 Okay, so if everybody's ready, then I'm very happy to introduce the speaker of today and the ICTP Math Associates Seminar, who is Sivaldo Núñez from the Federal University of Malangelo, who will speak about characterization of free boundary constant mean curvature surfaces in the ball. So, the floor is yours. Okay. Thank you very much. First of all, I'd like to thank the invitation, especially Zacharias and Claudio Rezo, to give me this opportunity to start in this new world of these talks. Okay, let me start with the plane of my talk. This talk will be divided in two parts. In the first part, I will explain some results about bounds on the topology of stable free boundary CMC surface in compact complex domain. And I will present a characterization of stable CMC surface in the unity ball. And in the part two, I will present a gap result for free boundary minimal surface in the unity ball, which was brought recently by me and by Lukas Zonbrosio. Okay, let me start with the basic notions on free boundary CMC surface. Consider compact remanement with omega with boundary, non-empty boundary. And inside of this ambient space, consider a compact hyper surface with boundary so that it has its boundary is inside the boundary of the ambient space. So we define a free boundary variation as a variation given by a variation of sigma given by the flow of a vector field that is tangent to the boundary of the ambient space. So let me draw your picture. If you have here, for example, the ambient space and here, your surface sigma, this variation different from the plateau problem, this variation permits, in this variation, the boundary of the surface can move, but it has to move all the time inside the boundary of the ambient space. We have a surface of the variation. The boundary of the variation can move, but only in the boundary of the ambient space. And we say that such a variation is volume present if all the surface of the variation enclose the same volume in s sigma in the ambient space. All the surface will enclose a volume in the ambient space. So in this type of variation, we assume that all this volume are constant along the variation. Finally, we can define a free boundary surface. The free boundary surface, in no matter, is just a critical point of the volume functional for all volume preserving free boundary variation of sigma. And there is a nice formula for the first variation of the volume, which is given by this identity. So if you suppose that the variation of sigma is given by a vector field x, so the first variation of the area is given by this expression. It's not difficult to see from this expression that sigma will be a critical point if and only if it has a constant curvature, of course, and the free boundary condition will be given by this property that the surface meets the boundary of the ambient space orthogonally along its boundary. Let me draw here a picture. We have the following situation. The surface has to be has constant mean curvature and this angle of contact between the surface sigma and the boundary of the ambient space is equal to pi bar 2. In the case that the mean curvature is 0, the surface is a critical point of the area. We have a minimal surface and the surface is a critical point of the area with respect to all rebounded variation without any volume constraint. Now let me explain a little bit the notion of stability. Since the free boundary CMC hypersets are critical points of the area function, it's natural to look at the second variation of the area. From this, it's natural to introduce the stability condition. We have the same context. We have a permanent compact manifold with boundary and now we suppose that sigma is a compact free boundary CMC hypersurface. That is a critical point of the area functional among volume preserved variation. So we say that sigma is stable if the second variation of the area is non-negative for all volume preserved free boundary variations of sigma. There is also a nice formula for the second variation. If you suppose that the variation is given by a normal vector field given by a smooth function, then the second variation of the area is given by this expression. Here we have the classical term which had already appeared in the closed case. But now you have this term which has to do with the geometry of the boundary of the ambient space. Moreover, it's known that such a variation given by a function is volume preserved if and only if this function has zero mean. So from this, we say that sigma is a stable free boundary CMC surface if and only if we have the following inequality, but this inequality holds only for function with zero mean. So this type of function functions having these properties are the test function for this problem. And this part is in this imply some difficult for treating this type of problem because this inequality holds only for functions having zero mean. So let me give now two basic examples of such surface. The first one is the total is your desk and the balls. In this case we consider the omega as the closed unity ball. And for any hyperplane press through the passing through the origin, we consider the, the intersection between the plane in the unity ball and this gives an example of a free boundary minimal surface we have, for example, like this picture. This is the intersection of the ball with a plane. The second examples are the circle caps. In this case, the omega is the closed unity ball. And you consider any sphere intersect the boundary of the main space of totally and consider the intersection of this field with the ball. So, we have now a surface called a spherical cap. In this case, let me give some remark about these examples. The first one