 So I want to thank the organizers for the invitation. It's surely a great pleasure to be here. So the thing I'll talk about is my ongoing work with Fernando Marquez and more specifically, the collaborations we had with Yevgeny Lukumovich and Kay Eri. So I'll motivate the problem, state the conjecture, state the theorem, and then I'll prove the theorem. So the motivation for this type of questions goes back, I would say, right to the beginning of the area. And so this was originally posed by Poincare. And the question that Poincare asked was whether we can always find a closed jodesic on any two spheres. So Poincare was actually only looking at the convex spheres. And these turn out to be a foundational question in the field, in mathematics in general, because closed jodesics for two spheres, and generally in manifolds, they have two distinct points of view. On one hand, you can see them as closed as periodic orbits to some Hamiltonian flow, in this case, the jodesic flow. Or you can see them as critical points for some functional. So the question that Poincare was asking is really do Hamiltonian flows in general have periodic orbits, or do functionals in general have critical points? And I would say that these motivated dynamical systems, but also Morse theory, variational methods in geometry, and a lot of mathematics, was started from this question. So the answer to this came from Birkoff in 1912. And he indeed showed that every two sphere admits a closed jodesic, and the method he used was variational. It's called the min-max method. Then some years later, Lucian and Schnierlman, and Grayson also had an important contribution here in 89, because he did the curve shortening flow part. So what they showed is that on every two sphere, we can actually find three simple closed jodesics. And this result is optimal on some ellipsoid. So on the ellipsoid, it is easy to see what the jodesic should be. So if this is an ellipsoid with its axis, then the closed jodesics will be the simple ones. It will be this one, that one, and the third coming from this guy here. And we can actually find ellipsoids where there are no other three simple closed jodesics besides this. And every other jodesic will have arbitrarily large length and lots of self-intersections. And so then from the point of view of existence, there's also an important result due to Pew and Robinson. And what they showed is that for a generic set of metrics on any given surface, the set of all closed jodesics is dense. So here, they prove this using the Hamiltonian point of view. They treat jodesics as being periodic orbits for a Hamiltonian flow. And Kairi, one of our co-authors, two years ago, we improved this result from C to C infinity generic. And he did this using, again, the periodic point of view, but also using a previous result in embedded contact homology due to Hutchison and some co-authors. So the story, of course, one that is natural to ask, can you go from C to generic to every possible metric? And this was answered by Franks and Bangerd where they showed that every two-sphere indeed has infinitely many closed jodesics, but of course, they wouldn't know whether they are dense or not. Nancy Hinkstein also had an important contribution here. And in this theorem, so Franks uses the Hamiltonian point of view, and Bangerd uses the variational point of view, and both are complementary there. So I did this extremely short survey of the story of closed jodesics, but this is a central chapter in differential geometry, which had really brilliant work from many, many people. I just cited some of them, which are, of course, Gromo Meyer, Gromo of Klingenberg. Margulius studied them under negative curvature and showed that they are equidistributed. And also, this is an extremely rich topic in mathematics. So now, of course, in the same way that jodesics, they are critical points for either the length or the energy functional, we can choose one. Minimal hypersurface, they are critical points for the area functional. And so if we now, minimal hypersurface, they're also central in not only geometry, but, of course, relativity and a lot of other fields. But if we step back to the 70s, the only known examples would be three or four of them, even on R3. I think it was the ketonite, the plane, and the helicoid. So there was an important result here due to Lawson in 1970. And so what Lawson showed was that for the round three sphere, then you can find a minimal embedded surface of every possible genus. And then after this, then, if you assume the space has a lot of symmetries, then, of course, now we know that in R3, there are a lot of complete embedded minimal surface. And if you assume symmetry on the space, you can always find. We have some techniques to find many of those. So Yao, in 982, he put this list of open problems in geometry. And they are divided by sections. And the first problem for the minimal surface sections, and he put it as a conjecture is that, a conjecture that every closed-tree manifold should admit an infinite number of distinct minimal surface. So I will review now a bit the progress that has been made on this conjecture. So I restate the conjecture there. And then I should make this. Of course, when the ambient manifold is reached in topology, so for instance, if it's n-dimensional torus, then you can use direct minimization to just find lots of distinct minimal hyper-surface because, for instance, if you do the three torus, the H2 