 to talk about curves and surfaces with constant non-local mean curvature. OK. Yep. So thanks, Yannick, for the introduction. And let me start thanking Francesco mainly for the invitation. It's very nice to be here. So the title says curves and surfaces with constant non-local mean curvature. But the talk is a little updated. And the main part of it will be a survey on non-local minimal surfaces. So it's the same equation. Well, it's the same operator, non-local mean curvature. But I will start with how it started. Surfaces with zero non-local mean curvature that we call non-local minimal surfaces. And later, at the end, I will present you some results that exist for surfaces with constant non-zero positive non-local mean curvature. OK. So and by the end, you will know everything that is known about non-local minimal surface. Not everything, but most of it. There are works, but not so many. OK. And I think it's an interesting topic. So let's start with, I used to work before, but it doesn't work anymore. So let me start just recalling what is the perimeter of a bounded smooth set in a REN. Well, the perimeter, you know, is the house of measure, the n minus 1 dimensional house of measure of the boundary of E. But you can define it also in the following way through integration by parts or in a distributional way. By the way, I can send these slides to whomever, or maybe they will be in the web page, or I can send them. It is a simple equality where the vector field X is free with L infinity norm less or equal than 1, and nu is the normal to the set. And then you look at this quantity and you integrate it by parts with the divergent theorem. And instead of integrating with an E, let's say that you integrate in REN the divergence of X times the characteristic function of E. And now you integrate again by parts in a distributional or weak sense. And you see, you get the gradient of a characteristic function in the distributional sense times X. X has an infinity norm less than 1, so this supremum is going to be the L1 norm of the gradient of the characteristic. It's what we call the W11 semi norm of the characteristic, right? So now I'm going to introduce fractional perimeter starting from this, like saying, an academic, very academic way. But I think it's OK for our purposes. So now I'm going to do the same, but instead of taking one derivative, I'm going to take alpha derivatives, right? So I'm going to consider the fractional sub-left semi norm with a fraction alpha. Alpha is going to be always between 0 and 1. And I simply take this and, well, the sub-left semi norms, fractional sub-left semi norms, have a nice expression, which is this one, through of a function. And you would put here the function at X, the function at Y, subtract, and make the double integral, the X, D, Y. And when you do this for a characteristic, you get this, right? If X and Y are in E, you get 0. If X and Y are in the complement of E, you get 0. So X must be in E, Y must be in the complement, or Y sub-left semi norm, that's why there is a 2 here. So this is the expression. And this is the definition of fractional perimeter up to a multiplicative constant that you put here, positive constant. That depends only on the dimension n and the fraction alpha. And this constant is put, for instance, so that when alpha goes to 1, this constant behaves like 1 minus alpha. It goes to 0 and makes the whole perimeter, the whole expression, to go to classical perimeter as alpha goes to 1. So you have these fractional perimeters with order alpha. And as alpha goes to 1, you recover classical perimeter. And this is the expression. Remember, E, E complement. And the novelty is that here you have single integrals. Here we have double integrals. For those of you who know about the fractional Laplacian sometimes here, instead of alpha, one writes 2s. And I will always write it like this, m plus alpha, alpha between 0 and 1, which means that s is between 0 and 1 half. And in fact, if you have, let me say, this fast for those of you who know, you have the fractional Laplacian. And you consider, for instance, the Allen-Cann equation. And you do a blow down or you put an epsilon. Some of you know, perhaps, that when you have the Allen-Cann equation and you make the limit epsilon 10 to 0, you recover minimal surfaces. So let me just say, with the fractional Laplacian or the fractional Allen-Cann equation, when you do this game, if s is greater or equal than 1 half, the limiting configuration is a classical minimal surface. And it's only when s is less than 1 half that you recover a new thing, which is the fractional perimeter. It's exactly this perimeter. And non-local minimal surfaces are going to be surfaces which minimize this perimeter, say, roughly saying. But still, this only makes sense when e and e complement are bounded, sorry, when e is bounded. Otherwise, if e is not bounded, normally you would get infinity here. So we'll come back to this later on. Let me just say, whenever you have a geometric functional, like this, you wonder if there is an isoparimetric inequality. It would be nice to solve the isoparimetric problem. So this existed for a long time. There was an answer by Garcia Rodemick. So maybe these are the years in the 80s. Garcia Rodemick discovered that the classical Schwarz rearrangement, the classical one, also works with fractional perimeter. You give me a set, like the one before in the previous slide, I consider the ball with the same volume. It will have less fractional perimeter. This was known, Garcia Rodemick. But then Frank and Sirenger quite recently proved the uniqueness. I didn't