 So it's a real great pleasure to start the afternoon session with Andromeda Chiodi from Warwick who will tell us about the polarization of surfaces with particles in the land. OK, thank you. And also, especially thanks to all the organizers for inviting me here. I mean, I'm really pleased to also have finally met Eugenio Calabi, so I'm very excited. So it's really, for me, a great opportunity to screw up a seminar. OK, so I would like to talk about some sort of long term project jointly with the different people. So Daniele Bartolucci, Alessandro Carlotto, who is here, Francesca de Marquis, and David Ruiz. OK, so let me start, as we saw yesterday, with the uniformization problem. So I will consider a compact surface sigma with metric g and no boundary. So a classical problem in geometry is to find somehow the best possible metric on the surface. So for example, whether you can deform the metric so that you get constant Gaussian curvature. OK, so due to the uniformization theorem, this is the case. And the sign of the curvature depends on the topology of sigma. So if g stands for the genus, for genus 0, you can get positive curvature. For genus equal to 1, 0 curvature. And for genus greater than 1, negative curvature. So this is an if and only if. And this is somehow how far I can keep up with Donaldson's talk. So I will stick to two dimensions. And I will consider this problem, which is kind of, sorry? Real dimensions. Real dimensions, too. Yeah, it's really, OK, I'll do my best anyway. So I consider a problem of fixing the surface and m points, p1 pm on sigma, some angles, positive angles, t1 and tm. And the question would be to deform conformally the metric to get constant Gaussian curvature on such an object, keeping somehow prescribing the conical structure at these points. So that would be my notation. So if you choose a conformal metric in normal coordinates near a given point, pI. So I call theta i the opening angle of this cone. And I call theta i, I write it as 2 pi times 1 plus alpha i. And alpha i will be greater than minus 1, so that we get positive angles, which can exceed 2 pi. So with this notation, somehow the singular structure is encoded in a divisor, which is just a collection of the singular points with the singular weights alpha i. OK, so this comes somehow to a PDE, a L'Uville equation. So in the smooth case, if you deform the metric conformally, and if you write the conformal factor conveniently as e to the 2w times g, then that's the transformation law for the Gaussian curvature. So it comes with the L'Uville equation. And kg tilde will be the new Gaussian curvature after the conformal transformation. So the problem is reduced to a PDE. And in the presence of a divisor, you get this singular version. So you simply subtract off some Dirac deltas with the weights alpha j. So I have a slightly more complicated expression for writing down the angles, but you get a cleaner equation here. And you get some Dirac deltas at the singular points. So if you take a standard cone, then you can get it just by folding a nicely piece of paper. So the Gaussian curvature will be 0 on the smooth part. While you get a Dirac delta at the tip with weight minus 2 pi times alpha. So in particular, for alpha negative, so this would be a standard cone. And for alpha positive, then you get an orbit fold with angle greater than 2 pi. All right, so there is a constraint on somehow the function to be prescribed by the Gaussian formula. So if we want the new curvature to become constant, then kg tilde has to be equal to some constant rho. And rho is assigned prescribed basically by the Gaussian formula, which you can get basically by smoothing out the singularities. So you round them off, and that's how you get this formula. So a useful example to keep in mind is the American football. So I will actually use this several times. And the integral of the Gaussian curvature over the smooth part amounts with this notation to 4 pi 1 plus alpha. Because in the Gaussian formula, you have to compensate the curvature of the two conical singularities, each of them carrying an amount of curvature minus 2 pi alpha. So you have to add the 4 pi alpha to the smooth part. OK, actually, other values of rho can also be interesting for actually physical reasons. In fact, singularly equation, so even regular ones, appear in physical models like Tron-Simon's theory, a little weak theory, and also as a mean field equation stationary flows. So in this case, basically there is no particular constraint on the parameter rho. So for those problems, it just plays the role of a physical parameter. But for the geometric problem, there is a constraint given by the Gaussian formula. All right, so this is a problem with the variational structure. You can somehow desingularize the equation by a change of variables. So first of all, you solve for the Laplace equation equal to Dirac delta. So you get the Green's function on the surface, which has a logarithmic singularity near the point. And if you just make