homology is very large, so you can for sure minimize on many distinct homology classes and get infinitely many distinct minimal embedded surface. So the problem becomes much trickier when there's no topology at all, for instance, when it's the three-sphere. Because then there's nothing you can minimize. And so the first result in that direction, this was Pete's thesis, but also Shane Simon, they had a contribution there to extend the regularity for higher dimensions. And the way the result can be stated is that if you pick any closed manifold, then for sure, any closed-tree manifold, then for sure you can find one closed minimal embedded hyper-surface, which will be smooth outside a set of co-dimension 7. So what the result is saying is that for sure, there's always one. And then not much progress was made for 30 years. And then some years ago, jointly with Fernando, we showed that on every closed-tree manifold, we can actually find n plus 1 closed-minimal embedded hyper-surface, which are smooth outside a set of co-dimension 7. And if we are willing to put some curvature condition on G, we improve this result. And we show that we restricted ourselves to the case where the dimension of the ambient manifold was neither too small nor too large, just because we didn't want to have to deal with singularities. So if the ambient dimension is between 3 and 7, and if the manifold has a metric with positive Ricci, then we can indeed find infinitely many closed minimal smooth embedded hyper-surface. And two years later, Shinzu, he removed the dimension restriction there. And so what he showed is that even for higher dimensions, you'll be able to find infinitely many closed-minimal embedded hyper-surface, which will be smooth outside a set of co-dimension 7. And this was pretty much all that was known regardless the Yao conjecture. And so the result I want to talk about is my new work with Fernando and Kayiri. And we show the following. So again, we don't want to have to deal with a singular set just to make our life easier. But that's no big restriction, because at least we can deal every manifold from dimension 3 to 7. And so our result says that for the infinite generic set of metrics on any given manifold, the set of all closed-embedded minimal hyper-surface is dense, meaning if you pick any open set U, you'll be able to find a closed-embedded minimal hyper-surface that passes through that set U. So in particular, we solve the Yao conjecture for generic metrics. OK. So what I want to say now is, of course, I will dedicate the rest of the talk to explaining the main concepts behind the proof. But let me say that. So to put things a bit more into perspective, then the only available technique one has to find minimal-embedded hyper-surface are variational methods, because the Hamiltonian, we cannot, or at least as of now, there's no interpretation of minimal-embedded hyper-surface as being some periodic orbit, some Hamiltonian system. So it doesn't seem to make sense in this scenario. So the only available methods are variational. And the problem is that from the variational point of view, a minimal surface or the same minimal surface with multiplicity, I chose three there, but should apply for any positive integer, this count has distinct elements. This count has distinct critical points. But on the other hand, they do not correspond to the distinct minimal hyper-surface because I just pick one I already knew and multiplied by an integer there. But the variational point of view does not distinguish between this one and the other one just says they are distinct, or it does distinguish between that and that and just says they are distinct critical points when in reality they are not. So as it happens, or as it always happens, if we make some sizable progress on what was known before, it means that we need to have some new input, some new idea coming back into the area. And in our case, the new ingredient we bring is this gromov. It's a viola for the volume spectrum, which was conjectured by gromov and was proven last year by Lukumovic, Fernando, and myself. So the existence of these, we call these gromovs viola, the existence of these gromovs viola in combination with variational methods is what allow us to show that minimally embedded hypersurface will be dense for a generic matrix. So the plan of the talk is to explain what is the viola for the volume spectrum, give one idea about the proof, and then I'll explain how the viola implies that theorem. So now let's instead of, so before I talk about gromovs viola, let's talk about just the standard viola. So this makes a connection with what was said on the first talk. And so let me just do a little brief introduction here. So the setup is the same. Just take some n plus 1, close, remind, and manifold. And now I look at the space. So w12, this is the space functions with bounded energy and bounded L2 norm. So now I'll present the viola or I'll present the definition of the Pythagian value in a slightly different way than you're probably used to it, just because it's going to be useful for the next slide. So I look at the space w12. I remove the zero element. And now I make an equivalence class where I identify a function with all its constant multiples. And I denote the resulting space by p. So intuitively, w12 is a Hilbert space. So we should think as being homeomorphic to r infinity, if it makes any sense. If I remove the origin and identify a function with its