 It's my great pleasure to introduce Camilo de Lellis from the Institute for Advanced Study, who will talk about the center manifold. The topic that I'm going to explain to you is actually quite complicated and it took me a lot of time to figure out how I could actually convey the basic ideas in this piece of mathematics without entering in too many technicalities. I hope I have succeeded in that. So we are going to actually look at area minimizing graphs. First of all, let me set up a little bit of notation. So we will use this notation for the graph of a function u. So this will be the set of points x u of x such that x is in a certain domain f and f will usually be some subdomain of rn and for our purposes u from some omega which contains f into rn will be always a Liebschitz function. Yes? Right bigger. Okay. Good. Okay. So first of all, let us make a very simple observation. So omega contains f. Very well. So first of all, we will always look at functions which have a certain precise Liebschitz bound. So first of all, we will use this notation for the Liebschitz constant. And secondly, in all our assumptions, the Liebschitz constant of u will be either bounded by 1 or bounded by 2, something like that. Okay? So it's not so important actually. One could figure out better constants. I mean, the important point is that it's always under control with some uniform bound. Okay? And the first observation that we the first observation that we want to make in the lecture notes, it's actually stated in the form of a lemma. And it's not a lemma that I'm going to prove because it's very elementary, is that if we have a Liebschitz function and we tilt the coordinates slightly, so not too much. In the new system of coordinates, the graph of the Liebschitz function still the graph of a Liebschitz function. Okay? So if we tilt the coordinates slightly, graph of u over some set f will remain. So I will use actually this notation for the new coordinates will be actually the graph of the new functions in the new system of coordinates. So here I have my original coordinates and maybe some Liebschitz function and then here I have a new system of coordinates. Okay? So this would be like x and y. Here I have a new system of coordinates x prime and y prime and in the new system of coordinates this will remain the graph of a new Liebschitz function which we'll call u prime. So if I take a point over here and I go down over here perpendicularly in the new system of coordinates this will be the point x prime u prime of x prime. Okay? And it's a simple observation that the Liebschitz constant of u prime is actually less or equal than the Liebschitz constant of u plus some universal constant c times the modulus of the matrix whose rotation I mean of the rotational matrix which is giving you the coordinate transformation from x prime y prime to x y. So the coordinates x prime y prime have then the relation that x prime y prime is equal to a x y. Okay? I mean this as long as you have this uniform bound of the Liebschitz constant which is like one or two. Okay? So this is nothing but by saying that since the Liebschitz function I mean the Liebschitz constant of I mean if you had a nice domain the Liebschitz constant is nothing but the maximum slope of the function and if you look at it in the new system of coordinates the maximum slope of the function can be at most the maximum slope in the original system of coordinates plus the angle that the new system of coordinates forms with the old system of coordinates. Hilbert Schmidt norm for a matrix operator norm it I mean it's a matrix so it's an element of a finite dimensional space so pretty much any norm you use actually it's going to make this statement happy. But okay so let's say in general when we I mean in general when I have a matrix and I put these two and I mean I look at norm of the matrix I'm always thinking about the Hilbert Schmidt norm if I'm not saying anything else. So the Hilbert Schmidt norm meaning the sum of the squares of the entries of the matrix and then you take the square root of that. Sorry? Right sorry exactly so the angle sorry I forgot to subtract the identity good point. Yes absolutely I have to change actually that in the lecture notes. Very well okay so thank you very much for the observation so now let me give you a definition so I will consider area minimizing graphs okay so I will define area minimizing Lipschitz graphs by the following definition so first of all the Lipschitz constant of view that's going to be less or equal than two okay and then so here there will be some some constant c0 which depends on m and n and it's positive okay and so for every system of coordinates with this rotation matrix which is close to the identity by this constant c0 so and so if u prime so if graph of u prime omega prime is the description of our surface in that system of coordinates okay then what I want is that the m-dimensional volume of the graph of any competitor v and omega prime is actually bigger or equal than the m-dimensional volume of the graph of u prime and omega prime and this is for every Lipschitz map v such that the place where v and u prime differ right is compactly contained into my omega prime okay and now of course this u is from omega into rm and we actually prescribe that the domain omega is open yeah rm okay so this means that every time that I make a perturbation which is a new Lipschitz graph but I allow my system of coordinates to be slightly changed so then the competitor must have strictly a bigger volume and what is this m-dimensional volume that I'm using so this m-dimensional volume is the usual m-dimensional volume of a surface I mean it might be described as the house of m-dimensional measure of the graph as a subset of the product space and we have a classical formula for this graph for the area of this graph which I'm now going to write down explicitly