a change of variables, then the new equation you get somehow is more regular. So the singularities are gone. But you get a weight in front of the exponential term. And so I'll write down the equation, so which is a problem total equivalent to the previous one, minus delta u equal to rho h of x e to the 2u, say, minus 1. And the weight function h of x is positive and behaves like the distance from the singularities to the power 2 alpha i. So it might blow up or vanish depending on the sign of alpha. And OK, so here's the variational structure. So you can find solutions as critical points of this functional, which has the Dirac-Leh energy. So we saw this function in various fashions in these days. 2 rho, so then there is a linear term. And then a tricky term which involves a logarithm of the integral of h of x times e to the 2u. And somehow, this is the most tricky term to understand. All right, and you do it using the Moser-Tüdinger inequality. So maybe differently from Boo's talk, I mean I will read the correct constant here, so I will really care very much about it. So that's the Moser-Tüdinger inequality. So with my notation, the constant is 1 over 4 pi. So I'm putting e to the 2u. And u bar is the average. I'm sorry? I forgot the logarithm. OK, well, it doesn't matter anyway. So you cared about finiteness, but yeah. OK. And u bar is the average of the function u on the surface, which makes somehow inequality consistent. And so the constant 1 over 4 pi is sharp, because you can insert into the inequality bubble functions, which have this form. So you fix a point x on the surface, and you are basically developing a spherical metric near the point x. So when the parameter lambda tends to infinity, they become more and more concentrated. And in fact, the conformal volume converges to 4 pi times the direct delta at the point x you consider. And the metric is becoming more and more round. OK, so with this choice, both sides of the equation diverge at the same rate. All right, so but here there is a weight function, which behaves in this way. And the best constant was found by Chen and Trianov. And there is also some rated work by Chen and Yang in, say, domains of R2. So here's the best constant. So it depends on the weight alpha j in front of the singular points. OK, so let me comment on this. When alpha is negative, which means, say, when you have locally like a standard cone with opening angle less than 2 pi, h of x is a singular term. OK, so you expect a worse constant because you have a diverging weight function here. OK, but when alpha is negative, you don't get a better constant because somehow you can consider one of those bubbling functions I showed before. And you let it concentrate at a point where h of x is not vanishing. OK, so you don't see any, say, advantage from the vanishing of h at the singular points. OK, you concentrate volume elsewhere. And so you get exactly the same constant as before. OK, so that's the reason why you see that constant there. And you pick up somehow the most negative weight among all the ones you have. Because that's where you pay the most price of having a singular weight. OK, so what's the consequence of this inequality? Is that when rho is sufficiently small, then you can easily control the logarithmic term by means of the Dirichlet energy. And therefore, your functional has a global minimum. So just apply the direct methods of the calculus of variations. All right, so then there are three geometric cases. So that's the value of rho constrained by the gas-monet formula. And there are three cases. So there's a subcritical case, which is the one I described before. So you have a nice coercive function which you can minimize. OK, coercive means that it goes to infinity when you diverges in norm. Then there is a critical case when you have exactly equality. So the functional is bound below, but not coercive anymore. So this can be sort of tricky. And the supercritical case when rho exceeds that chain trion of constant. So in this case, the functional is unbounded below. And there is no hope of minimizing. So if you compare to the classical uniformization problem, the subcritical case corresponds to k less or equal to 0. And by the way, when you are on S2, and if all the weights are negative, the subcriticality is also necessary. And the classical counterparts of the critical case is actually the sphere. So of course, for the classical uniformization problem, you can solve always. But in general, the picture of the functional could be something like this. So you might have a lower bound, but you're not guaranteed to be able to minimize, because you might have some kind of asymptotes. And the supercritical case, in fact, does not have a classical counterpart. And in fact, in some cases, you cannot solve for this problem. So one example, which is well known, is the field drop. So take the sphere with only one singular point. Then there is no constant curvature metric on such an object. And in general, if you take two singularities, then