 It's a great pleasure to have today Yevgeny Lyukovic from University of Toronto, and he's gonna talk about foliation of three manifolds of positive scalar curvature by surfaces of control size. So thank you very much Yevgeny for agreeing to give the talk and I remind that this is a shared seminar between the PD seminar and the Informal Geometric Analysis seminar. The floor is yours, Yevgeny, take it over. Thank you so much, Antonio. Thank you for the invitation. This is my first time at UNT, although digitally not quite in person. And I'm going to talk about a joint work with Davy Maximo. And the theme of the talk today will be how curvature, more specifically, positive scalar curvature, what kind of geometric features it determines about Riemannian manifolds. So let me very quickly give you an overview about curvature. So if we have a surface S in R3, we can pick a point and we can start intersecting the surface with various planes that go through that point and contain the normal vector to the surface at that point. And we can look at how these intersections, these various curves, how much they're curving and we can start recording these values. And if you look at the maximum and minimal curvature, which turn out to be perpendicular to each other, then the product of the maximum minimal curvature is here turns out to be a certain intrinsic invariant of the surface, something that only depends on the geometry of the surface itself and is known as a Gauss curvature. And so if our surface looks somewhat like this, then you can see that the curves are curving the same way. And so the Gauss curvature will be positive, but if our surface is, looks like a saddle and there will be a curve which is pointing one way like this and another curve that's just pointing the other way. And so the product of the two will be negative and then we'll say that it's a negative, negatively curved manifold. So this is the situation for surfaces. And you can see that curvature captures important information about geometry of the surface, but if you'd like to generalize it to and dimensional manifold, the situation is more complicated. The kind of full information about the curvature is contained in the Riemann curvature tensor, something that you feed it with three vectors in the tangent space of your point P and it outputs another vector. And Gromov has called this a little monster of multilinear algebra because even though it contains the description, all the description of curvature that we need it is very difficult to understand. So we may sacrifice some of the information that make this a little bit simpler by taking the trace. So if you have some vector x, we can consider a orthonormal basis of vectors perpendicular to x and we can sum over these scalar products to obtain something which is called the Ricci curvature of the point P. And so this already has a lot of important interesting things. It gives us a lot of interesting information if we know that this quantity is positive, as we will see in a moment. And then we can take another step and take another trace and then we will obtain a scalar valued quantity, something that just assigns a number to every point in our manifold. Here we have a function which takes in a vector in the tangent space and outputs a number. And here we have just the number which is just summing up the Ricci curvatures over an orthonormal basis of the point. So today we will be talking about this scalar curvature quantity and we'll try to understand the geometric consequences of this quantity being positive. There is one immediate interpretation that you can obtain by just Taylor expanding the volume of a small bowl in normal coordinates. You'll see that locally the scalar curvature determines how large your small tiny bowls around the point are. So if you're kind of in the positive curvature situation, then tiny bowls will have smaller volume than the Euclidean bowl. And if you are in a negatively curved situation, then tiny bowls will have slightly larger. So as far as supposed to be epsilon. That's right, thank you. But this holds at a small scale and it's not clear what's happening if we increase the sizes of the bowls and what it tells us about the geometry of the manifold. And somehow all of our success and understanding scalar curvature came from somewhat indirect methods, not from looking at this particular identity. Okay, so the theme will be that positive curvature implies that the manifold is small in a certain sense. And we have excellent results, classical results in this direction for Ricci curvature. We know that the Ricci curvature is positive, then we know in a very precise sense that the manifold is small. Particularly if you have a positive lower bound, which is the Ricci curvature bound away from zero, then we know the bound only the amateur of the manifold. And also if you look at our manifold M and we pick a point, here can we start considering geodesic spheres around that point, then their volumes and areas will be growing in a way that's not faster than the volumes and areas of geodesic goals on the standard round sphere of the corresponding curvature. And so we can sweep out this manifold by