multiple, then the space I should get should be something homeomorphic to rp infinity. If I do this on a finite dimensional vector space, then I do get a finite dimensional projective space. So this is the intuition here. And of course, a p plus 1 plane in w12, now it's going to project to a p-dimensional projective space on this space, which we'll call curl p. So I'll state the definition of the spectrum in terms of these space of equivalence classes instead of w12, as we usually see it. So the Pythagian value is given by the following expression. We take a p-projective space in this space of all in w12 with this equivalence class. And now for this projective space, I pick an equivalence class of functions there, and I maximize the Rayleigh quotient. So here I pick an equivalence class. Here I apply the Rayleigh quotient to a given representative. But of course, this is well defined because the Rayleigh quotient is scaling variant. If I apply on a function f, or if I evaluate the Rayleigh quotient on lambda times f, I get the very same value. So this only depends on the equivalence class. And this is exactly the min-max definition for the Pythagian value. So it was constructed. And so the most fundamental result learns about the sequence of Pythagian values. This is called the spectrum of the manifold is that, as p goes to infinity, the asymptotic behavior of the spectrum depends only on the volume of m. So what we saw in the first talk of the morning then was a more detailed refined analysis of what are the second and third order terms there. So this was proven originally by Vile in 1911. But his proof only works for domains of space. And then many years later, Minakshi Shudaram and Playaal, they gave a proof that applies for closed manifolds. And their proof is based on the asymptotic expansion for the trace of the heat kernel, as we saw it in the morning. So Vile proof is purely variational. But the second, this more general proof uses things which are very much related with the linear theory. And in this case, because one can compute the spectrum of highly symmetric examples like the sphere or a square or a torus, then we can know what universal constant is. We just choose this on a sphere and compute this constant there because for the sphere, we know exactly what all these values are. So now I want to talk about, so this is the Vile spectrum. I want to explain what is the volume spectrum. So the idea is the same. So I need a space. I need the space to have lots of projective spaces. And I need a functional on the space because these were the three things that we had on the previous slide to be able to get started on the Vile spectrum. So the space I'm going to use is the space of... So Robert spoke about this space yesterday. So I'm looking at the space of co-dimension 1, mod 2 cycles in a given manifold, such that these cycles are the boundary of something. So if you're not so familiar with this language of geometric measure theory, you can just think of the space I'm using as the space of all boundaries of regions in the ambient manifold. I put this restructuring because I want this to be the connected. So this quantity is requiring T to be the boundary of something. It's there to make sure that T is homologically trivial. So I want to work on the connected component of zero. So, but this is the base space. And this space has a topology. One puts there called the flat topology, which Robert also spoke about that. So Almgren in his thesis, he showed that he computed the homotopy type of this space and he showed that it's quickly homotopic to RP infinity, just like the space of functions if you remove the origin and make an equivalence class there. So if the space is weakly homotopic to RP infinity, then it means it has the same homology ring as RP infinity, in which case we know that there's only one generator. So let me write that. So if we do this, compute the homology ring, coefficient Z2, if we do a singular homology, dominance in Z2, then these are just polynomials with the Z2 coefficients with the only one generator. So this is our space and the space as topology. So this is the analog, this space is the analog of this space there, okay? So now we have topology on this large space so we can talk about P-projective spaces, right? And so this is the concept of a P-sweep-out. So if we take a continuous map from some, in the theory we use cubicle complex. If we look at continuous map from a cubicle complex into this space of more two cycles, code dimension one, more two cycles, and we say this space is a P-sweep-out if we take the generator, we raise it to the P-power, and if we pull back that with the map phi, then we require to get a non-trivial element there. So you should think as phi as just a map from a, there of course the space X can be anything, but typically we can think of it as just being a map from a P-projective space into there which is non-trivial, and what this means is that in some sense its image should detect the P-power of this form. So if this were, of course it doesn't make sense, but we can think of it as the P-power as being the volume form of some P-projective space which hits inside this space, and what we are requiring is to make sure that this map is detecting this P-power, okay? So that's the idea. So this is our replacement for P-projective planes, okay? And then, so these ideas were due to Gromov, and so then he did this in the 80s. So for every integer P, then Gromov