the weights have to be the same. So these results, you prove basically exploiting the Möbius transformations. And it's a sort of Kazan Warner type of abstraction. And in a paper by Shannon Lee, it was shown that if all the weights are negative, and if you are in the supercritical case, then necessarily the surface has to be an American football, if you can solve. And in a paper by Remenko, the case of three singularities on S2 was studied. And he derived the necessary and sufficient conditions for existence using basically monodromy methods. So patching somehow pieces of spheres together. OK. So I said, in general, when you are in the supercritical case, there is no chance of minimizing, but still you may hope to be able to find settled points. So I would like to describe a general variational strategy to attack the problem using variational methods. And that's, say, the general strategy you use in an abstract setting. So you're given a functional on some Hilbert space with values into R, so smooth. And a classical strategy to find the critical points of a functional is to look at the structure of the sub-levels of this functional. So here's one, maybe a simple example, just x squared minus y squared in R2. And in fact, you see there is a change in topology from positive sub-levels into negative sub-levels. So the sub-level transforms from connected to disconnected. So eventually, this is a complicated proof to show that the origin actually is a settled point for this function. And the general idea is the following. So if you're given two values a and b, a less than b, if there are no critical points in between these two values, then you can use a gradient flow to deform the higher sub-level into the lower sub-level. So you never get stuck in a sense. This is called the deformation lemma. And the consequence is that if you observe a change in topology between the two sub-levels, then there should be a critical point. So I would like to apply actually this general strategy, which is more theory to this functional, showing two properties that for this specific functional, high sub-levels are topologically trivial, contractible, while low sub-levels inherit somehow topological structure from a finite dimensional CW complex, which I call sigma rho alpha, which depends on the topology of sigma and the divisor alpha. So I will tell you exactly what this object is. So the idea is to show that by the deformation lemma, if this object, which describes the topology of low sub-levels, is non-contractable, then you should get existence, because high sub-levels have trivial topology. But you should be careful in making such statements, because asymptotes can give you trouble. So this is one example in which you have an asymptote and no critical point between C minus epsilon and C plus epsilon. And still, you're not able to deform the higher sub-level, which is disconnected to the lower one. So you need some kind of compactness criterion. The problem here is that if you follow a gradient flow for this functional, that will bring it to infinity. You lose compactness. And so you need some kind of compactness criterion to be able to apply such a statement. And this is done by blow-up analysis. So let me discuss compactness of solutions of the problem. And this is done using blow-up analysis. So as people usually do in geometry, so you rescale solutions and try to get some standard profiles. So here they are simple and only of two kinds. So we have the sphere with total curvature for pi and the American football. So I already remarked that with this notation, the integral of k over the American football is 4 pi 1 plus alpha. So when alpha is integer, indeed the American football is not the only profile. So there are also some sort of wiggly versions. But that doesn't affect the quantization of the curvature. Okay. So there was a result, indeed, about blow-up of solutions for this problem. So in the smooth case, there are results from the early 90s by Beziz and Merle, Lee and Shafir, mostly. And for the singular case, the analysis has been done by Bertolucci and Tarantello about 10 years ago. So you consider a sequence, Rhoyne, which is bounded, and a sequence of solutions of this equation. Okay. Then you have an alternative. Either you have uniform boundedness of the solutions, or if there is blow-up, solutions would blow up k spheres, plus possibly American footballs at the singular points. And this concentration phenomenon leads to some quantization, because the limit curvature somehow should be given by 4k pi, so which is the total curvature of k spheres, plus the curvatures you get from the American footballs. Okay. So you will blow up at some regular points, possibly, and at some subset of the singular points. All right. So if blow-up occurs, there is a quantization of curvature. So necessarily the limit value of rho must belong to a family of elements of this form, so which you can write down as sums of this type. Okay. So there is a quantization of the limit curvature