 be asked to speak at this. And what I want to talk about today is the tidal gap theorems for minimal sub-manifolds of spheres and as indicated, the main thing I'm talking about is some recent joint work with Tom Ilmenin. So the basic question is, consider all k-dimensional minimal sub-manifolds of the unit and sphere. So it turns out the one of least area is the totally geodesic k-sphere. And the question then is, well, what has the next smallest area? So if you take the infimum among all other k-dimensional minimal sub-manifolds of SN, what's the infimum of area? And just about a year ago, there was a, well, there's actually very few cases, almost maybe one case when we really know the answer. But anyway, there's a big breakthrough about a year ago. So Fernando Marquez and Andre Neves, as a crucial part of their solution to the Wilmore conjecture, gave the answer for k equals 2 and n equals 3. So they show that if you consider all minimal surfaces, all two-dimensional minimal surfaces, closed minimal surfaces in the three-sphere, the one of, as we said before, the smallest area is the totally geodesic s2. What they proved was the next smallest area is attained by the Clifford torus. And by k plus 1-sphere, it can't be of the same dimension. k-dimensional area in the k plus 1-sphere. Let's get k in the n-sphere. In the n-sphere. So n is 3, and that's n-sphere. Surface in the three-sphere. It's a k-dimensional surface in an n-sphere. Oh, so it's k-sphere there. Oh, it's right. Do I have the right thing? It's k-sphere on line 3. What is that? Right. It's on, oh, that's what you meant. Oh, yeah. That's right. It's the n-sphere. Oh, you know, it's a totally geodesic k-sphere. k-sphere in the n-sphere. Yeah. So we're working in the n-sphere. We're looking at k-dimensional submanifolds of the n-sphere. And of those, the totally geodesic k-dimensional sphere has the lowest area. And then if you're talking about two-dimensional in s3, the next smallest turns out to be the Clifford torus. OK, so, but there was another earlier, oh, and by the way, I think it's, I called it a gap, but it's really, I think it's a most natural look at the ratio. And so the ratio of the area of the Clifford torus, if you say what's the smallest ratio? The Clifford torus, you normalize that to be one. And then the next smallest thing is the ratio of the Clifford torus. The ratio of that is pi over 2, which is about 1.57. OK, well, it turns out Gene answered this question, or versioned this question many, many years earlier. In 1967, you could answer the question instead of asking about all two-dimensional minimal surfaces. You could just ask about minimal surface that are topological spheres. And in 1967, Gene answered that. So among other things, he proved that if you have a two-dimensional minimal sphere in Sn that is not totally geodesic, then the ratio of its area to 4 pi, of course, 4 pi is the area of the totally geodesic. That's the smallest possible. But anything else has to have at least three times as much area. And that's a sharp result. OK, so notice I stated the hypothesis in two ways. I said, let M be a two-dimensional sphere that's not totally geodesic or equivalent. You could say it's not contained in a linear three space. And the reason I said that is because Gene went on to answer a much more general thing. Namely, you could say replace not contained by a three space, not contained in a three plane by not complained in a k-plane. So you look at all minimal spheres that are not. So fix some k, like k equals 5 or 7 or whatever. Look at minimal spheres, two spheres in Sn that are not contained in any k-plane. And ask what's the least area among those. And again, Gene gave the sharp answer in that same paper. Really beautiful result. OK, but until recently, I think that's really all that was known about it. And I really want to ask, well, what about in higher dimensions? And by the way, you can make that go away. Yeah, that doesn't help. So one remark is you can reword the question in terms of k plus one dimensional minimal cones. So if you have a minimal sub-manifold of the sphere, you form the cone over it, all the rays from the origin that pass through that surface. And you get a minimal cone of one higher dimension. Or conversely, if you have a minimal cone with vertex at the origin, if you intersect with a sphere, then you get a minimal sub-manifold of the sphere. So you can ask it about minimal cones or minimal spheres, minimal sub-manifolds of the sphere. But anyway, that leads, if you ask it in terms of cones, it leads to the related question, which is really a sort of motivation, one motivation for asking the question. And it's where the starting point for Tom and me. So we originally were asking a slightly different question, or interested in a slightly different question, namely, take an area minimizing hypersurface in Euclidean space and take an interior point on the boundary. And then the density, so you have just some singular. So in general, if you take a boundary in high dimensions, you can always find the least area sub-manifold with that as boundary. But if you're in high dimensions, it's likely to have singularities. And you like to understand things of what those singularities are like. And sort of one of the most basic properties of a singularity is what its density is. So the density is defined as follows. You density to point x, you intersect your surface with a ball of radius r centered at x. You look at the area of that intersection and you normalize it by dividing by the area of the cross-sectional disc. So if you're talking about two-dimensional surface, you divide by pi r squared. So you take the area in the little ball, you normalize it by dividing by the area of the cross-sectional disc. And then you let r go to zero. That's by definition the density of the surface at that point. And of course, in general, if you're talking about general objects, you'd have to know the limit exists. You might have to take a lim super, a lim int. But for minimal surfaces, there's a monocyste formula that says this ratio is an increasing function of r. So therefore, the limit definitely