write it here. I should write it. Balls are the unique. So I wrote this reference because it's also important to have uniqueness. Frank and Sirenger proved that balls are the unique minimizers of fractional perimeter for a given volume. And this called the isoparimetric inequality. And then there is even more recent works with quantitative versions, Fusco, Mio, Morini, Ficali, and Francesco Maggi. So let's go into matter and let's say, what is the first variation? When you have a function, you do the first variation and you should get a PD, right? In this case, it's going to be a fractional PD or a non-local PD. And that's very easy to do. And this is what you can do it as an exercise. What you'll get is the following expression. You have your set E. This will make sense even if E is unbounded. So now it could be even unbounded. You take a boundary point, X. X is on the boundary. The boundary is sufficiently smooth. But the important thing, X is on the boundary. And this object, it's called the non-local mean curvature of the surface boundary of E at X. And it's simply this integral in the principal value sense. You center yourself at X because there is the typical power kernel, singular power kernel, centered at X. You're integrating Y. And Y moves everywhere in Rn. When Y is in E, you put a minus 1. And when Y is in the complement of E, you put a 1. So you see what you are doing? You are centered at this point, X. Think of spherical coordinates if you want because this weight is radially symmetric around X. And you count the number of ones, which are the guys outside E, the proportion with the number of minus ones, which are the guys in E. And these you ponderate with the distance of the points to yourself, right? X minus Y to a power. The points closer count more. The points farther away count less. And for those who know the fractional aplation, this is simply the fractional aplation of the function, which is minus 1 in E, 1 outside E, and say 0 at X, 0 at the boundary. And if you compute the fractional aplation of that function at X, you get this expression. So the non-local mean curvature is simply the fractional aplation of this function, which is minus 1 here and 1 here at the boundary point. And you put the value 0 there. OK. You have to analyze no problem at infinity. There is a cancellation at X equals Y because the ones and the minus ones, when you are very close to X, and if the set is smooth enough, it looks like a hyperplane. So there are pretty much the same ones and minus ones when you are very close. When you are very close to X, the set is smooth. Locally, it looks like a hyperplane. So at the first order, there are the same number of ones and minus ones that will produce a cancellation, will add a 1X minus Y that will make this integrable at the origin at X. Very good, but still this makes sense, as I told you, for an unbounded set. Of course, the simplest unbounded set is a hyperplane. So E would be what is below a half space. This makes sense, and it gives you 0. It has 0 non-local mean curvature. But still you can wonder what is the variational problem behind? Because the fractional perimeter or the classical perimeter of a hyperplane is infinity. It's infinity. So for classical minimal surfaces, what you do is, for instance, in R3, you take a wire. This is called the plateau problem. You take a wire, and you wonder what is the surface that spans the wire and has minimal perimeter? Well, minimal area, right? Minimal area. Within a ball, for instance, the set below would have relative minimal perimeter. Okay, that's the recled problem, which is called plateau problem for minimal surfaces. How do you do this? How do you do this in the fractional setting? This is the following. This is done in the following way. I have this unbounded set E, okay? It has infinite fractional perimeter, so this is not good. What I'm going to do is I'm going to try to count its fractional perimeter inside a domain, omega. So fix any domain, for every domain. Let me say it another way. Let's think of the plateau problem or the recled problem. What I'm going to do is, I'm going to take a domain and I'm going to consider a set given set E, which is the blue guy, okay? And I'm going to see, for the fractional laplacian, I'm going to see this part of E, which is outside omega, as given. This is given. And what I'm going to try is to minimize a fractional perimeter inside omega. So the game is this. Here, outside omega, you cannot touch anything. The set must be exactly this. But inside, you could be the blue thing or you could be the dotted black thing. You have the freedom to change it inside. And as a result, you have a set. And now, you would write the fractional perimeter as before. It has the interactions, points in E, points in the complement of E. But now there is this omega, so I can, let me write all the interactions. There are four types of interactions. You see, points here, let me write it just, let me do it with the pointer instead with my voice because it's a little, points here, with points here. So points in E and omega, points outside E and in omega. This is the first interaction. Second interaction, points here and points there. So in E and in omega and outside E, outside omega. Then a third interaction, this and this, points outside E, points in E, sorry, outside omega and points outside E inside omega. And there is a fourth that I forgot, this with this. But this is, since this is the unbounded part, this would give you infinity. But at the same time, this is fixed, so every competitor, the blue and the black guy, would have the same interaction