is that the, the, these two samples are stable. In fact, due to a cold skin sperm, they solve the isoparametric problem. We, we are a stronger condition than stability. When, when in the two dimensional case for surface, which proved that the surface are the only free boundary CMC disks in the closed unity ball. And recently, Fraser Chan proved that they, they are the only free boundary CMC disks in the closed unity ball for all N. Okay, as a, as a motivation for the, the, our first results, let me recall now a classical classification of the sphere in the Euclidean space given by Barbosa in the car. They proved the following results. They proved that the round spheres are the only match the stable CMC closed the hyper surface in the clean space. And let me remark that Barbosa, the car and Ashford also prove the analog result in the sphere in the hyperbolic space. So motivated by, by the result, it's a natural question to ask about the classification of stable free boundary CMC surface in the unity ball. So we'll have this question, are totally geodesic and balls and spherical caps the only message table CMC hypersurface with free boundary in the closed unity ball. The first try to answer this question was done by him. I was in Pergasta in 1995, where they prove the following result. If you consider, if you consider a free boundary CMC surface in the closing to ball, then if the signal is stable, then they prove it that the only possibility are the following. Either the first case Sigma is the a totally geodesic disk or a spherical cap, but there is a third possibility that Sigma has genus one with at most two boundary components, but in that paper, they don't know they don't present any example or an example of the existence of a surface having genus one and at most two boundary components. One of the main steps they they use to prove this result was a technique was to to give to give a logical bounds for the boundary stable CMC surface in the unity ball. Following this the following flows of the following philosophy. In some case, in some case, we have that is stable, stable free boundary CMC surface don't have a so complicated topology. In general, they prove the following topological bound for surface in compact in compact convex domains. They prove that if you have a immersed stable free boundary CMC surface that the only possible values for the genus of Sigma and the number of boundary components are the following. Or the genus is equal to zero or one and Sigma has at most three boundary components or the genus of Sigma is equal to two or three with connect boundary. And the main tool used by husband as the previous team was the so called her streak, which is a trick, which is a technique that give you a way to to found nice test functions for the problem. Let me just remark that so on. Prove similar results for surface in the sphere in the hyperbolic space and also so on, also stood the capillary case, the capillary case is the is the general situation where the the Sigma, the surface Sigma may make a constant angle with the boundary. Our case is this the special case whether the angle is five over two. Let me also remark that another technique for find good test function is the is to use. Harmon Q one form, and this was done by Byron 2008. And using these words was able to improve that topological bound, they prove that the only they, they, they using harmonica function he exclude this, this, this possibility. They prove that the genus is zero and with with at most three boundary components or genus is one with at most two boundary components. To state our first result. Our first result is an improvement of these topological bounds in the case that the geometry of the, of the boundary of the ambience page is sufficiently close to the geometry of the two sphere. Let me, let me state the theorem. If you have let, let omega be a smooth compact convex domain and suppose suppose that the principle curvature of the boundary of omega satisfies the following pinching condition. They cannot be bigger than three over two. In this case, if you have a CMC, I stable free boundary CMC surface, then there are only two possibilities for the topology of the surface or the surface is a disk or is a is an analyst. If you, if you look for this, this bounds using this pinching condition, you cannot have this possibility and this one. Okay, so as a consequence of this topological bounds. We can conclude that, that in the, in this theorem, due to house, this possibility, this, this possibility does not occur. Okay. So in this case, we can classify all the stable CMC free boundary surface in the unity ball. The, the, there are only the topology of the disk in the circle caps. Let me remark that using the same techniques. There was a proof that in higher dimension. Every stable CMC hypersurface in the unity ball is a star shaped with respect to the origin. And finally, very, very recently, one is she a totally solved the problem. I mean, even in the case of capillary hypersurface the prospect, the proof that the only stable capillary hypersurface in the closed unity ball are the the spherical caps in the totally stable. And in this proof, they, they, and therefore, follow follows the same spirit as the, the same ideas as the, the proof give the proof given by Barbosa into Carlos in the classical, in the classical result. For close the hypersurface in the clean space. They used a nice minkowski identity. Okay, let me, let me start explaining how Rosenthal Gasta uses a hash trick to, to, to give topological bounds for stable CMC hypersurface. Let me focus only on the case where the ambient space is the unity ball for simplicity. It started with a stable CMC surface