introduced the sequence, the volume spectrum, which is defined the following way. So I take a P-sweep-out, and then I maximize the volume functional on that P-sweep-out, and I minimize among every possible P-sweep-out. So this is the analog for the vial spectrum where I'm replacing the Rayleigh quotient by the volume functional. I'm replacing W12 by the space of motorcycles, and I'm replacing planes by P-sweep-outs, but the idea is the very same one. Okay, and so Gromov, in the same paper that he introduced this notion of the volume spectrum, he studied his growth, and he showed that you can find a universal, you can find a constant that depends on the ambient manifold and on the metric, such that the P-power grows like the N plus one roof, such that these elements, sorry, we denote this by the P-width, such that the P-width grows like the N plus one root of P for all P in N. So this is something I'm saying that the, like studying the growth of the P-tagon value without worrying about whether the limit exists or not. So Gromov, he first did in 86, and then Larry many years later, he proved a more general version. So Larry Goode, he does not restrict himself to just co-dimension one cycle, he studies cycles of any co-dimension and he gets the equivalent statement there, okay? So motivated by, I guess, the standard Vi law, then Gromov, he conjectured that the volume spectrum that he introduced should also obey a Vi law. So what this means is that there's a universal constant such that if you divide this P-width by this growth power, then we're gonna get that universal constant times, of course, just the volume, because this quantity scales, it's not scaling variance, so you have to put something there reflecting the way the scaling of this quantity. And so last year, jointly with Lukmavich, Marcus and myself, we showed that these Gromovs Vi law holds for every space where the space may or may not have boundary. So I will say some words about the proof just to highlight the sort of techniques one uses. And so I guess the first thing one observes is that we were able to prove the Vi law without knowing exactly what the constant is. So Vi law, of course, he computes the Vi law and he uses heavily the fact that he knows what the constant is a priority because he uses heavily the fact that he can compute the spectrum for cubes. In our case, one does not know how to compute this constant and so this was a bit of a surprise at least to us. So now the proof is divided in, again, the crucial thing is what technique you're gonna use to prove the result, right? If we do the analogy with the Vi law when the manifold is a region of space then we can use just variational methods to do it. This was what Vi law did and this is what we do as well. So when our manifold is just a region of space say like a ball, then the results follow from variational principles, namely from a super additivity formula for the P width in terms of the domain U and you can prove this using the same type of techniques that Gromov introduced and that later Larry Gooth explored. Of course, when M is not a region of space but just some closed manifold for instance like a sphere then this method breaks in the same way that it breaks for the Vi law and in the same way that it took maybe 40 years more to get the Vi law for the closed manifolds, right? So using these variational techniques we can only know that the limit is bounded by a universal constant, right? And you can bound from one way because you can easily fit cubes inside the closed manifold but you cannot fit the closed manifold inside cubes of the same dimension, right? The thing does not make sense, okay? So I will explain now what is the idea to prove this Vi law when M is just some closed manifold in space because we did have to crack our heads a bit here to get this right. So the idea to handle the general case is the following. So I take my manifold M, it could just be a sphere in which case the Vi law was also not known, this Gramo's Vi law. And so our method is to, we break the sphere, we divide the sphere into lots of regions which are almost Euclidean. So I could imagine dividing my sphere into lots and lots of cubes. They will have just some minimal overlap. It's not a big problem. And then I reassemble these cubes in space in order to get a region in Euclidean space of the same volume as M. So I just shot, I literally chop it up in cubes and put them back in Euclidean space in any way we want just to make sure that it's connected, right? And the reason why I do this is because for this guy I know the Vi law holds. I wanna conclude it for this one, right? And then, and it's on this part of the argument that using motorcycles becomes crucial because there's gonna be a lot of cancellations. I think Robert also mentioned that issue. If we were to do this with integer coefficients, we wouldn't know how to do it. So the upper bound really comes from using a lot of the structure about motorcycles. So now, given a relative cycle, T and U, for instance, a relative cycle means that now U has a boundary. So T will be a cycle whose boundary lies in the boundary of U, something like this. I choose a cycle there. And now I wanna find another cycle on the closed manifold which will have no boundary. And the property is that the volume of this new cycle will not be much higher than the volume of the old cycle. So actually it will not be more than the volume of the boundary of all the subdivision