surfaces of controlled area. So today I will focus on the question of what kind of geometric smallness can we obtain if we impose positive scalar curvature condition. So one thing that one immediately sees is that we cannot hope to get these kinds of results. So if you have a n dimensional manifold, and let's say for now that n is to let greater or equal than four, if you have manifold M, then you can define a manifold and then dimensional manifold but taking a product of an n minus two dimensional manifold with a sphere of very small radius. And because you're averaging over curvatures in various directions, if you pick this R sufficiently small than your scalar curvature, you can make positive and arbitrarily large. In particular, for example, you can take the n minus two hyperbolic space and multiply it by a very tiny two dimensional sphere with standard metric. And then it will have large positive scalar curvature but it will have infinite diameter. And if you take a point and can see that geodesic spheres around that point, then the areas of the spheres and the volumes of the balls will grow exponentially. So you don't have smallness in the sense that you have it for positive Ritchie curvature. And if you have three dimensional manifolds, then you can do the following construction. If you have two manifolds M one and M two, and there's a construction due to Gromov Lawson and Cheng Yao, where you can take a, connect some of those in a way that preserves the positivity of the scalar curvature. It will be still positive. So if you take any sort of graph, one dimensional graph, with bounded degree, maybe three or maybe more complicated graph, then you can construct a Riemannian manifold with positive scalar curvature by just associating to each vertex, some manifold with positive scalar curvature with just the sphere S three. And then corresponding to each edge, you can perform the Gromov Lawson, connect some this way you will obtain manifold and you can make the diameter as large as you like. And also it will be large in a sort of isometric sense. You will not be able to cut this manifold into two pieces of approximately the same volume by some hypersurface of small area. For example, if you take an expander graph, then those are very connected. And so the hypersurfaces you want to cut will have to be very large. Or if you want to construct some sweep out, the areas will have to be very large. So this kind of example show in what sense manifolds with positive scalar curvature can be large. But they also suggest a certain notion of smallness because all of these manifolds, when you look from far away, look like they're two dimensions smaller than they are. They look like something n minus two dimensional times something else or maybe a tubular neighborhood of some n minus two dimensional polyhedral complex in some larger space. And that's precisely what Gromov conjectured that they admit a map into a polyhedral complex of dimension n minus two. And the fibers of that map will be small in a certain sense. So in the 80s, he conjectured that they will have small diameter. More recently, he conjectured that they will have small area. And then he combined the two conjectures to make it as hard as possible to prove of both small diameter and small area. Okay, so I'm going to talk about some progress in this direction in dimension three. Let me maybe pause for any questions. Just a question. Here, when you say the bounds are on the wall fiber, so not so that in principle, they could be composing in component, in several connected components. So the conjecture is, so the diameter should be bounded on all the fiber. Right, right. But because we have freedom in how we choose this complex, we can modify it so that it's really just the connected components. Yeah, yeah, sure. It's a kind of a re-graph construction that you can do. Okay, thank you. So we proved that there's just a foliation of any closed remanifold with positive scalar curvature, which has nice bounds on its area diameter and also its genus. And in some more specific situations when the manifold is three-sphere or it's one-crossed two, then actually we have foliation by spheres. Okay, so here, the theorem is dated in terms of a Morse function and connected components. So it's kind of a choice for aesthetic reasons. So it's people are more used to Morse functions, but you can also formulate it as a statement about some maybe different map G from M into some one-dimensional graph, gamma, this is my gamma. And then we're looking at the bounds on the areas, the ameters and genus of just fibers, not the connected components of the fibers. Then you can replace the connected components with just fiber. And in fact, in this case, you can improve this to six. Bound to the genus. And right, so these bounds are not sharp. The way we prove things, it's tricky to produce sharp bounds while controlling all of these three things. But if we drop one of those, if you drop the diameter, then it's easier to get closer to the sharp bounds here. If you just think about the area, then in the case of S3, there's a Morse function whose fiber, whose connected components of fibers have areas of most 24 pi