here. For instance, if you cut it with a bigger ball. So you just forget about that, right? Because everybody would have the same infinite energy, right? So you forget about that, it's crossed here. You see, you don't put it and in this way here, always one of the sets, at least one is bounded and this makes sense. So this is the variational problem or the fractional plateau problem, say, right? And this was introduced in what we call the seminal paper in non-local minimal surfaces. This is the first paper where this functional is written like that and it's a paper by Luis Caffarelli, Jean-Michel Romjofka and Obidiu Sabin, 2010. So they write this variational problem and also the non-local mean curvature, okay? So the guy that minimizes this problem that exists, I will tell you about this later, it's going to be a surface with zero non-local minimal, with zero non-local mean curvature. It's called a non-local minimal surface, okay? In fact, the non-local minimal surface, sorry, the non-local minimal, no, sorry, the non-local mean curvature, it appeared in a previous paper by Luis Caffarelli and Taki Suganidis, 2008, where they studied threshold dynamics to approach motion by a new mean curvature, which is not the classical, it's non-local. They discovered here, in fact, the non-local mean curvature. But let's keep this slide and let's go more into matter, all right? So this slide is about the results in the paper by Caffarelli, Rocgioffre Sabin. They consider that variational problem and always they consider surfaces which minimize. You see, when you are with minimal surfaces, you could, many people define minimal surface and I will use this terminology. A minimal surface is a surface that has zero mean curvature and on local minimal surface, we can say that it's a surface that has zero non-local mean curvature. But sometimes when you make the theory, you start with minimizers because they have more properties, right? Minimizers and then I will say that they are minimizing minimal surfaces or minimizing non-local minimal surfaces. So in this paper, they study minimizers and they prove many of the things that exist for minimal surfaces now in the non-local setting. And this is very nice because minimal surfaces, maybe it's a unique equation, right? That may, for which there may be, there exist so many things as the one I will list now, okay? It has many, many properties because it's a geometric equation. Well, a non-local mean curvature has most of the same properties and this is very nice. I think it says good things about this whole field of research. For instance, first thing, well, we already saw that. Definition of the variational problem, good. Existence of minimizer, very easy. Lower semi-continuity, some compactness. Density estimates also exist and they prove them here. The Euler Lagrange equation in viscosity sense, this is the Euler Lagrange equation, but for smooth sets, the minimizer a priori, you don't know if it's regular or not. And that's the goal at the end to know if it's regular or not. So you have to derive an equation in a variational sense or in a viscosity. In a variational sense, it's easy because the problem is variational in a viscosity sense. This is in part one of the, and the equation is going to be this with the fractional Laplacian of ones and minus ones, but in the viscosity sense. And that's one of the toughest parts in the paper. This section here, it's quite involved and the proof is quite involved. And I think there is no simpler proof yet. Then improvement of flatness as in the classical theory, then an extension problem and a monotonicity formula. Monotonicity formula is very important, it's crucial in the theory of minimal surfaces and also it will be here in some applications, I will tell you, some results that will come later. And they prove it here, it's great to have this. The funny thing is that the monotonicity formula is not known, we only know of a proof and it's in this paper. And the proof, the only known proof, uses the extension problem that I will describe for you in the next slide. There is no proof of the monotonicity formula working in Rn, you have to go to the extension problem that you will see in the next slide, okay? And a classical argument on dimension reduction using the monotonicity formula in classical minimal surfaces also works here and all this allows them to prove the theorem in the paper, the main theorem say, you need most of this or all of it. And it says, if you have a minimizing non-local minimal set in omega equals big one, so we are playing the plot, we are playing this game, the plateau problem in big one, in particular this set, okay? This set has zero non-local mean curvature but only on big one, not outside because outside it was given arbitrary, right? Then in the interior, the surface is c one alpha except for a closed set of household dimension hn minus two of household dimension n minus two which means that for instance in the plane, in the plane this would be a set of zero dimension would be for instance points, isolated points could be singularities, all right? By the way, when you play this game in the plane is very interesting already, not for minimal surfaces because when you are in the plane and if you have a minimal surface which means a minimal curve, it's curvature is zero so it has to be, it must be a straight line. So if you play the plateau problem in the plane, you get lines, right? Or a combination of lines, straight lines but when you play the plateau problem, the plateau game, say the plateau variational problem in the non-local setting and you give me this