exists. And that's defined to be the density of m at the point x. OK. And by the way, well, much of what I've said at the beginning, I've said area minimizing and hypersurface. What I've said so far, it doesn't have to be area minimizing. It would just be a critical point for the area functional. And it doesn't have to be a hypersurface. It could be general co-dimension. So for the reason I said area minimizing and hypersurface because that's the category in which we know we can prove something interesting. So this density, well, notice if you have a smooth embedded sub-manifold, then the density is 1 at every point. Because in a very small ball, the manifold looks almost the same as the disc. So it's almost like a disc you're dividing by there, the cross-sectional disc. You get something close to 1. So for any smooth manifold, the density is 1 everywhere. If it was smooth and immersed, for instance, then the density would be 2 wherever you had a self-intersection. And density plays an absolutely fundamental role in the regularity theory for minimal varieties in many ways. But one of the most important is the regularity result going back to Georgian, then generalized by Allard. But it says that in an area minimizing or even a minimal variety, the density is always greater than or equal to 1. At every point, the density is greater than or equal to 1. And you have equality if and only if it's a regular point of the surface. So I already mentioned that it's sort of trivial. If you have a regular point of the surface, the density is 1. But the converse is a deep theorem that conversely, if the density is 1, then it is a regular point. You have this minimal variety. You have a point where the density is 1, then there's a little neighborhood where it's a smooth minimal manifold. So that's the celebrated to Georgian Allard regularity theorem. OK. And in fact, there's some gap. There's a, so if the density is equal to 1, it's regular. So any singular point has to have density strictly greater than 1, but there's even a gap. There's some delta. So some positive deltas, the density has to be, at a singular point, has to be greater than 1. There's some delta bigger than 1. So the density of singular points are bounded away from 1. It's not just that each one is bigger than 1, but. The gap depends on the K and N. Yeah, a prairie, right. Yeah. So the question, Tom, and I were interested in is, what's the smallest possible density of singularity? So just consider all area minimizing hyper surfaces of any dimension. Look at the singularities. Say what's the least possible density of such a singularity? Some is bigger than 1, but what is it? And as I say, if you work over all dimensions and so forth, it's not even clear there is a minimum. So it's attained. So the precise question would be, what's the infimum of densities of area minimizing hyper surfaces at singular points? What's the least possible density of any area minimizing hyper surface at any singular point? And again, you could ask this about other categories of minimal surfaces. You don't have to ask about hyper surfaces. If you ask about more general things, you might get a different answer. Or if you just looked at minimal surfaces, there's things that are critical points for the area functional. Instead of minimizers, you could get a different answer. If you ask for stable things, depending on what class of minimal varieties you're looking at, you might get a different answers question. But the one we wanted to study is area minimizing hyper surfaces. So that's sort of the key question we were hoping we were trying to make progress on. So first, there's a simplification. And that is, you can simplify to the special case of cones from arbitrary area minimizing varieties, just a case, a special case of cones. And the way is the follows. Well, you take, let x be a singular point of any area minimizing hyper surface. Well, it turns out there's, if you dilate about that point x, if you take a limit of dilations, that will converge to a cone, so-called the tangent cone to the surface at x. So you take a sequence of dilations about x. If you pass to a subsequence, that's guaranteed to converge to an area minimizing cone, called the tangent cone to element x. And that cone, if your original point x was a singularity, then the cone will also be singular at that point. If you start with a minimizing hyper surface, a singular point, take the tangent cone. The tangent cone will also be singular at its vertex. And the two densities will be the same. And it's convenient to, so the density of a cone at its vertex, just say that's the density of the cone. When I say density of a cone, I mean density of the cone at the vertex. So instead of working with all area minimizing, asking what's the least possible sum of singularities among all area minimizing hyper surfaces, you could just ask, what about area minimizing cones? And you get the same answer by this reasoning. So question two is, what's the infimum of density of a cone among the class of all singular area minimizing hypercones? So it simplifies it. You just have to look at cones. You have to look at arbitrary minimizing hyper surfaces. But then there's a second simplification. Reduction, you can reduce the case of simple cones. So I'll say a cone is simple if it's smooth, except that it's vertex. So or equivalently, that means if you intersect the cone with the sphere, you have a smooth sum manifold. But possibly immersed. But immersed, perhaps? Well, it turns out, in area minimizing, it'll automatically be embedded. But in general, if you're talking about minimal things in general, it might be immersed, yeah. So if you look at things like soap films, if you take something like this, you see this kind of cone as a tangent cone in soap film. So that's an example that's singular all along this edge. This is not simple. Or even more complicated, you also see things that look like this and cones that look like this in a soap film, kind of a cone where if you intersect with a sphere, you get a kind of tetrahedron made out of totally geodesic arcs. So that, again, would be not a simple