over lambda zero. That's the area of an equator in the standard S3. So that I expect to be sharp. And we get a similar bound for S2 cross S1 into, for fibers of a map. Let's do cross S1 into S1. And more generally, we get the slightly worse bound for the areas and we get bound of genus two, which I expect to be sharp because two is the Higor genus of some spherical space forms. And so when we do the fallations, we'll have to pass through those. So two has to have to be the sharp bound here. One can use this result to construct another kind of singular fallation, in this case, by closed curves. So we have a fallation, a singular fallation by surfaces. And then each surface, you can also fully it by curves. And this way you obtain the map from M into R2 and the bounds on the areas and the genus of the surfaces translates into a bound for the length of curves in this fallations. So we have a family of nice curves, short curves sweeping out our manifold. It would be really nice from here to conclude the existence of a short closed geodesic in our three manifold. But unfortunately, when you do the mountain pass argument on these types of things, you don't quite get a closed geodesic. Instead you get something that's called the stationary geodesic network, which is eternalization of closed geodesic to graphs. So basically you get a graph, something like that, that has geodesic arcs and then the vectors at the vertices sum up to zero. So here, for example, they would have to form 120 degree angles. So it's a stationary point of mapping of a graph into the manual. Okay, so this was a quick overview of results and now I'll go into the proofs, but let me maybe quickly pause for any questions. Okay, so the main technique in the proof is to use minimal surfaces. So it was a very important observation about scale curvature that one of the ways to get at scale curvature is through minimal surfaces. So ambient positive scale curvature using stability operator gives us information about smallness of certain minimal surfaces in the manual. Mainly, there are the following diameter bounds from the arguments of Shane Yellen and also Gromov Lawson, that if you have a stable surface with boundary, so let's say you have some curve in your manifold with positive scale curvature and then you feel that curve by some minimal surface, let's say this and then you take any point on that disk, then the distance from that disk to the boundary will be at most to pi or square root of lambda zero. So it's going to be controlled. Actually, one can slightly improve on this. Well, this is not sharp, but it's good enough for us because some of the other arguments are not sharp in any case. And then from this argument, one can also conclude that if you have a closed stable minimal surface, closed stable minimal sphere, then it will have to have controlled diameter. And you also get the bound on an index one minimal surface because if you have some index one minimal surface, you can look at two points which are at the maximum distance and you can go halfway between them so that both parts are kind of borderline stable. So then you will have bounds on half on the length of half the distance between these two points. So you get bounds for the diameters of index one and stable surfaces. And you have bounds in the area. So this is a kind of a classical bound, but Marcus and Nevis proved also a bound on the area of the index one minimal sphere using some clever computation with stability operator and the harsh trick. So if you have index one and stable minimal surfaces then you can control their size in a nice way. Now we'd like to go from that, from nice control for minimal surfaces in our manifold to existence of these nice fluctuations. And generally speaking, why do we even expect to have some minimal surfaces in our manifold? Well, if our surfaces, if our three manifold is something like S one cross S two, then you can imagine that we have some non-trivial two spheres and so we can minimize those two spheres in the commodity class. And then we will obtain some minimal two sphere. So we know that it's small, okay, but it doesn't kind of tell us much about the rest of the manifold. And if we have something like S three, then well, we don't have any reason to believe that if we minimize something we will get any minimal surface. So instead we need to do a mountain pass construction where instead of minimizing surface in its commodity class we consider a whole one parameter family of surfaces. So a sweep out of S three by two spheres and then we somehow try to minimize them simultaneously and try to minimize the maximal area over surface in this family. And so there is a theory that deals with this problem and kind of using this intuition. It shows that this is possible. So it's a kind of, it's called the Simon Smith version of Albin-Pitts-Meanmax theory. And in this particular setting, sorry, my daughter just woke up. So in this particular setting we can obtain minimal surfaces that come from the Higert splitting. So if you have a three manifold then we can decompose this manifold into two handle bodies whose boundaries are, is a surface. And if they know that it's a spherical