in case of blow-up. And an immediate consequence of this result, which has been also refined by Chen Lin and Zhang, okay, but I will care about the quantization mostly, is that if you avoid those critical values for the curvature, then you stay compact. Okay. So if rho doesn't belong to this set, then you have compartments of solutions, so they will stay uniformly bounded. And also, in fact, the deformation lemma holds true. Okay. So in fact, if you can pick up some difference in topology between different sub-levels, then there will be a critical point in between. So the second statement is actually non-completely trivial and relies on some entropy inequality proved by Struve at the end of the 80s. Okay. So topology of high sub-levels, I claimed that it's trivial. Okay. So how do you show it? Well, if you have compactness of solutions, then you can basically deform by gradient flow the full functional space you're working with, so which is a vector space and therefore contractible, onto some high sub-level. Okay. So you have something like this. So for the saddle function, which I showed before, so you can deform the whole plane into the interior of two hyperbolas. Okay. So you would obtain some contractible set. And therefore, being high sub-levels, the formation retracts of vector spaces, they both have trivial topology. What about low sub-levels? So let me recall the quantization property of blowing up solutions. Okay. So if you blow up at a regular point, you get an amount four pi of curvature. While if you blow up at a singular point, you get an amount four pi one plus alpha. Okay. Somehow the main tool in our analysis is that we can prove a similar result basically for arbitrary functions. Okay. So the principle is the following. So if you concentrate any finite amount of conformal volume near a regular point, then you get an amount at least four pi of curvature. While if you concentrate near a singular point, then you get an amount four pi one plus alpha of curvature. Okay. So somehow you wait the points differently. And the general principle is the following. So the total curvature which is available is rho, so dictated by the Gaussian formula. And if you keep adding volume at those regular or singular points, eventually you will exhaust all the available curvature. And you have to stick to finitely many concentration points. So that's the general principle which I will try to illustrate. Okay. So based on the principle, you can consider a waiter cardinality as follows. So you wait a regular point four pi and you wait a singular point four pi one plus alpha. Okay. Depending on the weight at that point. And then you extend this cardinality to finite set just by additivity. Okay. And the set of say admissible configurations you would expect would be somehow the unit measures supported on finite sets for which the waiter cardinality is less than rho. Okay. So you allow concentrations which do not exhaust all the possible curvature. Otherwise it would be too much. Okay. So elements in this family can be written as formal sums of direct deltas with some weights and which are called the formal body centers of the surface. So the points Xi are in Sigma and the weights Ti are positive. And you can normalize so that it becomes a unit measure. Okay. So you obtain a finite dimensional object. Okay. So what's the structure of this object? Well, it's a family of measures. So somehow the natural topology to be used should be the weak topology of distributions. Okay. So which by the way induces a metric on this object. So what's the structure of this set? Well, it's a finite dimension but it's not smooth indeed. Okay. And it's a stratified set. So union of many faults with or without boundary of different dimensions. And the problem is, okay, you can write down elements of this set, Sigma Rho Alpha in this way. Okay. So they seem to depend smoothly on parameters. But there are first of all equivalence relations because somehow different combinations of this type can represent the same element. And also there are rules dictated by the weight of cardinality. Okay. So we want that the sum of the weight of cardinalities should be less than rho. All right. So take this example. Okay. When four pi is less than four pi times the sum of alpha one and alpha two. So on the torus with two singularities which is less than rho and less than four pi two plus alpha one. Okay. So what is happening here is that you can consider these combinations and you're allowed to count one regular point. You can count two singular points. So take a combination of the two but not a regular point and a singular point. Okay. So what you get in this case is the surface you started with with a one dimensional handle. Okay. And in the case of three points, for example, if the weights are negative you can combine the three of them to obtain say a torus with some sort of sale attached at three points. Okay. So this set sigma rho alpha looks like this. So this was the case for say negative weights. For