curve outside, unit ball, what you will get inside is not a straight line, it will bend because it has to compensate for the ones and minus ones that come from the guy outside, right? So it will bend and you wonder, is it infinity or not? At this point it was not still known, it could have singular points, right? What is the extension problem? Maybe most of you know it, let me remind it for some of you who don't know it for sure. We are in a Rn, X in a Rn and we are going to add one more variable, Y and we are going to have a half space, right? S is alpha over two, okay? Alpha is two S, like in the first transparency. But here I will take, here I should change a little this slide, sorry, I just made it today fast. It should say one half because of what I will tell you later, one half. What you do is you take your function, U, which is minus one on E and plus one, or the other way around, one in E and minus one on the complement of E. It's a singular function. I mean, it has a discontinuity in E. You take this function, constant one, constant minus one. And you are going to extend it with a function V that also depends on Y, V depends on X and Y and it's going to solve this elliptic equation in the half space. It's like the Laplace equation, but with a weight here, with a weight which is the distance to the half, to the distance to the boundary of the half space to this power. And to be able to extend minus one one characteristic function, and so that everything is well defined, you need S less than one half, okay? And then you can prove there is a unique extension. V is a function between minus one and one. You see? It's like you go up here, you are flying, you are in the sky, now this is the floor, what we call the bottom. Now you are here in the sky. And you simply, you are in this point, you look down to the earth and you see how many minus ones and ones you see. From up there, maybe the ones are all below you, there are a lot of ones and the minus ones are very far, so I will not be one, I will be 0.8, okay? And if I go farther closer to the ones, I will be 0.9, but this function V is going to be between minus one and one strictly, okay? And looks for the quantity of ones and minus ones on the bottom. And then the theorem of Luis Caffarelli and Luis Silvestre is that if you consider this, the flux of this function when y goes to, when y goes to zero, I should rewrite this better, adapted to non-local minimum surfaces. When you compute this flux on the surface, on boundary of V, where the minus ones and the ones change, the flux here, that's going to be the fractional application of this characteristic function and that's going to be the non-local mean curvature, particular for a non-local minimal surface, this is going to be zero on the boundary of V, right? So let me just say that once again, you here is the bottom, you have the surface boundary of V, you put minus ones, you put ones, now you make this harmonic extension and then you make the flux but on the surface and these flux must be zero. It's like on the surface, the ones and the minus ones compensate and the flux is zero. So this is the second way, now we have two formulas for a local mean curvature. The first on the first transparency, making an integral of ones and minus ones and this is a second way, second way to compute a non-local mean curvature, okay? And this is used to prove the monotonicity formula. So now let me go into what is known, the main results about non-local minimal surfaces. So this is more recent works, 2012, 2013, Begoña Barrios, Alessio Figali and Rico Valdinoci. If you have a minimizing non-local minimal set and if you know that it's lipchitz, it's going to be C infinity. So the difficult thing is going to know if it's lipchitz or not, okay? And there is only a complete answer, there is a complete answer only in dimension two, up to now. So we are still in the beginning and this is a beautiful paper by Savino Valdinoci. If you take a minimizing non-local minimal set in the plane, it's going to be a half plane necessarily. Now I'm assuming, I'm not playing the plateau, I'm assuming that they have a curve in the plane and that it's minimized, it has zero non-local minimal curvature, mean curvature, sorry, local mean curvature. I'm assuming that it's minimizing in every ball. When I play this game of the plateau problem in the ball of radius one, but also in the ball of radius two, three, any ball of any radius, you're assuming that your surface, your curve is a minimizer, what we call a global minimizer, is a minimizer on every ball, okay? And then the theorem says that necessarily it must be aligned, right? This has a corollary that says if now we play the problem in omega in a bounded set, and you give me an any curve outside, any curve, and I get the winner, the set or a set that minimizes inside omega, a consequence of the theorem and blow up is that this surface is going to be seen infinity. It's not going to be aligned, I told you before, it's going to bend, but it's going to be seen infinity, okay? As a consequence of this global theorem, you get the local theorem also. Let me explain more why this theorem is interesting, because here you have, you see some enemies here. The simplest enemy is the cross, what we call the cross, is two perpendicular lines, straight lines, and the set E is the blue guy here and the blue guy here, you see? So I claim that this set at any boundary point, any point in the red part, it has zero non-local minimal curvature. Oh my God, today, non-local mean curvature. Sorry. Why? Take this point, for instance, compute the non-local mean