because it will be extremely useful in a lot of our future computations so the m-dimensional volume of the graph of u over some set omega and actually in this case this omega does not have to be necessarily an open set actually we can make this computation on any Borel set okay so what is this this is this has an explicit formula so it's the integral over omega and then I'm going to take the square root so here I have one plus modules of du squared and then I have the sum and in this sum I have all minors of order two times two so and I have I have to take the determinant of these minors and then square it okay so m alpha beta of du is a or is any k times k minor and our convention is that modules of alpha where alpha is a multi index is exactly the length of the multi index so this means alpha is a collection of numbers which are going to be the rows that I'm selecting in my in the matrix representation of the derivative of u beta is going to be the columns and here I'm selecting k equal modules of alpha rows and k equal modules of alpha equal modules of beta lines and then I take the determinant and square it so this complicated this complicated formula of course becomes rather easy when you have a function which is taking value in the real line because when the function is taking value on the real line this term over here drops and you only have the square root of one plus modules of gradient of u squared okay so this formula gives you the house of measure of the portion of the graph which is lying over the set f okay so it's it's it's a well-known fact that area minimizing graphs with a sufficiently small constant are actually are actually real analytic so this is a fact and we are going to see actually at least a partial proof of this so if u is a leap sheets area minimizing graph with sufficiently small constant then u is real analytic actually in co-dimension one so if your graph is taking value into r into the real line you don't need any restriction on the leap sheets constant and there are several ways of doing this so the most classical way would be if you are for instance in co-dimension one to write down a system of partial a single partial differential equation for your minimizer just by writing down the order Lagrange equation using the Georgian-H theorem to infer that actually the derivative is herder continuous and then use the Schauder estimates to infer sort of further regularity okay and then okay so it's a classical fact that the equation that you get which is an elliptic equation and which is non-linear but it is real analytic in the in the gradient the second derivative any new if it has I mean any smooth solution then will be real analytic in higher co-dimension actually the bound on the lip sheets constant is important so there are examples of solutions of the Euler Lagrange equations and these examples are actually due to Ostermann and to Lawson and Ostermann which are just lip sheets but don't gain any further regularity so they're not going to be c1 alpha in fact of course as soon as you have herder regularity for the derivative you can apply Schauder estimates so if you have c1 alpha regularity even in the higher dimensional case you can carry on the usual program and then get to real analyticity okay so we're not going to follow that I mean we're not going to follow this plan but we're going to prove another theorem of the Georgi in this particular case so it's it's not the most general version of this theorem of the Georgi because the most general version works without assuming that the surface minimizer is a graph but it's particularly interesting to see how the proof actually works in the graph case and then the general case is kind of more technical generalization for which you have anyway references in the lecture notes okay so here's then the first theorem that we will prove in this lecture I mean roughly two of the lectures will take to prove this theorem over here so this is the Georgi's theorem for area minimizing graphs and here is the statement so for every alpha between zero and one there exists constants and these constants which are epsilon zero and c are going to depend on alpha the dimension and the co-dimension of the domain such that if u from the ball of radius one into rn is a area minimizing leap sheet's graph with constant less than one then actually in half the ball u is a c1 alpha function and the helder norm I mean the helder semi-norm of the derivative of the function oh no no no wait wait wait I need another assumption sorry so if it is a leap sheet's graph and very important one quantity which will be called the excess which is the volume of the graph minus the volume of the disk so which we we will denote by omega m so omega m is there a back measure of the one-dimensional ball in rm okay so if this one is less than epsilon zero then as I was saying the surface is actually c1 alpha and the helder semi-norm of the derivative of u in the ball of radius one half is controlled by this constant that I gave you and the square root of this e which we will often call the excess okay so you know this over here that I'm taking the volume of the graph which is sitting over the unit disk and I'm subtracting the volume of the disk so this is so this quantity tells you how much the graph exceeds the area of the disk of course since it's coveting the disk it must have area which is at least the area of the disk and if this excess were equal to zero it's always a good thing to check whether you know the theorem is at least trivially true when you have a trivial assumption if the excess is equal to zero then the only possibility is that the function is completely flat and if the function is completely flat actually the helder norm of the derivative is just equal to zero because the derivative is constant I mean the derivative is actually identically equal to zero in this case okay so you see that