we made. Initially, this is large, but because eventually these guys will go to infinity, this term will be knowledgeable, right? So how can we do this? So I hope you can see it. There's a color code there. So I go, I look at the way the original cycle intersects every cube. And there on every cube I completed just to be on a cycle, meaning we'll have no boundary, right? So I look at this red part and then I add that piece, that piece and that piece to get a cycle. I do the same thing on this cube there, on that cube there. And so I can add pieces of the boundary to get now good cycles on each cube. And now I just put them back on the original manifold and I don't even worry about the order, right? If I do this, of course, I'm gaining a bit more on the area, but what I gain is never higher than this. Many of them will share the same boundary, but because it's more to cycles, there's a lot of cancellations, which make sure that this term is uniform and independent of the cycle T. So this is an absolutely crucial point. And then the punchline is that we can do this in a continuous way, meaning if we have a continuous family, a continuous family of p-parameters of cycles, of these relative cycles, I can construct another family of p-parameters on the closed manifold. And I get that the p-width for the closed manifold is bounded by the p-width of this region in space plus something which is bounded, this term is of this order, right? But now this is fixed. And so, of course, when I divide by the growth power and make the limit go to infinity, I get that this asymptotic limit will be less or equal than that asymptotic limit and that we nailed out already using the variational methods, okay? So this is in a very broad stroke, the basic idea to get this Vi law, okay? So now I already explained what is the volume spectrum, I explained what is the Vi law, I need to make the connection with a theory of minimal surface, okay? And so this connection is done using what we call a min-max theory. Or it's generally, it's known as min-max theory. So using the Andren-Pitts min-max theory for homotopy classes and the Morse index bounds for min-max hypersurface that Fernando and I proved maybe three or four years ago, then we can show the following theorem. So restrict ourselves to the dimension between three and seven just because we don't wanna deal with singularities. And then the theorem is that for every integer p, there's some closed embedded minimal hypersurface with possible multiplicities which realizes the width. So let me make several comments about this. So the first comment is that this does not follow from the theory left by Andren-Pitts because Andren-Pitts, they always work on a fixed homotopy class and dispute with they are defined in terms of homology classes. So it has a much larger class of competitors than for the Andren-Pitts min-max theory. So if one were to use only Andren-Pitts min-max theory, then what the result would say is that we can find a sequence of embedded minimal hypersurface whose area converges to the p-width. But of course, even if you have a sequence of embedded minimal hypersurface with bounded area, they might converge to something singular, right? It's easy to construct examples just on the three sphere. On the other hand, Fernando and I, we proved that even when one works on a homotopy class, these minimal embedded hypersurface one gets from the Andren-Pitts min-max theory, they have a uniform bound on their Morse index. And so if they have a uniform bound on their Morse index, now when we take the limit, we know that we're gonna converge to something that's nice, smooth and embedded and that's how we wanna obtain this result. So the second comment, which was something I alluded to in the beginning is that the sigma p can be of course, twice some sigma prime and three times some other minimal surface where they left disjoint supports. And as I said in the beginning, it's the issue of this existence of multiplicities that makes the existence of embedded minimal hypersurface to be a hard problem because from the variational point of view, all these elements are distinct while if this is the case, they are obviously not. And then the other comment is that using this theorem, one can at least compute the p-width for some symmetric spaces. So the ones for the three sphere, they do mimic a bit the eigenvalues and then of course as p gets large, the problem becomes much harder and it's hard to compute. So the first four-width of the three sphere, they are four pi, then the next three. So this always has to be realized by the volume of some minimal hypersurface. Here is the equator. The next three, this is the Clifford Torres. And so this was done by a student of mine and of course to know this, he had to use these what we call the conformal families that they were introduced by Fernando and myself. And then the eighth-width, we know that it's between, is bigger than two pi square, but smaller than eight pi. And of course computing that is probably related with solving the Wilmore conjecture for genus two surface. And I would say the conjecture is that the eighth or the ninth-width, it should be realized by the area of the loss and genus two minimal surface, which we don't know explicitly what it is. So now we have all the ingredients, I'll explain what is the, how to prove the theorem from the Vi law and min max. So the theorem, just recalling again is that for a generic set of metrics, the