in the unity ball. And the note by the genius in the number of components of its boundary. And the first step, they do is to, to, to glue some disk in the boundary components of the original surface to construct a closed surface. They start with a surface with boundary with some boundary components and some genus. And they glue some disk to form a closed surface. So, next they, they, they use the, they use that there is a conformal map from a closed surface to as to with the Dirichlet energy bounded from this quantity which depends on the genus of the surface in particular the energy Dirichlet energy of the restriction is less than the same quantity. And now they, they use the hash trick, the hash trick. The, the, the main, the importance of the hash trick is that the, the guarantees that we can assume that the, the three coordinates of the dysfunction has zero means. And this fact, the, the main gradients to, to, to, to apply the, the hash trick, the hash trick use the conforming strict of the as to in the fact that the Dirichlet energy is a conforming variant. So you can, without loss of generality, suppose that the, this holds. So, now we can use this tree's function. I want C, C1, C, Psi 2, Psi 3 as test functions for the problem. I can prove, prove this function in the stability and equality and sum over i. And now you have this estimate. And so we can use the, the, the, this fact in the Gaussian theorem to get the following inequality. And finally, we can also estimate this left side by six, six pi. And from this, we can conclude that the genus of the, the, the surface is zero one with at most two boundary components. And finally, in the case that the, the genus is zero, they prove that in this case, the sigma is totally your desk or a spherical cap. But they can, they were not able to, to deal with this case. Okay. Now let me explain our, our approach to, to improve this topological bounds. We start with a closed, a stable free boundary since the surface. And our main, main result is this, this theorem, which says that if sigma is stable, then sigma has zero zero, the genus equal to zero. And the idea is to use the modified harsh balance argument. And the difference instead of taking a conformal map from that closed surface to as to we take directly, we use this fact that that there is a conformal map from the original surface with boundary in the unity disk. Okay. Since the, the disk, the unity disk is conformal to the high M sphere, we can suppose that this, this conformal map is from sigma to the M sphere. Now, one problem which has here, we have here is that now, since the, the, the conformal group of the conformal group of the unity disk is smaller than the conformal group of the, the, the, the, the twist here. We can only balance it at this time to function to coordinate function. So you have, you will have yet to deal with this third function. But so you have this natural question, is it possible to use Psi three as a test function. And the answer is yes. And this is done by this lemma, this is a fundamental lemma in that, in that paper. Which says the following. If you have a compact, a table compact free boundary. In a convex compact domain, then by definition, the stability means that we have this inequality for all functions with zero mean. Then our lemma implies that if sigma is stable, then free boundary stability implies plateau, strong stability, which means that these inequality holds for all function with vanishing at the boundary of the surface. Note that we are only assume that the function P is vanishing on the boundary of the, the surface. It's not necessary that this function have. We are not assume that this function have a zero mean. Okay, does sense our sense this third coordinate. Vantage on the boundary of this, the surface sigma, we can use Psi three as a test function. So you can now apply this three function in the stability inequality and some over I. And after this, you can use the same ideas as before to get this inequality. And from these, we get that the surface, in fact, has genus equal to zero. So is that the way that we can improve that bond at a logical balance. Okay, now let me start. Let me start to the second part of the, the talk. And this, as I said, is a jointly work with Lucas and Brazil. And let me start here with the closed unity ball center at the origin. And let me, let me. Right here, some basic examples. You have, of course, the equatorial disk. Another basic example is the critical cat anode. The critical cat anode is the following. If you consider the one parameter family of cat anode in at three. And there is only one parameter such that this cat anode meets the boundary of the sphere, the boundary of the unity ball or talking about me. Let me draw here picture. You have here, your, your, your one parameter family of cat anode. And there is only one such cat anode meeting the, the, the boundary. The three sphere. So, when doing the intersection with the ball, you'll get the critical cat anode in the unit ball. This is an example of a free boundary minimal analyst in the unit ball, something like this. Okay. Let me give some remarks about this to the two examples. These remarks are classification characterization of the equatorial disk and the critical cat anode. Some characterizations. The first one. And this is the following. The equatorial disk is the only free boundary minimal disk in the unity ball. And recently, Frazier generalize this result to higher dimension. The equatorial disk is also the only free bounded minimal search in the unity ball with more index one. The most index is a number associated to the variation of areas with counts, the, the, with counts the independent