positive weights, usually you are allowed to take less points because the weight would be greater than four pi. So they're more easily forbidden in a sense. And as say a row increases and as the number of singular point increases somehow the structure of this set becomes really very complicated. So it could be a complete mess. So that's our result in terms of this set sigma rho alpha. Okay. So let me recall the set, the values of the possible blowups for the curvature. Okay. So which can be written in this way. And so that's a theory which is actually work in progress but somehow close to completion. So if a row does not belong to this say critical set lambda alpha and if sigma rho alpha. So this family of measures is not a contractible set as for example in the pictures we saw before then you can solve for the problem. Okay. The proof as I said uses most theory. Okay. And there are two assumptions here. So row shouldn't belong to this set lambda alpha. So that's used to apply the formation lemma. Okay. And as I said, high energy sub levels are contractible while I claim that the structure of low sub levels has to do with this family of measures sigma rho alpha. Okay. So I would like to mostly explain that in the rest of the talk. And since we are assuming that this set is not contractible then okay we apply most theory as I said. All right. So some comparison to some more recent results. So using this approach, we were able to prove this result in some special cases. So still with Alessandro Carlotto in the case of negative alphas, in the case of positive weights and positive genus with Daniela Bartolucci and Francesca de Marquis and for say positive alphas, but low values of the total curvature in two other joint papers with the roots and Bartolucci. And none of these results actually dealt with the weights with different signs with the exception of the result by Aromenco who studied three singularities on the sphere. Okay. And there is indeed a program by two people in Taiwan, Chen and Lin, to compute the degree of the equation. So they have a paper appeared in 2010 for the case of integer weights and this was actually done for the regular case so with no singularities in 2003, still by time. And I would like to say that in general, you would expect to get stronger results using most theory because most theory can detect solutions even when the total ratio of the degree of the equation is zero. Okay. So you would expect to get more general results with this approach. So in the rest of the talk, I would like to persuade you that somehow considering this weird object, sigma or alpha, is actually maybe the right thing to do for when studying this problem. And I will give you a couple of reasons. So we showed that the non-contractability of this set of measures is sufficient to obtain solutions of the equation. And you may wonder whether it might be also possibly necessary. Okay. So which is not known. However, there is an interesting result by Alessandro Carlotto, which is the following. So if you fix any surface with a metric G and some points P1PM, then you can find weights for which the problem is not solvable. Okay. So previous non-contractability results, which are the ones I mentioned before, were only available on S2 with at most three singularities. So mostly using the Möbius action. But here there is no restriction either on the genus or on the conformal class. So these are the first non-existent results in for say positive genus. And so they're inspired, in fact by this characterization. Okay. So in these cases, in fact this set of measures would be contractible and you would sort of tend to believe that there might be non-existence. Okay. And so in this paper, he also gives necessary and sufficient conditions for the non-contractability of this set of measures when all the weights are negative. Okay. So in general, it's hard to find optimal conditions for the topological characterization of this set. And especially on skiers. So on Torai you can play somehow with the first fundamental group, say. And I mean, it would be really nice to get help from some topologists to be able to find clean conditions for the non-contractability of this set. So this is another persuasion attempt. So to try to convince you that this set is natural. So as I said, you can write elements of sigma, alpha in this form. So as weighted sums of direct deltas. Okay, which are normalized. And I justified this set looking basically at the quantization results for blowing up solutions. Okay. So you know that a blow up of solution at a regular or singular point carries a fixed amount of curvature. While here, we're putting some real weights in front of these direct deltas. So I'm taking the weights to be arbitrary. But in fact, I'm not claiming these holds for solutions, but only for functions with low energy in a sense. Okay. So there should be more flexibility. And in fact, it is possible to find some test functions which are parametrized on this element which somehow destroy well