set of all minimal embedded hypersurface is dense. So let's introduce a tiny bit of notation. So given a metric G, I denote by MG, M is from minimal, M of G. This denotes the set of all minimal embedded hypersurface with respect to G. A priori, we know that it has at least N plus one elements, but that could be it. So a priori it could be a finite set, right? And now we fix some open set U on M and we look at the set of all metrics, such that for that given metric, I can find a minimal embedded hypersurface that intersects U, okay? So this could be empty for almost every metric, okay? And so I also require that the minimal hypersurface that passes through the set U is non-degenerate. So non-degenerate, it means that if we do second variation, there will be no non-trivial element in the kernel of the second variation. So the fact that it's non-degenerate, it means that this set is open. So it could be empty, but if it's non-empty, then at least, well, the empty set is also open, but each time you pick a metric G there, then for a tiny neighborhood of these metrics, they will also satisfy this property. And the reason we know this, it's because of these non-degeneracy conditions. So if we have a set U, if I have a minimal surface that cuts that set U, then if I wiggle a bit the metric, if the minimal surface is non-degenerate, then I can wiggle the minimal surface to get a new minimal surface for the new metric that still passes through the set U. So the set is U, I wanna make sure that it's dense because if the set is dense, then I take a countable intersection of open dense sets, and that's generic, okay? So the first remark is that it suffices to see that the following set is dense. And the following set is just the set of all metrics for which there's a minimal surface that intersects the set U. So this set is a tiny bit bigger than that one, but their closure is the same. So if I pick a minimal surface for some given metric G, and if the minimal surface is degenerate, then we can do a tiny perturbation of the metric G which makes that minimal surface non-degenerate. We can even do that with a tiny conformal deformation. So this set is bigger, but the closure is the same. So it might as well work with the bigger set, right? So I will, so the next slide will be the last one. It's gonna have four steps, and I will explain why this set has to be dense, right? Okay, so let's just remind the notation M of G. This is the set of all embedded minimal hypersurface with respect to G, and G of U, this is the set of all metrics, so let's erase here. So there's my manifold M, and this is my set U. And G of U, this is the set of all metrics for which I can find the minimal surface that cuts U. Okay, that passes through U, okay? So of course the proof is by contradiction, and so suppose that the set is not dense. So what this means is that I can find an open set in the space of all metrics, which avoids this set, right? So I pick an element in this open set, and then I can see that the following one parameter family of metrics, so what I do is I pick a positive function with compact support defined just on my set U, and I do a little bump on the metric there. So if I were to make a picture, so let's suppose that our set U is like this, and the G of T would just make a tiny bump, right? This is the family G of T. So when G equals zero is like this, then ST increases, I just make a tiny bump there, right? Of course, because the set is open, I can assure that all these one parameter family is still within the open set, right? Because I can make this as small as I want by choosing the size of the bump as small as I want, so no big deal there, right? So let's unwind what does it mean for this one parameter family to be in the open set V, okay? So what it means is that for all these metrics, there's no minimal embedded hypersurface that passes through you. But all these metrics G of T, they coincide with G outside the set U, right? Because I just did a tiny bump on the interior of U. So what this means is that the minimal surface with respect to the metric G of T and with respect to the metric G will be the same because no minimal surface cuts you and the metrics only differ inside U, okay? So I change the metrics, but the minimal surface, I have not changed, they all remain there, right? Then the third step is to say that there's an important theorem of Brian White which allow us to pick the initial metric to be bumpy. Bumpy means that every minimal, every embedded minimal hypersurface will be non-degenerate, meaning you'll never find elements in the kernel of the second variation. The advantage of doing that, it's because there's a result here of, why I call this Ben Sharp, come back in the theorem, so Ben Sharp, it was a postdoc of mine many years ago in London. So in the case where the metric is bumpy, what Ben Sharp proved is that the set of all minimal embedded hypersurface is countable, okay? When the metric is bumpy. So what this means is that I have this one parameter family of metrics, the minimal surface do not change and it's a countable set, okay? So if it's a countable set, we can look at the way the P-width changes in terms of the parameter T and this is a Lipschitz continuous function, the width. In the same way that the spectrum is Lipschitz continuous, the volume spectrum is also Lipschitz continuous in terms of the metric. And so this function is continuous in terms of the metric. If we only have a