direction, you can deform the, the, the surface, decreasing the, decreasing each area. Also, the cat critical cat anode, we have this result, which says that the critical cat anode is the, is the only free boundary minimal analyst in the unity ball immersed by the first stack loss again function. The example of a game function is then if I give functions for the following problem we have an harmonic function with the normal derivative on the boundary is a is a constant times the function. It's a, it's a, it's a, it's a fact. It's a well known fact that that when you consider a surface in the unity ball. The three coordinate function functions has tricked it to the, the surface are a stack of a game function with a, a, I gave, I gave, I gave value equal to one. So, Frasian Shane proved that if this functions are the first one are the first stack of a game function then in the surface is an analyst then the surface has to be the critical cat. There is also a result characterizing the critical cat anode in terms of its Morse index and the critical cat has Morse index equal to four. The only free boundary minimal analyst with Morse index for this was done by Tran and the viewer independently, and there is a conjecture that this in fact this condition can be removed. You can just assume that we have a, you can, the conjecture is that the critical cat not is the only rebound the minimal surface in the unity ball with Morse index equal to four. And let me just mention one more result of classification result is due to my graph, and they prove that the only rebounded minimal analyst in the ball which is simultaneously with respect to the coordinate planes are the critical cat anode. So, our result our gap result is is a try to add a new classification characterization results to this list. Let me give, let me give our main motivation, our motivation is. Sorry, before the motivation let me just, I'd like just to mention one very basic and fundamental question about free boundary minimal surface in the unity ball, we are not totally answered yet. The question is the following. If you consider two interest DNR. The question is, is there a free boundary minimal surface close in the unit ball with genomes are and are boundary components. There is some partial results. These are existing results of rebounded minimal surface. First is presentation prove that the existence of surface with no zero in any boundary components. After that, fully apaca in Zolotareva prove the existence of surface with no zero or one and number of the boundary component sufficiently large. Get over prove it the existence of the office of rebounded minimal surface with genome sufficiently large and three boundary components couple layers and Lee proved the same type of results using different techniques. Couple layers in vehicle prove that gave examples of surface with genome sufficiently large and connected boundary. And finally, Carlo to friends shoes has simply proved that. In fact, here. I'm sorry, here is for any gym for energy. And in boundary component and connected boundary. Okay. So the, the, the, the completely question is not solved, but there are these nice examples of free boundary seems to surface. Now let me let me state the our main motivation. Our main motivation is this classical result due to 10 to come in Kobyash and Lawson and Simon, where they prove the following gap result for minimal surface in the unit is fear. So consider a closed minimal hypersurface in the unit is fear. It's satisfying this following condition. Then we have the only two possibilities or the surface is totally geographically in the surface is an equator, or the square norm of the second fundamental form is constant equal to N in the surface is a clip or hyper surface. So motivated by this result. We can state our gap theorem, which says the following. If you have a free boundary minimal surface in the closed unit ball. Assume that for all points of Sigma. You have the following this following inequality here you have the square norm of the second fundamental form. And here we have the square of this support function with the function given by the position vector x times the normal amount. So in this case, we also have the only two possibilities. The first one is that all this quantity is identically zero. And in this case, the surface is a flat actorial disk, or it's, it's quantity has to attend to at some point. And in this case, the surface is a critical Catanoia. So let me give just a little brief of the proof. The first step is to define this following function with with is the distance of the off a point of surface to the already squared or two. So the, the, the area of his fashion, which are given by this by this. And the point is that our gap condition is equivalent to the fact that the ration of this function is no negative. Now define the, the, the, the minimal values, the minimal points of the, the, the set C, which are the minimal points of the function F. And if it's not difficult to see that since the, the function is convex. This, this set has to be totally convex. Now, if you suppose from this, from this info, if from this fact, if you suppose that this set has only one point P, then the surface has to have has to be a topological disk. If you assume that we have strictly in a quarter here, your function has to be strictly convex. And this implied that the, the set C is only one point, which implies that the Sigma is a disk. So you can now use the, the Nietzsche results, which implies that the Sigma is an equatorial disk. Now, if you suppose that the, the, the set is not one, only one point, then we can only, we can also prove that it can, can prove in this case that the surface