instead of assuming that the lipschitz constant is small because here I'm saying actually that the lipschitz constant is less or equal than one I'm assuming in a sense that the oscillation of the tangent planes is not too large in an even in an integral sense right so this is more kind of an integral control okay so that's the george's theorem and let me I mean before going on to state the other theorem which is actually the real main goal of the lectures and which is also an illustration of a much more general theorem in a very simple setting so let me just make the following remark so this is also left as an exercise to you but it's certainly a more difficult exercise than this change of variables okay so we're going to do the same so we're going to do the following so assume this denotes the tangent space to the graph of the function u which is lipschitz and therefore differentiable almost everywhere I mean when I mean actually ignoring the domain of the graph I mean when I'm in this notation when I'm ignoring the domain of the function it means the domain does not play such a big role sometimes I will just drop it okay the tangent space comes with a very natural orientation right so if you take a standard basis e1 em standard basis of rm if you then take em plus 1 till em plus n standard basis of the other factor okay then you can write down explicitly a basis for your tangent space and the basis for your tangent space is going to I'm going to denote it in the following way so it's going to be ei plus the differential at the point p of f at the point ei I mean on the computed on the vector ei and this writing over here is nothing but the sum for k equal 1 to n and here you will have df k dxi of course at the point p sorry oh f is a function and the function there is called u yeah let me call it f actually okay and here I'm putting the em plus k vector okay so that's what I'm what that's what I I understand is this sum so this sum is going to be a vector in rm plus n and as I take i which is running between 1 and em I actually get a basis an oriented basis of my tangent to the libschitz graph which gives you a canonical orientation of the tangent and of course this canonical orientation of the tangent is kind of the only orientation which is compatible with the idea that if the function is identically equal to zero and you are completely fat right so the canonical basis becomes e1 em of your tangent space right so just the basis that we have fixed of the of the first factor okay very good so now if I give a name vk to these okay vi to these vectors okay I can use multilinear algebra and this k vector so I take the wedge and then I normalize by the modulus and I'm going to tell you what is the modulus of the of the of the m vector in a couple of seconds okay so this is what we will call t vector p graph of u so this is the m the unit m vector which is orienting your tangent plane okay and what is this object over here so this object over here has a very neat geometric interpretation this object over here you can define it as the m dimensional volume of the parallel of the parallelogram which is spanned by v1 vm that's one possible definition but you can actually endow the space of m vectors with a scalar product the scalar product being the following so if you have two simple m vector the scalar product between them is simply the square root of the determinant of the matrix that you obtain by taking the scalar products of the various vectors and then this m dimensional volume is actually nothing but the norm which is associated to this scalar product that you can define on all m vectors so I'm defining it only on the simple m vectors but you can actually extend it by multilinearity to all m vectors okay very good so now there is an interesting formula so let me give you this as a proposition maybe it's actually a lemma in the lecture notes so take a leapschitz graph it's just a computation so you don't need actually any bound on the leapschitz norm it's not going to be an estimate okay then the m dimensional volume of the graph of u over omega minus the measure of omega okay which we know by a simple geometric observation is actually a positive number right just because we have said you're covering omega so you only have hope of getting at least as much area as omega so this is actually one half and then you have to take the integral over the graph u omega and here you take the distance between the m vector which is tangent to the graph of u and the horizontal m vector so for which we will use often this notation squared okay and here you're integrating in the house in the house of measure and no no no sorry no no no sorry sorry no no I got I got actually mixed up no this color product does not have any square root sorry so thank you for pointing this out I got mixed up so the modules of the m vector right is of course the square root of the scalar product of the m vector with itself okay so of course the square roots you only you only take it when you're interested in computing this guy yeah yes okay and so what is this pi zero with a vex with a vector on top so this is the notation that we will use for the standard m vector which is orienting which is orienting your horizontal plane okay so pi zero is equal to e1 wedge em okay so now this gives you a sort of explanation why in the estimate I mean in the george's estimate of the c alpha norm of the derivative of u you had a square root of this excess okay so if you interpret it in this way right this quantity this excess is a kind of square of an l2 norm right so this looks like the square of an l2 norm okay so now one of course you can observe one thing if you are a leap sheets graph as soon as the left hand side is equal to zero the right hand side is equal to zero and vice versa because you have this identity but if you have a general surface the thing on the left might be equal