking at the linearized equation which is dila plus equation so let's focus on dila plus equation in the proof of the nonlinear case will come in so what we look now is a generalized harmonic functions in this case of of multiple bands function so what we try to do is to define harmonic functions as the minimizers of the Dirichlet energy so let's introduce the generalized Dirichlet energy I mean now we have everything ready will be just the integral of our domain of the u square and let's a definition for this energy we can define the boundary value problem and say that u is harmonic or in this context is called dyr minimizing if this is a solution of this variational problem so the form of definition is u is called dyr minimizing and this one keep in mind is a synonym in this no smooth case of harmonic if the integral of the u square in omega is equal to the minimum of this integral of df squared in omega where the minimum is taken among all the function w12 from omega to aq such that the trace is the same as the one of u so which means so we define the trace in terms of the composition with a distant function so for every t the trace of the composition u t is equal to to the trace of the composition of f so this equality is meant in the usual sense of traces and we test for every with a distance to every point on on the space of q-points and sometimes I don't need I don't know if we need later but we will call we will just write this condition as f on the boundary equal u on the boundary but this one will mean that all the compositions are the same on the boundary so a function minimizing such energy with a given trace is what we call the minimizing and now what are the basic theorems about this minimization problem so at least we need to at least one of them which is the existence of solution so distance of diminimizing we would like somehow to show that we are able to solve this Dirichlet problem for such an edge so given g in w12 omega aq that exist u in the same space w12 aq did minimizing u at the boundary equal g at the boundary of omega t which I mean I remember the boundary we just look at the composition is nothing esotic there so the first theorem is on the existence and then we have a first basic regularity theorem which is how regular are the harmonic function in this harmonic function in the classical setting analytic here they are they are not analytic but they are just just continuous so this one will be first result and the second one is the regularity every u is older continuous with exponent alpha which depends on the dimension and the number of values in the interior so this one is the starting part of the theory so actually we can find solution to this problem and they are somehow more regular than just being sober function they are continuous but nothing more than continuous and the reason why they are nothing more than continuous you see also with the examples of harmonic of complex varieties because the complex varieties they are examples of minimizing currents but actually there are also examples of de minimizing function where they can be written as a multiple covering of a plane so this one of course is more than older continuous but we we could have parameterized this complex variety with respect to the wrong plane in that case what we see it's something like this which is continuous but here it's not differentiable because the slope here is going to infinity this one is kind of the same curve parameterized with respect to the wrong plane this one gives an example of a de minimizing function and it is just older continuous so also this result here is sharp no so this one for example is an example where it is not liptured it looks like so w here it's like equal plus or minus I mean you have a different number of comments but it looks like z to the two thirds here you have three sheets coming there now we have just few minutes left so let me just comment on this first part and then tomorrow we will start with the regularity part and the first part I mean now I can go somehow very fast because it's an application of the direct method in the calculus of variation which Alessio spoke yesterday so for this class of function we have all the ingredients and for this energy we have all the ingredients for applying this method so these are really the last comments so more than a proof will be just comments on the existence of de minimizing and the point is that one can apply in some sense the existence the use via the the direct in the calculus of variations and in particular the ingredients where that we have the compactness the sequentially weak compactness for sequences of function which means whenever we have fk sequence of function with the trace of fk at the boundary fixed and the energy equibounded then there exist a subsequence and an F such that we have convergence in L2 so this one is the classical compact sobolev abandoned so sobolev norm I can find the subsequence converging L2 the proof is very elementary and just pass through the composition because this sobolev function are defined in terms of composition with the distance so for every composition I can apply kind of the same argument and then pass to the limit so it's not 100% exactly like this but the basic idea is this the second element is that actually the trace is continuous in this limiting process so let me write here and actually the trace is kept and that's again the trace operator for classical sobolev function was continuous under weak convergence and here I mean the trace is defined already in terms of composition with the Lipschitz Bapp so this one pass in a natural way into the limit and the second element was the lower semi-continuity under this this sequentially weak convergence lower semi-continuity of the Dirichlet energy so which means that the Dirichlet energy of the limit is the Dirichlet energy of the sequences as cables to be very few comments also about in this point in the notes you find that the proof also of these also here the ingredients are very elementary and to prove this lower semi-continuity one of the best way is to use the first definition I gave of df squared which was the supremum of all the norms of this partial derivative on the composition of functions then of course this one is a supremum on a countable set but by the approximation you can reduce as a limit on a supremum on a finite set and now for a finite set you have on each set composition but on that set that composition is converging weekly so you may apply the usual lower semi-continuity and then some back to reconstruct the entire energy so again also at this point the philosophy is that since the energy is defined in terms of composition and the compositions are lower semi-continues then this property remains as a function of course you have to pay a bit attention because the lower semi-continuity you have to test on open set so you have to approximate properly your partition where you put the different compositions but apart from these technical details here the proof is really the classical one so df is the soup of a composition of leptin functions so the graded of regular sovereign functions so once you have this ingredient with the mechanism as I explained yesterday so you take a minimized sequence you have compartments so you extract a converging subsequence and the energy of this subsequence will be the minimum energy so you solve the existence of this the minimizing function for today I will stop here and I will give you time for questions and tomorrow we will start seeing some details on this regularity result that is functional a bit more than so but they are continuous and that's actually the best we can do so if you have questions that's a so that's I don't know they should be on the web page so they are already there other questions so let's have the coffee break