countable number of minimal, if we only have a countable set of minimal embedded hypersurface, because the image of this lies in a linear combination with positive integers of the volumes of elements would live in a countable set, the image of this lives in a countable set, independently of T. And now if you have a continuous function whose image lies on a countable set, the function has to be constant, okay? And so we get that this function is constant, okay? Now we really get the punchline, okay? And the punchline is that the volume of the final guy is bigger than the volume of the initial one. And by the Gromov-Weiler that we proved with Yevgeny and Fernando, I know that asymptotically this guy will grow, will have a bigger value than that guy. So there's some integer P where the P-width for the final metric has to be bigger than the P-width for the initial metric. And so it's impossible for this function to be continuous, which means that the set has to be dense. And so the set of all minimal embedded hypersurface is dense for generic metrics. Okay, I did only 50 minutes, as I promised to Ziller. So I'll just stop here. The case of the Trisfer, do you know the genus of the minimal surfaces? Okay, so for the round Trisfer, we know, I mean, we know the genus of the Lothian construction. No, no, I have a conjecture that the genus grows like P. So because this is an important point. So a lot of the way we guide ourselves here, literally, we just follow what the intuition one gets from the vile spectrum and argues the other way, right? So the P-th minimal surface should have index P. And then I would hope that if you have a minimal surface of index P with the space of positive reach, then the genus should be proportional to P because the index P, you know, having something with more index P, it means that you have P directions in which you can decrease the area. The picture I have in my mind is P-necks, which you can, so if you have P-necks, you can pinch them and decrease the area in P of them, so you should have genus P. But that's not proven. But that is the hope, yeah. So if you look at this, I would repeat as a sequence, is the asymptotic behavior, or for example, we can consider the, say, the data function for this sequence. So does it, so data function tells the, like the geometry behind it? Okay, so of course we can have a zeta function associated with it, just the same way that you can have a zeta function with a spectrum. The problem is, okay, so a difference from the spectrum, and this I would love to know is that I don't know what the second order term is, okay? Nor do I know what the optimal constant is. So these properties become more useful if we could nail down the second term in the expansion. But that's a line of research, of course. So what kind of technique, you know, will potentially provide to you take that second term guy in your mind? So far none, right? So the heat kernel, the trace the heat kernel, it doesn't make sense in the case of minimal hypersurface. You could try to think in terms of Brachyflow, but even Brachyflow doesn't seem to be helpful here because the Brachyflow is not continuous in the flight topology. So if you have two guys which are initially close to each other, if you run Brachyflow, they can be farther apart. So I don't think Brachyflow will do it. Of course, I would love to know that. Yeah, I would love to know that. I would love to know what is the analog with these trace of the heat kernel? I cannot make product of eigenfunctions. I cannot make product of minimal surface. Maybe there's some regularization one could use there. That's what I would think. You can use some regularized kernel and try to do that and take it as a limit. Are you any guess for the constant? We, yeah, I do, I do, I do. It should be, I do, and I compute it. It should be the one. So, okay, so the guess, I would, so the guess is based on the following. If I do this on the tree sphere, asymptotically, these minimal surface, they should approach nodal sets of the eigenfunctions. And as soon, if we assume this, then you can get the optimal constant from craft and formula. So, the guess would be the one that comes from looking at zeros of homogeneous harmonic polynomials restricted to the tree sphere. Okay, yeah. So, that gives you a educated guess. I guess that's the optimal one. I don't know how to prove it. Question, do you think that any counter examples in general? For the, for the density? Yes. The, I didn't think about that one. The, It's not a closed dressing problem. What is the limit? It's, it's, it's like, it's, it's a lose, lose situation because of the following. Of course, we know that generically, there will be no counter examples. So, we have to work with them, example you can work with. All the examples we work with, just by the fact that we can work with them, they're gonna have symmetries. And as soon as they have symmetries, the denseness is there because you can just start applying symmetries and just rotate things. So, I have no idea about that. Do you know, do people know that for? For the closed dressing, I don't know. Oh, yeah. I have no idea. For the closed dressing. Does that know? Metagons, too, such that, but the opposite, such that no closed dressing was true. For negative curve, it's not true, right? Of course. So, people know that, yeah. For puzzles with curve, yeah. I don't know that. It truly exists. I don't know how to prove it. Yeah, yeah, yeah. I don't know.