to zero and the thing on the right might be actually positive so if a surface gamma so so if a surface gamma let's say for surface sigma a general surface sigma has no boundary in the cylinder omega times rn right which is kind of the situation in which we are working if we have a graph so if we have a graph the situation is something like this right so the boundary of the surface is all lying on the lateral surface of the cylinder so you don't see any boundary inside so in a more general situation in which the surface sigma has no boundary in omega times rn it's going to be a multiple cover of what you have down right but you might have more than one than one sheet okay so you might have for instance several sheets okay then if you have several sheets the right hand side for instance really measures how much these sheets are horizontal right so the right hand side would be equal to zero if these three sheets in the example they're all flat but on the other hand the left hand side is specific of assuming that you have one single sheet because the left hand side is going to be the volume is going to be three times the volume of the horizontal projection then you subtract the horizontal projection and you just have something positive okay so you might wonder whether there is a general the georgi type statement which tells you even if I don't assume that it is a graph and so the surface might be multiple sheeted if I assume that the right hand side the excess written with the moduli is small can I conclude that my surface comes of course not with a single graph with c1 alpha estimates but with a bunch of graphs right so with a sheeting so of a certain number of graphs which will depend somehow on the total area of the surface to start with which are all c1 alpha okay and I mean this is not a you know it's not a generalization for the sake of generalization so if you were looking at general of it at the theory of general surfaces which are minimizing the area for which a priori you don't know that there is a graphical description at any point right because they might be very rough this is the kind of theorem you're actually looking for because since you don't have any a priori assumption that you're locally graphical right so you want to work in the most general situation and now here it comes a very interesting point so in co-dimension one this sheeting theorem is actually correct so in co-dimension one if I have a generalized surface and I tell you that this quantity is small over a certain cylinder and I have a bound on the area of the surface then if this quantity is sufficiently small then I know in half of the cylinder I'm actually a certain number of sheets which are c1 alpha and they are all separate graphs which don't intersect so in co-dimension one this kind of sheeting version of the georgi's theorem is correct but in higher co-dimension is completely incorrect so it's I mean it's it's it's a famous example which is due to Federer and based on the computation of Wirtinger and I can actually I cannot give you the proof but if you take the the following surface so you take the complex if you take the the complex space c2 which you identify with r4 and you take the following surface so you take z square w square equal to z to the power 3 okay so this is what is usually called a a algebraic variety in in in c2 well locally this is an an oriented area minimizing surface so in the sense that it's an oriented surface it has a singular point in the origin but if you sort of discard the singular point it is an oriented surface such that if you replace a portion of this surface with any competitor the competitor has to go has has to have larger area okay it is very flat in the neighborhood of the origin in this sense so you see if I were allowed locally or I am allowed of course locally if z is different from 0 to take a square root and then on top of every z I see somehow two copies and I will see z to the power 3 over 2 so I will have very flat tangents and this quantity will become actually very small now we will see that's the the the quantity that we have to look at is not really this one but it's it's average so if you want the quantity which is really relevant for the epsilon regularity theory is actually the average of this so I have to normalize this by the volume of the horizontal projection but even if I normalize by the volume of the horizontal projection these the tangents to this surface will be uniformly close to the horizontal tangent when I take z very small and nonetheless in the neighborhood of the origin I'm not able to say that there are two sheets which are c1 alpha right so if I take I mean if I make a turn around zero I will pass from one branch of the square root to the other branch of the square root and my surface is always connected actually whereas if I had a sheeting theorem like the one I'm hoping in the generalization of the georgies theorem I would have to have two separate sheets which are disconnected okay so and this actually has given big big headaches in the regularity theory of minimal surfaces for a lot of time but there is a fundamental theorem by Andren which is actually I mean which I cannot quote in its generality but for which I can give you at least a kind of an idea so so the theorem of Andren just says that if this quantity over here is small so let us call it boldface e so if e is small and the mass and the total area of the surface is under control so if e is small for not for a graph I mean but for a general surface so if I replace this graph of you with sigma and the volume of sigma is under control okay then up to some small errors my surface is a cover of the horizontal omega so is a say a q cover where q just gives you the number of points that I meet on top of my horizontal point at most places is a q cover and although actually I might have a singularity at the origin as it