space form then we know the three manifold topologies tell us that if it's a land space then the handle body will be a torus. And for other ones we will have a genus two handle body. And so we can run, we can construct a sweep out using this Higert decomposition. And we can construct a minimal surface and it will be index one minimal surface which is kind of natural since it comes from a one parameter sweep out. And it will be isotopic to the Higert splitting of M. All right, in case if it's not RP three. So now if it's RP three for M, RP three we have a dichotomy. So either it will be index one torus either we will get an index one torus which of course we will not in, no, that's the truth we actually do. We might get an index one torus or we get an RP two stable minimal RP two with stable double cover. We'll have one of these two possibilities in case of RP three. Okay, so now we have these index one surfaces and possibly some stable minimal surfaces and we want to interpolate between them. And the idea will be to use mean curvature flow to flow the index ones until they hit the stable minimal surfaces. So let's try to understand what's happening topologically. So we know by Perlman's work by classification of manifolds or free manifolds in this case of positive scalar curvature that our free manifold can be decomposed into pieces one, two, three and so on where each MI here so these are minimal two spheres in fact, minimal stable two spheres we can cut it into pieces and each piece will be a spherical space form. Then we might get as one crosses two but if you get as one crosses two you simply cut it by another stable minimal sphere or we might get a RP two process one. That's another possibility. And then in each of those pieces we can do mean max and we can obtain stable figured splitting sorry, index one figured splitting. So maybe it's or it's original steward sphere of that manifold, any questions? So we can formulate this as follows. So the pieces into which to cut. I'm sorry to interrupt. I have a question. I'm probably asking at the wrong point though. Yes. I'm wondering about the whether there's any issues of regularity with the minimal surfaces, you know it's like automatically smoother. Right. So that's a very important question. And it's a very complicated question results of me, a lot of work of many people. So for the mean max construction only in the sixties formulated the very weak notion of minimal surfaces that can be obtained via these methods. And then with the work of Shane Simon, Shane Simon Yau and the kids, they were able to obtain regularity in some dimensions. So in co-dimension one, we have regularity or manifold of dimension less than eight. So in particular in the situation we have. Now starting with dimension eight and up we will have some singular set. It will not be smooth. That's a very good question. Luckily in dimension three, we have results that give us smoothness of this minimal services. We have curvature estimates that will guarantee smoothness. Yes, thank you for a great question. So in this particular situation, we decompose our manifold into pieces such that one boundary component it's going to be some index one surface. And then all the other boundary components are going to be minimal spheres. So this is going to be stable minimal as twos. And this is going to be index one minimal surface of genus less or equal than two. So it's index one minimal surface. So the fact that it has more index one means that there is one direction where you can push it and decrease its area. So if we push it in slightly, it's going to become mean convex. And then hopefully we start applying mean curvature flow to it and the surface will flow. Sometimes it may disconnect into pieces and eventually it will converge to the boundary minimal twos spheres. And because we already cut a maximum of this joint collection maximum joint collection of minimal twos spheres it will not encounter any other minimal surfaces in this process. And it will fully the whole manifold. Now mean curvature flow may have some singularities. And so if you want to have a sufficiently nice flowation, we will not be able to have a smooth flowation. We will have to have some singularities. But for example, we want only more style singularities. We won't only have finally many singularities which are of more style. Then we need to do some extra work, but luckily not that much because mean curvature flow is very well understood. So there are words of Brendel and Heisken and Hasselhofer Klein, Hasselhofer Beethoven and Buzan and Hasselhofer and Hirschkevitz which describe mean curvature flow with surgeries. And so they give a very detailed description of how singularities happen and how one can cut them away and how those regions where you cut them away look like. And basically the idea is that you will have some regions where your surface will become very close to cylinder and then potentially it will pinch off but right before it pinches off, you can zoom in onto this region and in the region where the singularities has to happen actually want to zoom in on this region. So right before the singularity has to happen, we can map it into