curvature here, and do, for instance, instead of doing a spherical polar coordinates, do fubini, vertical fubini. When you integrate on this line, the minus ones and the ones cancel, because the kernel, that is that power kernel, is symmetric with respect to this point. Now take another line here, the same thing, the two distances cancel, a minus one and a one. And the same for a line here, now the minus ones become ones and the ones minus ones. So because you see this point, the distance to x is the same as the distance of the reflected point to x. And therefore the kernel is the same at this point and at this point and then everything is canceled. The same for any regular polygon. If you take a boundary point and now you do fubini perpendicular to this line, you will see the cancellation. So all these guides have zero non-local mean curvature. They were being candidates for being minimizers. They are not infinity because they have a singularity here. They are not leopards graphs at the vertex. But the theorem of here says that if they are minimizing, these guys are not minimizers. The only minimizer is the half line globally and locally are infinity. Okay, so R2, everything is understood. Minimizers are infinity, right? What is known in R3? This is the main theorem that exists up to now. It concerns graphs. So now I'm going to take a set in R3, which is the graph, the boundary of E is going to be the graph of a function phi of two variables. So sets in R3 could bend like this, right? Could bend and this is more involved geometry. But I'm considering the simplest ones first are sets which are below a graph. E is what there is below a graph E, okay? And then Alessio and Enrico proved, well, graphs are only always minimizers. That's very easy to prove. If you have a graph with zero non-local mean curvature, it's going to be a minimizer because you slide it and it produces a polyation, okay? Like in the classical theory. So any graph with zero non-local mean curvature in the whole space R2, the whole plane, is a plane in R3. So it's straight. It's equivalent of the thing. So this is the Liouville and the rigidity theorem. In R3. So let me show you a third expression for the non-local mean curvature. We saw the integral with ones and minus ones, the extension problem. And now I will give you the expression for graphs. And this is the expression. Suppose that you are in Rn plus one. Now I changed from Rn to Rn plus one because I prefer my function phi, which is going to give a graph to be a function of n variables, okay? Then it's very easy to compute the non-local mean curvature doing Fubini, vertical Fubini, as I did before. You have this graph, take a boundary point on the graph, don't use the spherical coordinates. Instead, use vertical Fubini to reduce everything to an integral in Rn. And, well, there will be minus ones, there will be ones, and everything will depend on the height of this function, which is u of, no, phi of x, phi of x. And a simple computation, Fubini, simply, gives you this expression, look. A non-linear function that looks like this, it's a bounded function, which is concave here, odd, concave here, and tends to a constant at infinity. And it's the identity near the origin. A non-linear function of the incremental function, oh, this u should be a phi, u equals phi, okay, sorry. U equals phi, phi of x minus phi of y over x minus y. So, the incremental quotients, but then you make this non-linear truncation, say. And then you have the typical power kernel, dy. And F is explicit, pretty much, almost explicit. You see, if F was, capital F was the identity, then this would become the fractional appellation of u, or of phi, u equals phi. And F is the identity pretty much when the incremental quotients are bounded around here, which is the same as saying that phi is Lipschitz, okay? But you don't know if phi is Lipschitz or not, okay? The graph could be anything. This expression is very nice because it's a very simple way to show some properties like the following. If you give me an, if this is zero, because you have a non-local minimal graph in v1, in the unit ball, I can prove you, for instance, an L infinity bound for the oscillation of phi. Independently of what the tails are at infinity. Independently of what phi grows at infinity, right? And this, not even the Laplace equation has this property, the Liouville theorem for the Laplace equation says bounded harmonic functions are constant, but if they are not bounded, this is not true. So this is a property that minimal surfaces have, but it's quite mysterious for minimal surfaces. For instance, for those of you, now let me go a little fast, for those of you who know what is the Simon theorem says, if you have a minimal graph in Rn, classical mean curvature, and it has, you have a minimal graph in Rn, and n is less or equal than seven, then it has to be on a fine function. And that's quite amazing, right, the theorem, because there is a priority, there is no control on the behavior of the graph at infinity. And instead, in the non-local setting, it's quite easy to see that the fact that capital F is bounded at infinity, and this is zero. Independently of how is the graph at infinity, in the unit ball, I can prove you, and very easily, in fact, an L infinity bound for the graph. Right, independently of the tails, right? And this you will find in a paper by Enrico, and Ovidio, and Serena, probably, the L infinity bound, independent of the tails. Okay, so it becomes less mysterious, in fact, than classical minimal surfaces. Let's go ahead. Then there is another paper by Caffarelli and Valdinoci