happens in this case however if I take the average of the sheets the average of the sheets is more regular than the whole object itself yes yes yes I'm taking it in the cylinder and I'm assuming and I'm assuming it has no boundary in the cylinder yes yes sorry that this is going to be a little bit vague but I mean in a few seconds we actually arrive at a new theorem which is going to have a very precise statement so I mean all of this discussion is trying to summarize things that you can actually read on surveys so for what is the reason why this theorem is important in the regularity theory and so on but then I mean in like five minutes we arrive at a precise theorem on graphs and then we will focus on that somehow and then my definitions will become precise again so and and the average of the sheets is more regular in the following sense there is an approximation so there exists an efficient and approximation which is c3 beta for some positive beta now efficient means the following so this approximation let us call it say g then efficient means that the c2 alpha norm of the derivative of g c2 alpha c2 beta sorry so not c beta like before but two derivatives more is actually bounded by a constant times the success to the power one-half and in which sense this is an efficient approximation this is an efficient approximation meaning the following in every region in every small region you look at the separation between the the two sort of sheets which are over the region and which are kind of further away and this approximation of the average sits in between okay so it's an approximation of the average meaning that at every scale the maximum separation between the sheets that you have in your object is comparable to the distance between these sheets and this average okay so under this definition you could actually go back to the original situation of the geology and ask yourself what happens if you have a single valued graph so if you have a single valued graph i'm telling you that in each region your approximation of the average of the sheets has to have distance between the two sheets which are most further away among each other which is comparable right but that's only one sheet so this the distance between the the sheets which are further away is exactly equal to zero so in the case of a single sheet your approximation has to be the single sheet everywhere and if it has to be the single sheet everywhere you actually conclude that your area minimizing graph is c3 beta okay so so let me quote this theorem by Angren and so as you see it's just a baby version of a much more general situation which is much more complicated to prove but my hope is actually to give you good ideas on how you can actually get the correct estimates in this simplified situation so if say as before u from b1 into rm has slip sheets constant less or equal than 1 and graph of u is area minimizing okay so here at the beginning i probably should have said that there exists two constants epsilon zero which depends on m and n then another constant c which depends on m and n and now the constant beta which different from alpha alpha in the georgi's theorem was any constant between zero and one so you could get you could get almost c11 estimates now beta is just some small number so these are all positive such that if you have this u which has lips is constant less or equal than one and the graph of u is area minimizing and which is the important assumption on the excess so again i write it as the volume of the graph of u minus the volume of the unit ball which was omega m so this has to be less than epsilon zero and this is your excess e okay then in the ball of rough radius the derivative of u has c2 beta estimates which are controlled by a constant times this parameter e to the power one half okay so now just i mean if you read the lecture notes you you will get a description of why somehow this theorem is important and so on so let me just give make two comments so this is just a very i mean it's a toy it's a toy problem i mean once you have c1 alpha estimates you can write down the pd for your function u and by shoulder estimates you can actually get that conclusion not only you can get that conclusion with c2 beta but you get actually that conclusion with any derivative okay so the important point is that we will prove actually this theorem without using any shoulder estimates without using any pd or any Euler Lagrange equation for your critical point in particular because the interest of such a theorem of having such a proof is that when you are in the multi-sheeted situation and you cannot write down a pd because you have an object which is not regular okay you will not be able to carry on the proof via shoulder estimates of this okay so the other comment is the following so Andren's theorem in its full generality is a step of a major work that he did back in the 80s on the regularity of oriented surfaces in co-dimension higher than one okay so it was a monograph of a thousand pages and this theorem over here i mean the bigger theorem i mean not this one which is the corollary the bigger theorem which i sketched over there as a 500 pages proof okay so i have some works with Emanuele Spadaro in which we have recast this theory in a more manageable framework and in particular we have a proof of this center manifold construction which is around 50 pages and it's much more flexible meaning that people meanwhile have picked up these constructions and they've proved other theorems whereas Andren's paper was almost unused so for us this observation that in the single-sheeted situation you can get actually C3-beta estimates without writing any shoulder and without writing any Euler Lagrange for your function U was the starting point to get this simplified proof okay and what i'm going to do is in the proof of this i'm actually following a small paper that i wrote with