R3 by some one plus epsilon by Lipschitz map and we will see something very close to the standard trial cylinder. And then we can basically by hand describe that if you have something very close to standard trial cylinder, then you can cut this away and you can construct a family of surfaces with exactly one singular point of voliation in here. In the process, we may lose the correct sign of the mean curvature here. We may slightly increase areas or diameters but because it's happening on a very small scale, whatever we lose will be really time. And so basically the description that they give about mean curvature flow with surgery lets us take any problematic region where the singularities has to happen, zoom in, see a nice Euclidean picture, cut it, decompose it and then restart the flow. And then we know from their estimates that the flow will have to run at least for some time before it happens again. So we only have to do it violently many times. So basically we end up with the proposition that there exists a more voliation of a geometrically prime region that starts to the index one minimal surface and that ends up with a bunch of stable minimal spheres. And the areas will be smaller than the area of the original index one minimal surface. That pretty much proves the result and we need just to put all of this together and then we'll have a voliation of the whole. So just a question. So that's for me to see in the if I understood. So you first decompose and you get these boundary curves. Then with the mean curvature flow you have a propagation of the area bound. So is the, but I'm a bit confused on the bound on the genus and on the... So they get propagated as well with the mean curvature flow or you need... So the bound on the genus does get propagated because the mean curvature flow if you have a surface with some genus, what can happen? You may get something pinched off, right? So you may lose some genus here but you will not gain... No genus will appear, no handles will appear during the mean curvature flow. So the genus can only go down but the diameter bound you lose you cannot say anything about the diameter. That's what I'm about to say next. Okay, you need to treat after the... Okay, okay, sorry. So this procedure gives us nice bounds for the area and the genus. Now, what about the diameter? So that's indeed an excellent question. How do we get the bound on the diameter? So let us now go back to an argument of Gromov Lawson in the 80s that gives the bound on the diameter for filiation for filiations of simply connected or homology zero, first or manifolds the zero first homology picture. So the argument is really nice but it is slightly technical but I still want to kind of give you the description of this argument. So let's pick a point P and let's consider geodesic spheres around that point. Some point that geodesic spheres may have more than one connected component. So Gromov claims that each connected component of the geodesic sphere will have diameter at most 12 pi or square root of lambda zero. So let's for simplicity take the scale of coverage equal to one. We want to claim that the connected component will have diameter at most 12 pi. And by diameter, I mean the extrinsic diameter. And this is also actually in our theorem, the diameter that we have is the diameter in the ambient metric, not in the intrinsic diameter of the surface. So let's use a different color. Let us pick two points X and Y. Let's assume that they're at the maximum distance from each other. So the distance between is equal to the diameter. And let's consider a minimizing geodesics connecting these two points. Now we may assume that the distance between P and X and Y is large because if it's say less, if it's less than six pi, then obviously the diameter is less than 12 pi. So let's say that the distance here, the distance here, let's say is some distance L, greater than six pi. And let's go back a little bit and consider a geodesic sphere. Oh, that's at the distance L minus two pi plus epsilon. So we can see the geodesic sphere a little bit closer to P. Now, okay, so that's, so now we do the following. So we can connect Y and X inside of the level set because they're in the same connected component. So we can connect it inside the geodesic sphere. And we obtain a closed curve, the triangle here, and we can feel that triangle with the minimal surface. If the manifold is simply connected, we can contract the curves, in particular, if there exists a disc, we can minimize among all discs with that boundary. And so we obtain a minimal surface. Now, this minimal surface will intersect this level set. Let me call that level set boundary of BL minus two pi minus epsilon. It will intersect a geodesic sphere in a bunch of curves. Some of them may be circles, but there will be a curve which connects these two endpoints. And because it's a disc, there will have to be an intersection with this boundary point, with this boundary point, we cannot have anything like that because those are different distances. So there has to be a curve connecting these two points which is the intersection of this geodesic with this boundary sphere and that geodesic with that