the low energy sub levels. Okay. So they're parametrized on this family of elements. And if you consider the formal, sorry, the conformal volume associated to such test functions, okay. So this converge weekly to the elements in the space of measures you started with. And also they have the property that the energy goes to minus infinity uniformly for the choice of sigma in this set, sigma or alpha. Okay, so what you can do is to embed copies of this topological space into arbitrarily low sub levels of the function. Okay. So I would like to see also somehow a sort of converse characterization. So I saw that you can pass from these family of measures into low sub levels. And I would like to show actually the opposite characterization in a sense. Okay. And so this is done. So I saw that I can embed copies of this object into low sub levels. Okay. I would like to somehow give an opposite characterization. And I would like to show the following that for energy sufficiently low, okay, then the conformal volume has to look like one of these objects. So it has to be close to an element of sigma or alpha in the distributional sense. Okay. So that would be a sort of optimal characterization of low energy levels. Okay. And to do this, the general philosophy is that say a spreading of conformal volume of the surface gives indeed a better constant in the weighted MOSF2-inger inequality. Okay. And then you proceed in this way. Okay. So you use a contradiction argument as follows. Suppose you have spreading of conformal volume. I claim that you get a better constant in the MOSF2-inger inequality. And if you get a better constant, it means that you can more easily control the logarithmic term in the functional. So the tricky term. Okay. So that will give you a lower bound on the energy functional. Okay. So you can somehow use this chain of implications in the opposite way. So suppose that the energy is low. So namely that the lower bound on the functional fails. That means that you don't get a better constant in the MOSF2-inger inequality. Which means that you cannot have spreading on volume. So this will favor concentration of conformal volume. Okay. So I would like to give an idea of how this works more precisely. And this is done actually using some improved MOSF2-inger inequalities. Okay. So this is the MOSF2-inger inequality in the classical case. So when there is no weight and the constant you get is one over four pi. Okay. So this holds for your arbitrary. But the idea is that if you satisfy some extra assumptions then you can maybe get a better value here. Okay. So here are a couple of very well known examples. So one by Moser back to 1973. So if you consider even functions on the sphere with the round metric, then in fact you can push the constant from one over four pi to one over eight pi. Okay. And another well known example is due to a band still on the standard sphere. So you can push the constant up to almost one over eight pi provided the function u is balanced. So what does this mean? That if you embed the sphere into a tree and if you look at the conformal volume given by u so which is e to the two u. Okay. Then the center of mass has to be at the origin of S3. Okay. So this is another case in which you can push down the constant here. All right. So actually this theorem of a band relies on say a more general principle which was exploited by Chen and Li basically by localizing the Moser to the Injuring Equality. So that's the statement. So if you are able to spread the conformal volume over a certain number of subsets of the surface. Okay. So if you find say L plus one sets each of them containing a finite portion of the total volume and assuming that they have positive mutual distance. Okay. So then you can push down the constant from one over four pi to one over four pi times the number of regions you're considering here. Okay. Up to an epsilon. All right. So what is the consequence? Is that somehow spreading of conformal volume gives you a better constant here. All right. Which means a lower bound on the functional. On the contrary. So if the energy is low then you should expect concentration of volume. So suppose that the constant rho so the total curvature is less than four k plus one pi. Okay. So then this lemma tells you the following. So give me an epsilon and an R. So I can find an energy sufficiently low so that almost all the volume up to an epsilon will be contained in k balls centered of radius R centered at some points of the surface. So this means that if rho is less than this number here so four k plus one pi and if the energy is sufficiently low then the conformal volume has to concentrate near at most k points of the surface. So it will look like a sum of direct deltas. Okay. From the point of view of measures say in the distributional sense. Right. So using a covering argument so this is kind of easy to explain heuristically. So if the volume cannot be spread into say k plus one regions then it should be concentrated near at most k regions. Okay. So