saying that if the dimension is less or equal than seven, like in the classical theory, but alpha is sufficiently close to one, remember that when alpha goes to one, fractional perimeter goes to classical perimeter. So non-local minimal surfaces go to minimal surfaces, roughly saying, well, here they do this rigorously, and they prove that in Rn, if n is less or equal than seven, and if alpha is very close to one, minimizing non-local minimal surfaces are smooth, and minimizing non-local minimal cones are flat, right? And this is proved through a compactness argument, because minimizers, you can prove that they are compact in some norm, in the L1 norm, for instance, as alpha goes to one, and this theorem is proved through a compactness argument, it doesn't tell you how close alpha must be to one. And after that, there is one more important paper by Juan Davila, Manuel del Pino, Junction Way, where they start, so you see there are answers in dimension two, full answer, some answers in dimension three, and what we expect is that the same type of smoothness should hold up to dimension six or seven, but everything is pretty much open in dimension three, four, five, six, seven, except for this case, when alpha is sufficiently close to one, and what remains to be done is what I'll explain you next. And it started, for instance, this is a paper where they start to study cones in higher dimensions and trying to see if they are minimizers or not. It's very difficult to see if something is a minimizer or not in general. But it's simpler, first you test the stability, because we will talk about stability later and you will see you have a formula, say, an inequality to check if something is stable or not. So that's what they do there. And these start, for instance, with the simplest cones, circular cones, sorry, the simplest cones, for instance in R3, imagine a circular cone, cones can be much more general than these, but start with the simplest cones, circular cones, and then it's easy to see that for every n and alpha, there is an opening of the cone that makes, that the ones and minus ones cancel. So given n and alpha, the opening of the cone is chosen so that the ones and minus ones cancel and the cone has zero non-local mean curvature. That's easy to prove, yet now you want to know, can this guy be a minimizer or not? That's difficult. So let's look first if it can be stable or not. And this is what they do there. I'll show you later how they do it. Well, they do it numerically because there are many hypergeometric functions coming in, but I'll show you the formula that they use. And the answer seems to be for circular cones, seems to be that six or seven will be the right dimension. Six, one less than here when alpha is very small. So I think there is one of these guys which is stable in dimension six when alpha is very small, if I remember well. They find the analog of the non-local catenoid, the helicoid also has zero non-local mean curvature, but I don't have time to go into this. And let me describe, yes. This is a result that extends the guan of Savin and Valdinoci here that used the minimizing at some point. So the proof worked with minimizing and then Eleonora Chimdi, Joachim Serra and Enrico Valdinoci improved it very recently, you find all these papers in archive. And they prove that in the plane also a stable, not minimizing, but a stable, which is less. Stable means second variation, not negative second variation of fractional perimeter. Instead of minimizer, you only assume that that you are a minimizer, but for small perturbations. You have a stable non-local minimal cone in R2, it has to be a straight line, okay? And for these they improved, they improved the argument of Ovidio and Enrico. And their argument will have another important consequence later. But first let me tell you what a paper of me with the Eleonora Chimdi and Joachim Serra, it will be in archiving within one or two weeks, okay? We are just reading it, it's almost ready. And what it says is we classify stable non-local minimal cones in R3 for alpha sufficiently close to one, like in the paper of Caffarelli and Valdinoci, but now where they had compagnet through, because they studied minimizers, now we study stable configurations, stable cones, and we prove that they must be flat if alpha is sufficiently close to one and you are in three dimensions, okay? And here, and if it's not a cone and it's a global surface, globally stable, say, stable in the whole space, it must be a plate. Here, with stability, you don't have compagnet, because a family of parallel planes, for instance, infinite parallel planes, these are not minimizers. If you play the plateau problem with this boundary data, you will not preserve all the planes. They have too much area if the ball is big enough. You will join some of them or do something or bend them. All right, and that's not a minimizing configuration, but instead, parallel hyper planes is always a stable configuration because you are only allowed small deformations and the fractional, if you perturb one of them or any of them, the fractional perimeter, small, with a small perturbation, is going to increase. So here, we don't have compagnet. In fact, proving compagnet is as difficult as proving this theorem. And we have to do a proof where, in fact, we can tell you how close alpha is close to, should be close to one, okay? We could get a value, but we don't compute it because it depends on too many constants, okay? But it gives you a quantitative value. It's not through compagnet. So I want to tell you what do we use to prove this theorem, okay? Here, and