boundary sphere. Let's give these two points names y one and x one. And the claim is that these two points, y one and x one actually close to each other. So let's pick a point on the curve in between. That is halfway between this, so let me call this gamma x, call this gamma y. So we pick a point, let's call it q, which is halfway between gamma x and gamma y. Same distance between gamma x and gamma y. We know by the estimate for a minimal surfaces that this point q has to lie close to the boundary of this disc. So if we draw a minimizing geodesic to the boundary, a minimizing geodesic to the boundary, notice that this minimizing geodesic to the boundary has to go to either gamma y or gamma x. It cannot go to this curve because this curve is at the distance to pi plus epsilon and this point has to lie at the distance to pi. Because it's the same distance from gamma x and gamma y, there will be another geodesic of the same length that goes to the other curve. Okay, this way we obtain yet two more points, y2 and x2. So this is now getting confusing. So that one was x1, this is x2. Okay, so now let's look at some inequalities. So we have that from p to y2 plus the distance from y2 to q by triangle inequality is greater or equal than the distance from p to q. The distance from p to q is exactly l minus two pi minus epsilon. And we know that the distance from y2 to q is at most two pi. This is less or equal than two pi. So we get that p y2 is greater or equal than l minus two pi minus epsilon minus y to q. So that's greater or equal than l minus four pi minus epsilon. Okay, so the distance from here, from here to here is greater or equal than l minus two four pi minus epsilon which means that the distance from y2 to y1 supplies that the distance from y2 to y1 has to be less or equal than two pi. And because y1 lies exactly at the distance two pi plus epsilon from y. This gives us a bound on the sides of this arc. And well, that means that the distance from y to q which is bounded by the distance from y to y1 plus y1 to y2 plus y1 to y1 plus y2 to q can now bound that. So this one is two pi plus epsilon. This one is at most two pi and this one is at most two pi and we get six pi plus epsilon. The same we get for x, for the distance from x to q is at most six pi plus epsilon. So we conclude that the distance between y and x is less or equal than 12 pi plus two epsilon. But the epsilon goes arbitrary so we get that the distance between y and x is at most 12 pi. So this is a really nice argument that tells you that this in radius bound for this disk somehow gives you a bound between these two on the distance between these two points. Okay, so this was slightly technical but feel free to ask question. One can sort of remember the main idea of the argument by saying that if you have a disk and you know that every point is close to the boundary and then you somehow decompose it into the boundary into four parts and these two parts are sort of far from each other then these two parts have to be close to each other. That's essentially what's in that argument. So we want to imitate the same kind of argument and but in a somewhat more general setting now we measure distance not from a point but from a surface but this surface has control diameter. And we are looking not at level sets of the distance function, not just the geodesic spheres but surfaces which are constrained which are trapped between two geodesic spheres of comparable diameter. So if you have some surface here and it is trapped between two geodesic spheres and it's connected, actually it's a statement not even about the surfaces. It's a statement about the connected component of your manifold trapped between two geodesic spheres at the distance which is different by 10 pi or a square root of lambda zero then by pretty much the same argument we get the bound on the diameter of the surface. You can relax this argument to saying that if you are like this and this is your geometrically prime region and your boundary here has control diameter then if you're trapped between two geodesic spheres and you're connected, then you have to have small path. So the rest of the argument now works out like this. We start with the index one surface and then we start flowing it by mean curvature flow. In the process of flowing it, the diameter may get distorted and also it may become long. It's distance to our original surface sigma may become large. And then whenever this happens, whenever the distance, the difference between distances of the closest point to sigma and the furthest point from sigma become large whenever it kind of stretches in space and reaches that threshold of how much it is allowed to stretch in space, we cut it. We can see that a geodesic sphere which cuts it halfway between these two points which are furthest and furthest away. So this is now a surface as let's say two dimensional surface we intersect it with other two dimensional surface we obtain a bunch of curves, one dimensional curves and then we fill those curves in the isotope class of one of the two halves. I did it the wrong way. That's exactly what we are not doing because this region we already swept out. We are