this I hope this sounds kind of reasonable. All right. So what is the roles of the role of singularities which I haven't treated so far. Okay. So let me recall the chain trial of inequality in the presence of a weight. So the constant you get is the minimum between four pi and four pi one plus alpha j. So in a first paper with the callot we extended the channel ease inequality to the case of negative weights still using some kind of localization argument. Okay. But for positive weights so you might hope that still if the volume measure is concentrated near the singular points then you might have some might see some effects of the vanishing of the weights near these points. Okay. So I recall that h of x behaves like the distance from the singularity to the power to alpha. Okay. So you might expect that if the volume gets close to there so e to the two u then you might see some effects. Okay. But you should be careful in such a statement because you might still concentrate a bubble arbitrary close to a singular point and have the energy go to minus infinity. Okay. So you would need to pick up some sort of more refined or microscopic structure of the conformal volume near these singular points. Okay. So you want to be more careful. So you might expect that something like a center of mass condition could be the right one. So that's somehow a nice guess but it's actually more subtle than just that. And here is actually one version of the improved inequality using sometimes kind of angular moments on condition on the volume measure. So we proved this proposition. So to simplify things, consider just the unit disk over two. Okay. With one singularity at the origin. So just the simplest possible case. Okay. With positive weight. So the proposition is the following. So suppose the weight is positive and fix any positive epsilon. Okay. Suppose the conformal volume satisfies this vanishing moment condition. Okay. So you look at the conformal volume times x to the two alpha which plays the role of H of x. And you assume that if you multiply it by say some Fourier mode. Okay. This product is vanishing in C for J running from one to K. Okay. So the center of mass condition would be just J equal to one. Okay. So suppose you're given this extra conditions. Okay. So then you can actually improve the constant from Chen and Li's which is one over four pi to almost the American football one. Okay. So you can get very close actually to four pi one plus alpha. So precisely you get one over four pi times the minimum between one plus alpha and one plus K. So for K large enough, you get in fact the American football constant. So you can think of this as a sort of non-linear Fourier improvement. So if this were a linear problem, so if you had say the two normal view here and the Dirichlet norm here, then you would just use Fourier. Okay. But this is a sort of non-linear version. So I think I really no time to say anything about the proof of this result. So which takes mostly a lot of pain. And yeah, I like to continue. So some comments. So the main new feature of this inequality is that in fact these are scaling invariant assumptions on the conformal volume. Okay. So somehow if you take your conformal factor and you dilate it, say close to the origin. So it might look very much like a Dirac delta. So it's very much concentrated microscopically. But still you might get an improvement in the inequality here. Okay. And so the scaling invariance is the main new feature of the inequality because you might allow, say, any rate of concentration. And the result is kind of sharp. Okay. So you find counter examples if in fact one of these conditions fails. And this also recovers some improved versions of the inequality, which were given by McCowen back to the 80s and Du Bois de Vantagantello. So McCowen was creating the case of functions with zk symmetry. So some discrete symmetry group. And they were treating the case of radial functions. Okay. So adding symmetry somehow helps to get bounce from below on this kind of functionals, as usual. And so the main comment is that in fact those assumptions are just say two k constraints. Okay. I'm assuming the vanishing of say k numbers in C. Okay. So these are two real, two k real constraints. Okay. So, I mean the improved inequality would be satisfied in the function space in a set of finite co-dimension. And the sets of finite co-dimensions you can intersect using transversality arguments or say degree theory. Okay. So you can apply the min-max machinery once you have attained such a condition. Okay. So if you require symmetry, of course, everything is gone because it's just infinitely many constraints. But these are finally many constraints. Okay. And you can actually apply min-max theory to prove existence of critical points. And let me say that in general, so I just treated one example, but in general, so if you really want to see the structure of the full set of measures, sigma or alpha, you might need to improve the constant even more. So you