that's also in the original proof of the paper about minimizing lines by Savin and Balinocchi. The trick is, you know what you do. You take your set, the plane, and you translate it in a direction. You fix a direction, you translate it in this direction, in this direction, and also backwards. You make a cutoff function not to touch the tails and you compute the fractional perimeter of the translated, forward, backward, and you compare it with two times the fractional perimeter of yourself. And when you do this, you are comparing the fractional perimeter of yourself with these two competitors. Here, the stability, you will be able to use the stability to get some information. So it's through the morphisms. Here, we are going to use, for first time, we are going to use, well, it's also used here numerically to check the stability. We are going to use the formula for the second variation of fractional perimeter. So I still didn't show you what is a formula for the second variation of fractional perimeter. And this is here. This is our first ingredient. And this was found independently by, it's not a simple formula to find. This was found by Figali, Fusco, Maggi, Migio, Morini, and independently, I think, by Davila Del Pino way. More or less different ways, but independent ways. Okay, this is the formula. If you have a non-local minimal surface, which is stable, that means the second variation is non-negative, you will have this inequality. And this is, okay, this is like a fractional aplation but computed on the surface with the ambient distance in Rn. So you have this surface boundary of E of n minus one dimension in Rn. You are going to compute distances, but in Rn, not on the surface. And this will call the fractional Laplace Beltrami operator in boundary of E. We can call it like that, all right? Even that it's not exactly a fraction of any Laplace Beltrami, but okay. Or say a non-local Laplace Beltrami, maybe better. Non-local Laplace Beltrami operator on the surface E. Always using the ambient distance. And so this is the Laplace Beltrami. And then if you bring this to the other side, you will have here. Well, this is the quadratic form, of course, you see. It's a quadratic form associated to an operator that it's going to be a single integral psi of x minus psi of y, okay? And that's the non-local Laplace Beltrami. And then what you bring to the other term is this c square. So in fact, the second variation of energy is what we call the, in the classical theory, it's called the Chakobi operator, okay? And here I'm going to call it if you want the non-local Chakobi operator. And it's going to be this fractional Laplace Beltrami of order alpha, let me write it like this. This is what there is on the right-hand side, the quadratic form. And then simply here, a zero-order term. X is on the boundary. Here, c square E. And this is a zero-order term. And this is some fractional derivatives there. And what is c square? C square is defined at the point x on the boundary. It's defined in this way. Normal to the set at x minus normal at y squared. And then the kernel and you integrate in y, okay? So this is what Davila and collaborators use to check if circular cones are stable or not, all right? And there are a number of integrals to be done numerically, in fact. And this is what we used in the paper with Chinti and Serra. We also use something, a more classical ingredient. The behavior as alpha goes to one of the optimal constant in the fractional hard inequality. And the third ingredient is very recent and it's very surprising and it's a very powerful tool that surprises when it was found. It's in this paper of Chinti, Serra, Valdinochi that says if you have a stable, same situation, a stable, non-local, minimal surface in a ball of Rn, sufficiently smooth. Then in the interior of the ball, half the ball, you can bound the classical perimeter. This is not the fractional perimeter. This is the classical perimeter. Using that is stationary and it's stable. You can bound the classical perimeter by the right thing, R to the power n minus one. A plane would have Rn minus one. And this constant that depends like that as alpha goes to one. And here the non-locality, the fractional perimeter is crucial. So we are realizing lately that stability in the fractional setting is a stronger thing in a way that in the classical setting. And you will see a theorem presented by Alessio Figali on Friday, or Thursday on Friday, or Friday on Friday, about the fractional Allen-Cann equation, which is very beautiful result with Joaquim Serra, that for the classical Allen-Cann equation still is unknown, as he will describe, and about stable solutions of Allen-Cann in R3. And related to the conjecture of the Georgie. So, stability is stronger in the fractional setting. It gives you more things, okay? And now how we use, let me tell you in one minute how we use these ingredients. We have this cone in R3. We take alpha very close to one. You have this cone in R3. And we know that it satisfies this inequality. Well, a cone in R3 is just a curve in S2 and then you project, no, not circular, but you project it through the, from the origin, right? It's like an ice cream. You have this and then you do like that. But maybe it's not circular. Then, well, we use this because you can reduce your prime always to S2, but listen. What I'm going to do is use this, I'm going to flatten the cone. I'm going to look locally at the cone and I'm going to pretend that it's flat. Well, I'm going to flatten it. This could be very dangerous if the cone had a lot of perimeter, because then I would not control this