Emanuele Spadaro before we actually completed our program of recasting Andren's theory in a more manageable way and which in some sense it's a small self-contained work in which you you can give an idea of what are the key estimates in this central manifold construction now in the paper that i have with Emanuele Spadaro that is not the assumption that the function U that the surface is a graph okay rather than assuming that the surface is a graph i'm actually only assuming this condition over here now this condition over here when i mean since i'm taking the whole volume and i'm subtracting the volume of the disk if this one is very small it's certainly telling you that you cannot have two complete sheets on top of the other guy right so it's a weaker theorem than the one of Andren not only because i'm assuming that the surface is graphical but also because i'm assuming that this quantity is small which is right so now if i drop the graphicality assumption and that assume that this quantity is small i will not be able in a first step to say that the function is a graph but it will be a graph of a single valued function i mean the surface will be the graph of a single valued function on a very big region at least okay so the paper with Emanuele is sort of in between the general theorem and this theorem over here but it's still very very elementary compared to the to the big Andren's theorem okay so now what what is the plan of the lecture so the plan of the lecture is to give you first the proof of the georgie and then a proof of Andren in the in the way i have defined things over there okay and hopefully by the end of the second lecture tomorrow or if not by the end of the second lecture tomorrow not too deeply in the third lecture we will be finished with the georgie's theorem and then although we are not using shoulder estimates some of the conclusions of the georgie's theorem will be functional to actually prove this theorem over here okay so then therefore from now on we focus on that theorem over there no on the theorem of the georgie which i stated before and let me come to one first okay so let me say proof of the georgie's theorem so this is just the start we will do some preliminaries today and so it's i mean it will contain a lot of details but some things will be left as an exercise at least the things which are doable so first of all we are going to define this excess which is wrote down in a couple of very peculiar example in a very general situation so first of all we will define the spherical excess for a surface sigma on a ball of radius r centered at p so when i actually use this bold face ball i'm meaning a ball in the product space r m times n and with respect to a vector pi an m vector pi which is orienting a certain m plane so this excess is going to be the following quantity it's going to be one divided by omega m r to the power m so this is just the volume of the m-dimensional disc and then here i integrate over the ball of radius r centered at p intersected with sigma okay so the distance between the tangent to the surface sigma and your vector i mean your your your plane that you have fixed okay so in picture this means you just have a fixed vector a fixed plane pi a fixed m vector and you are measuring the l2 norm of the angle that you do between this pi and this tangent so here it's pi so here's your surface right and what you're doing is at each point here you're taking the tangent plane so this will be tp sigma this will look like this okay and essentially this quantity over here it tells you how big is the angle between this tangent plane and the vector and m vector pi that you have fixed oh yeah yeah yeah sorry thanks yeah this i want it to be q right and here i'm integrating in the house of measure in q thanks a lot yes so p is the center of the ball and i definitely don't want this point to be called b i want it to be called q yes thanks okay then you see that i fixed an arbitrary vector pi if i want to have an idea on the l2 oscillation of the tangents i can actually optimize on all possible pi okay and this will be called the spherical excess so if i'm not writing down here any plane then it means i'm minimizing over all possible unit k vectors okay so that's the excess on the sphere or on a ball and we will actually work often with this excess on the ball and i want to make a very important remark which is very elementary but nonetheless it's very important so the problem has a certain scaling invariance which we will often take advantage of so scaling invariance okay if i take a function u and i make the following transformation i define the new function u r and u r of y is going to be do we have it maybe over here uh no yes and you find a new function u r and this new function is one over r and then you compute u of x plus r y and you subtract u x okay so you you notice one thing the Lipschitz constant of u r stays the same as the Lipschitz constant of u right this u r is just i mean the graph of u r is just an amortity of the graph of u so if u was area minimizing then u r is also area minimizing the Lipschitz constant stays the same and this excess is scaled by this power in such a way that here in this amortity i transform balls of a certain radius into balls of a certain other radius so the excess on the corresponding balls stays the same okay so let me just summarize this discussion in the following so both the Lipschitz constant the area minimizing property and the excess are invariant under these scaling and translating so this means anytime that i will have a statement on some ball of radius r centered on point b i can actually apply this idea and get back a statement on the ball of radius one centered at zero okay and this will simply simplify a lot of computations okay and then there is another couple of elementary remarks that i want to make so another elementary remark is the following so