minimizing it that direction. So the part to the left is what we already swept out here and we are minimizing it to the inside to the part that we haven't swept out yet. And so we have good diameter bounds but for this feeling by the diameter bound estimate for minimal surfaces. And then we cut the surface into two parts. This color now to obtain here a surface to one side and another surface to the other side. So this is a minimal surface. This is the original surface is a mean convex surface and then we have a corner when we smooth out that corner we obtain a mean convex surface. And so we can restart the mean curvature flow and we can restart the mean curvature flow here. And this way we shortened the distance we shortened the difference between distances to sigma of different points and therefore we put the diameter under control until again it drifts off and then we cut again. So now when we do this procedure we have to keep control of both area and the genus. And this is where we have to start need to make some difficult choices because suppose this is our situation. Maybe we have the genus all on one side and most of the area is on the other side. And we now want to cut and decompose it into two surfaces. So when we cut we have two choices. Either we can either we minimize in the isotope class of this or we minimize in the isotope class of the other part. And then we whatever choice we make then if you make the first choice then we will end up with a surface here after we cut it into two pieces we will end up with a surface of genus four we'll double the genus. And if we end up if you make the show the and here we'll end up with nice surface of genus two. But if you do the other choice then we will end up with a surface of control the area on one side. But on the other side maybe when you minimize here the air doesn't go down at all or just by a little bit. And then you effectively double the area of the right side. So when you make this choice you either have to sacrifice the genus or have to sacrifice the other. But the observation is that there is an asymmetry in terms of these two regions. There is one region on the left side which after we resume the mean curvature flow will remain trapped. You see this this region is squeezed between two minimal surfaces. And so as we flow it it will remain squeezed between two geodesic spheres of comparable distance. And so we will not need to perform this cutting procedure on it ever again. But this one will move to the right and potentially maybe it will become very long and we will need to cut it again and again and again and we don't know how we actually need to do it. So we always know how to make this choice so that the thing on the right has both area and genus controlled while the thing on the left may double its area or may double its genus but we won't need to do it one time. And so we don't need to worry about it. So that's why we have sort of an increase in area and genus bounds in our final result. So thank you very much. Thank you for that beautiful talk. Are there any questions? Yeah, there are also virtual applause. So are there any questions from the audience for the beginning? So meanwhile, I had a question and this is related to a comment that you made along the way. So you were mentioning that this is the extrinsic diameter, right? But in principle, when you are building, I don't know if I'm getting the right intuition about it but when you are building this, this falliation so the issue is that these fibers may oscillate a lot and this would give you a problem in controlling the intrinsic diameter, right? So there is no way to try to control the oscillation on the falliation. Let's say I'm mentioning really a sort of adding to the process also a sort of pull tight procedure where you try to straighten up. So if you try to run a pull tight procedure along the way, now I don't know exactly at which point and you try to straighten up these fibers. At the end, the two things, the extrinsic and intrinsic diameter may become comparable. So this is what I was wondering. So have you tried this or do you think it may hold or? I think that's a great question. I guess we can check sure that for the result should be called with intrinsic diameter. For the minimal surfaces, both index one and stable, we have bounds for intrinsic diameter. And but when we start doing mean curvature flow and this argument was comparison, Rome of Lothan argument, we lose that. And it would be nice. One of the, and it would have some interesting applications. I think it will be helpful for obtaining a short close geodesic, for example. So with this methods, we only update a stationary short stationary geodesic network. But if you have a bound on the intrinsic diameter, I think one would be able to obtain the short close geodesic. And pulling tight, I think it's a nice idea. It's not immediately clear to me how to make it work. But I think that it should be possible. I see. Thank you very much. Thank you. Are there any questions for Yevgeny? Okay, so if not, let's thank Yevgeny again for the video. Thank you. Yes, so Yevgeny, if you want, we can chat a bit more. Let me stop the recording just a second.