change of variables. But the cone has controlled perimeter. So when I flatten it, say locally, I flatten it and I write things, because I want to write things in R2, that cone is two-dimensional. So I want to write things in R2. I flatten it. I will control the Jacobian of this transformation thanks to this estimate. And then I'm in R2. And you know, when you write a geometric operator like this, like the Jacobian operator for a cone, it becomes, it's something homogeneous. The cone is something homogeneous. When you write it in flat space, it becomes a hard inequality, because this is going to rescale the same as that, okay? You don't have an upper bound on the area for stable sets in general. Well, in R2, yes, but in R3, but in R4, it's still a non-local in a way, yes. Yes, exactly. When you write this and your guy is a cone and you write it in coordinates, for instance, in R2, you are going to get another quadratic form whose operator is the fractional Laplace. It's not going to be alpha. If I remember well, it's going to be this other power, but don't forget, let me just write beta. And then here you are going to get x to the power two beta and x is going to be in R2. And this is the operator and a constant, sorry, and a constant, very important constant. This is going to be the operator that whose quadratic form gives you the hard inequality. So now what are we going to do? We are going to take a radial function in R2 that saturates the fractional hard inequality, so that it almost achieves the fractional hard inequality. And knowing how constants behave with respect to, as alpha goes to one in the hardy here and there, we are able to get a contradiction if alpha is close to one, unless c squared is zero. But if c squared is identically zero, you are a plane, and that's what you wanted to prove. So this will be in archive within two weeks, something like that. And what I was going to tell you, let me just give you just a couple of minutes. The second part of the talk was going to be about surfaces with non-constant, constant, but non-zero, non-local, minimal, non-local mean curvature. And let me just tell you the results which are the equivalent of these two. The four sets, this is a classical mean curvature. The four sets with surfaces with constant mean curvature are called CMC surfaces. There are two, there are a huge amount of theorems, but the first two theorems, and you see they are quite old, are very important. The first one by, well, the second one by Alexandrov, saying if you have a smooth enough bounded connected set with constant mean curvature, it must be a ball. And then the one of Deloné years before, more than a century before, that says in our three, he proved in our three, but also in our N, there are unbounded sets which are periodic cylinders with constant mean curvature. And in fact, you can compute them, they are called andoloids, and they look like this in our three. And this guy has constant mean curvature, classical mean curvature. So the same things are true in the non-local setting. And this is the Alexandrov theorem, it was proved by myself and Mustafa Foll, Senegal, Joan Solamorales in Barcelona, Tobias Webb in Frankfurt, and independently by Chiraolo Figali Maggi Novaga. It says that if you have a smooth enough bounded set with constant and local mean curvature, you must be a ball. And even they prove here in this paper, they also prove a very strong quantitative version of the Alexandrov theorem. That by the way, you can prove with the Alexandrov reflection principle, and it's even simpler in the fractional setting. It's much simpler than in the local setting because strong maximum principles are cheaper. And then in my paper with Mustafa, Joan, and Tobias, we also prove the equivalent of the De Launay cylinders. We prove, it's a bifurcation result, so we only prove that for a certain small parameter from a straight cylinder or even in the plane from a straight band that obviously has constant non-local mean curvature, there are also, from the straight band, a family of periodic bands bifurcate. And they all have the same non-local mean curvature. Later, we prove that the same holds in R3 and in R4. We even compute them numerically, the bands. They look like this. And here we follow them a little more. At some point, this cylinder and this should touch the picture is not very good, sorry, should touch and produce another type of configuration and I finish here, which is periodic configurations. That is my last paper, you find it in archive with Mustafa and Tobias that says that in the plane or in Rn, you can find periodic configurations of bounded sets, repeat it, but that you have a bounded set which is close to a ball, but not exactly a ball. It's like a lentil, and when you repeat it in a periodic setting that you give me a priori, that you give me a periodic lattice, if I repeat it like that, the set inside the balls has constant non-local mean curvature, okay? And also, we also find them at least when the distance is very large. We find them when these balls are very, very far, near balls are very far from each other because we use the implicit function theorem. And to use the implicit function theorem, you have to analyze the linearized operator to non-local mean curvature, which is, the Jacobio is the second variation of perimeter. So we have to analyze this operator, but on a ball, in a single ball, or in a straight cylinder in the previous paper, and compute its spectrum and see that you can bifurcate, you can apply the implicit function theorem. So sorry for the extra eight minutes, and thanks for your attention.