if i so if i have a ball of radius r so if i fix an optimal plane so fix a pi such that your excess is the optimal here okay and now take a smaller ball say take a b rho q which is contained in b r p okay now i can i can simply make the following comparison so i can say the overall excess of sigma in the ball of radius rho and center that q is of course less or equal since here i'm optimizing overall planes pi then the excess with respect to that particular plane okay and now since indeed that if in the definition i have an integral right if i look at the excess in the ball of radius r centered at p which contains the ball of radius rho centered at q well the integral can only become larger but then of course i have a normalizing factor and i'm paying that normalizing factor so i can write here a very simple inequality which tells me okay this is then less or equal than r to the m divided by rho to the m and then here i will have the excess in the larger ball with respect to this plane pi but then this one is the optimal plane so of course i have this inequality okay so now with very similar arguments you can actually get other similar comparisons and let me just state a proposition here here so this is a proposition that i leave you as an exercise and it will be used very often so so u is a map from omega into rm the lipschitz constant is less or equal than 2 and you have a point p which is equal to x u of x and then another point q which is equal to y u of y okay so then there are constants c1, c2, c3 these are all geometric constants and you have the following facts so first of all you can make a lower and upper bound with respect to r to the m of the volume of the graph of u in any ball centered at p should map into rn yes and that is going to be true for every ball which is contained in the cylinder so the total volume that you have on the ball given that you have a lipschitz constant bound is controlled from above and from below by the total volume of the unit disk or of the disk of radius r okay now you can use that so you can use this first very elementary remark to actually have the following control so i can measure the distance between two planes with the excess of sigma in the ball of radius 2rp okay so that's the graph of u on b2rp with respect to pi1 plus the excess of the graph of u on b row of p with respect to pi2 and this provided i have that the ball of radius row is in between the ball of radius 2r and the ball of radius r and then of course i have to be less or equal than the distance between the point x and omega so this is simply telling you if this excess is small and this excess is small it's not possible actually that the two planes are too tilted right and then finally another very similar estimate so again for pi1 and pi2 but this time i'm actually changing the the central points so i can also say pi1 minus pi2 squared is certainly less or equal and here i will have a constant c3 and then i can put the excess of the graph on a certain ball so it's brp with respect to pi1 and then the excess of the graph of u again in the same ball with respect to pi2 sorry not in the same ball but in in in in in in this other point that i have over here and that as long as the distance between q and p is comparable so in fact we will need it in the following situation it's equal to r and of course again the distance between q and the boundary of omega yeah here's was always the boundary of omega actually yeah distance between x and the boundary of omega and the distance between y and the boundary of omega it's both time less than r okay so let me just give you a very just a very simple explanation of 30 seconds and then we finish this lecture so so what is the first estimate coming from well the first estimate is here over here is coming from the fact that the ball is contained in the respective cylinder right the ellipse constant is under control and then in the respective cylinder somehow the projection down is a disc of volume omega m r to the power m and you know over you have ellipse graph the area cannot become much larger what is what is the second bound actually coming from well the second bound is coming from the following fact so if i take a ball of reduce r here i have a central point and here i have my ellipse graph right well certainly there will be another cylinder which is slightly smaller maybe of radius b of r divided by 2 and the ellipse graph over this cylinder is entirely contained inside the ball okay and the projection of this cylinder down is going to be the disc so possibly this constant over here you can actually compute it as like 1 over 2 to the power m yes uh yeah exactly sorry and then when is this uh comparison here coming from well for instance let us take the second estimate so the second estimate i could simply say the following so take the intersection between brp and brq intersected with your graph of view by the first estimate this is certainly bigger or equal than some constant times some r to the power m okay okay and then if i average b1 minus p2 squared over here right uh i divide by this constant i will discover that this one controls pi1 minus pi2 squared and here i have 1 over a constant r to the power m okay now i with the triangle inequality i sneak in the tangent to your graph so i take pi1 minus the tangent to the graph and square it of course i do the same with pi2 and square it and then here i'm integrating on this intersection instead of integrating in the intersection i integrate once on the ball of reduce r centered at p and the other times on the ball of reduce r centered at q okay and then i get the excess with respect to pi2 and the excess with respect to pi1 well in the excess i was dividing by one over two i mean i was dividing by two here i'm dividing by some constant so then you actually get the constant that you have over here tomorrow we start over and then um go