have some how to interpolate between microscopic and macroscopic structures, in a sense, regimes. Okay. So I would like to conclude with some open problems. So as I said, it would be very nice to have a clean algebraic condition in terms of the divisor and of the genus of the surface to guarantee actually the non-triviality of this set of measures, okay, which bioactive relates to existence. And as well, it would be nice to find somehow general non-existent results. Okay, so ideally, I mean, it would be great to prove that maybe this might be a necessary and sufficient condition. But maybe the non-existence result would be even harder, I guess. And in fact, in many situations, the parameter rho, so which is the integral of the curvature of the smooth part, might actually belong to the critical set lambda alpha. So we were avoiding those values for compactness reasons. And you might see that this set belongs to lambda alpha just by the Gauss-Bohne formula, okay. And in this case, so if you still want to attack the problem variationally, you might need some more refined blow-up analysis with some sort of refined geometric information on the blown-up manifold. So like a sort of positive mass theorem, for example. And in a spectacular work by Lin and Wang, so the case of flat-tor-i with one singularity with weight equal to one is treated, okay. So what you're doing here, so you're considering the, say, a flat-torus, okay, not necessarily the square-torus, and you're concentrating somehow all the positive curvature on the smooth part while the negative curvature will be concentrated at the singular point, okay. So what they prove, and this is really spectacular, is that you have existence even only if the Green's function of the Laplacian of the torus, so depending on the geometry of the flat-torus, has five or three critical points. So if you have five critical points, you get existence, otherwise, nonexistence. And they use a really deep theory of elliptic functions, and I mean, it's a theorem which I find really spectacular and astonishing. So I mean, it would be also very nice to somehow interpret this theorem, so to find a good explanation. And well, as I said, these equations also rise from physics, and the scalar case would arise from abelian models, and when you have non-abelian models, then you're led to some coupled systems of the equation, so-called Todda systems, which also have a geometric interpretation because they describe holomorphic curves in CPN. And in general, this system is actually quite challenging, so it would be nice to find generally existence results, so in this case. All right, so I stop here, and thank you for the attention. And also one more, say, happy birthday to Eugenio. So what was the wiggly American football on page 14? Oh yeah, so you can get it, I cannot draw it, in fact, but I can write it as a conformal deformation of the Euclidean metric, okay? So the standard bubble would be something like this, so you have one positive parameter, lambda, and you consider the logarithm of lambda divided by, so in this notation, I guess it's, I might scrub the exponents, but it's something like this, mod x squared, and maybe everything is squared. Okay, so this would be you. The American football metric would have a one plus alpha here and a one plus alpha here, okay? So it's a nice radial function. In fact, when alpha is integer, so this depends basically on only one parameter, which is lambda. When alpha is integer, you somehow get a different polynomial of x here, I can draw a picture of the solution. So this would somehow be like solitons, okay? And when alpha is integer, then you would get sort of two peaks, okay? When alpha is equal to one, you get two peaks, and when alpha is integer, you get, say, K peaks, which are distributed along the vertices of a polygon. Okay, and this somehow matches very well with our condition, so on, on vanishing moment. So in a sense, if you have enough symmetry, then somehow this condition is automatically satisfied, in a sense, and somehow if you, if you fail to get zero kth momentum, then you start plugging one of those functions to show that you do not get improved inequality. So that's somehow why I believe this improvement is almost optimal. But you don't have a picture of the solution? No, no, unfortunately not, yeah. But I carry the same total curvature, so, yeah. Do you think there could be a rigid approach? Yeah, in fact, I discussed this with Peter Topping, and as I said, okay, there are cases in which you shouldn't expect existence of constant curvature metrics, but he would suspect that there should be possibly always a richly soliton, with some maybe nice, say, asymptotic conditions in their singularity. And if that is the case, maybe it might be a way to prove a non-existence of constant curvature, maybe. Yeah, but it's a very interesting question, yeah. By the way, I wanna talk back in, someone correctly, just say North America. North America. Yeah. North America. North America. North America.