is homeomorphic to an annulus. And that, in fact, this set C is a closed geodesk in the, the surface. In particular, if, if the, this point is equal to two as for some point, then we are in the situation situation that this set is not only one point, and we can conclude that the Sigma is an annulus and C is a closed geodesk. And finally, we can prove that C is in fact a great circle, a great circle in the normal vector along this great circle is equal to the, to the position vector. And from, from this point, we can only, we can just use the, the solution of the, the only problem for a minimal surface to conclude that the, the surface is to the critical catanide. And this finish the proof, then this is the line of the proof, the line of the proof. Let me give now some remarks. The first remark is that recently Lee and Sean proven the analog results for free boundary minimal surface in a geodesk ball of the sphere in the hyperbolic space. We also have a result due to Barboza in Vienna. We also have the result to higher co-dimension. We also have a result due to Cavalcanti means in the tutorial where they prove a topological gap for free boundary minimal hypersurface in any dimension and co-dimension. We also have a result, a recent result due to Barboza, Cavalcanti, Imperera, and Android Barboza Imperera, where they prove similar results for free boundary hypersurface. And let me just mention that the semi-gap condition was used by mixed pairs in rows to characterize the plane and the catanide in our tree among probably embedded minimal surface without boundary. Okay. So, I think I will, I will stop here my talk. Thank you very much for your attention. Okay, so thank you very much for a very interesting talk. So, are there any questions? You're very welcome to simply unmute your mic and ask. Or you can write in the chat with the other question. Let's see. May I? Yes. Go ahead. I was curious to see, to know whether, is there kind of an Aleksandrov type also argument to deduce instead of just the, instead of the classification, maybe just to get to, for example, rotational symmetries of these minimals, free boundary, minimal surfaces. I mean, in fact, at the end, you prove much more. I mean, if you, could you, could you try to imagine some kind of reflection arguments and things just to prove rotational symmetry of the solution? Yes, yes, there is, there is a, yes, there is, there is a recent result due to, due to Ezequiel Barboza and others, where they, they classify, they classify a CMC surface in the CMC, CMC animals in the unity ball using Aleksandrov reflection techniques. They, they, they assume that the boundary, the boundary of the surface, they assume that the surface is embedded and that the boundary has main symmetries and they can use an argument like Aleksandrov to, to, it's, it's so possible to use in some case. Okay, thank you. So, are there any other questions? Well, yes, actually I have another curiosity, but can I? Yeah, sure. What, I mean, how, how flexible is your technique to get results for domains? I mean, in fact, not even compact domains like a, not, not a ball, but for example, in fact, something like the complement of the ball, for example. The complement of the ball. I mean, the, the, the, the one important feature of my technique, for example, do you, do you, do you, are you referring to the first result or to the second one, to the gap or to topological bounds? I was thinking to the topological bounds, but I'm curious about. Okay, about, about to the topological bounds, one, one main feature of the techniques is that we use the convex of the boundary. And if you, if you consider the, the, the complement of, of the ball, now the boundary of the ambient space is not more, is not convex, is concave. And we cannot apply our technique in this case. We have to, the boundary of the event space is to, to be, has to be convex, to convex. But if you have, if you are, for instance, inside of a paraboloid, you have a compact surface free boundary CMC inside the region bounded by the convex region, bounded by a paraboloid. Our topological, our topological bounds roots for any ambient space, which is, which has bound a convex boundary, the ambient space can be compact or not, or not compact. The surface has to be compact, but the, the, the ambient space need not to be compact, but its boundary has to be convex. But don't you think that it should be a similar result for kind of the opposite of, I mean, of convex, like concave? I mean, don't you think, do you think it should be true something similar for, for example, the complement of a ball? Okay, a nice question, nice question. Yes, maybe if it's possible to prove by that, I don't know to prove because, because I, in, in, in this theorem, I, I use that lemma with the two problems, two different problems, the, the free boundary problem in the plateau problem. But I, I can, I only know to prove that theorem, if the boundary of the ambient space is convex. I don't know how to relate these two, two, two, two, two problems, the free boundary and the fixed boundary without the convexity of the ambient, the boundary of the ambient space. So we have a question from Elaldo Lima in the chat wondering if it may work for cylinders, your results? I'm not sure which it's referred to. Yes, maybe in the similar, you can conclude something because the, the, the boundary of the similar is weakly convex. You can, maybe, maybe we can conclude something. Okay. Very good. So are there any further questions? Seems not. So in that case, I'd like to thank the speaker again very much. Thank you for an excellent talk. And we'll see you all next week for the talk of Alicia Dickensday. Okay, thank you very much.