one. OK, so a simple cone is a cone that's smooth except at the vertex. And there's an important principle in the regularity theory, so-called dimension reducing. So it's supposed to take any singular area minimizing hypercone, take a singular area minimizing hypercone. Suppose it's not simple. Take one that's not, you'd like it to be simple. Suppose it's not simple. Then dimension reducing tells you you can find a simple cone whose dimension is strictly less and whose density is no bigger. OK? OK, so take a cone. Maybe it's not simple. But if it's not simple, you can find another singular cone of lower dimension that, well, in fact, a simple one of lower dimension that has no more density. But how is C-parm related to C? How is it related? Well, for this question, it doesn't matter, but I mean, I can kind of show you in a picture, let's see. So what we could do is, right, so let's take this example. So you can show if it's not simple, that means it's actually got a ray of singularities, right? The cone, if it's not simple, it doesn't have just the singularity of the vertex, but there's a whole ray of singularities. So in this picture, here's the example of a ray of singularities. Now take the tangent cone to the tangent cone at a point along the ray. If you do that, like in this case, if you do that, you'll get this guy. When you do that, you always get one that's translation and invariant in one direction. And so if you slice, you get something of lower dimension. You just keep on going until you get something simple. And the monofocity formula guarantees you, you'll get, as you do that, of course, the dimension goes down, but also the density can only go down when you do that. That's where this comes from. So question three, which is equivalent to the other two, is you can just say, well, it's enough to consider simple area minimizing cones. So question three is, what's the infimum of density among all simple singular area minimizing hypercones C? So as I've explained, questions one, two, and three are all equivalent. So reduced to this case of just cones, instead of general surface cones, and it's enough to look at cones that are smooth except at the vertex. OK, so Tom and I haven't been able to settle that, but we can give a sharp result under one hypothesis. So the main result is, suppose you take a simple singular area minimizing hypercone. So the extra hypothesis, you have to assume, is that it's topologically non-trivial, which I'll explain in a minute what that is. But the theorem says, if you take such a hypercone, if it's topologically non-trivial, then the density is strictly greater than square root of 2. And furthermore, that is optimal. In other words, that really is the infimum among all such cones. So all simple area minimizing topologically non-trivial, the infimum of density among all such cones is exactly square root of 2. OK, and the definition here of topologically non-trivial is such cones always divide the ambient euclidean space into two components. And so for us, topologically non-trivial means at least one of those components is non-contractable. And I just mentioned that there are now many, many known examples of singular area minimizing hypercones. Of course, one way you could ask this question is if you could classify all the cones, then you'd just see which one in your classification has the least density. But such a classification seems, you know, it's probably hopeless. It's probably totally impossible to do that. But still, people have constructed many, many examples of singular area minimizing hypercones. And all the known examples are topologically non-trivial. OK. So why is the square root of 2 optimal? Well, that is shown by the so-called Simon's cones. So if you're just considering an r dimension to have, I guess, you know, an rn plus 1 cross rn plus 1, you look at all the things where the norm of the first n coordinates is the same as the norm of the ds. The first coordinates is the same as the norm of the last coordinates. Schematically, it's like this. If n were equal to 0, it'd be exactly this picture. This would be the Simon's cones. This would be, I guess, what I call it, cnn, yeah. So c11 would just be these two cross lines in the plane. But anyway, this cone is, it's a minimal cone for all n, but it's area minimizing for n greater than or equal to 3. Oh, we can see c11, maybe. Sorry, this would be c00. Anyway, well, cnn. Anyway, so these cones are generalizations of the Clifford-Torres. For n greater than or equal to 3, all these Simon's cones are area minimizing. They are topologically nontrivial. And their densities goes to square root of 2 as n goes to infinity. So what Tom and I prove is the density for any such cone has to be greater than square root of 2. And then the Simon's cones show you can't do any better than that, because there are cones whose areas are as close to square root of 2 as you like. Now we also proved some sharper bounds in terms of topology. So remember the definition of being topologically nontrivial is that one of the components is non-contractable. Well, if a component is non-contractable, it means it has some non-vanishing homotopy group, some nontrivial homotopy group. And so you could then try to ask for bounds in terms of which homotopy group is nontrivial. Instead of just saying nontrivial, you could say, well, suppose it's not simply connected, or suppose pi 2 of it is non-zero, or so forth. And it turns out our methods let you answer that question as well. So take a simple area minimizing hypercone. And suppose one of the components of complement has nontrivial k-thomotopy. So of course, by being topologically nontrivial, it means this high path will be true for some k. But now suppose you fix the k. In terms of the k, you get the bound well. It has nontrivial k-thomotopy. Then its density has to be greater than an explicit number, which I've written down here. And dk, a kind of odd looking number at first. But anyway, and again, the constant is optimal. So for instance, if you take the infim among all cones where at least one of the components of complement is not simply connected, and then the infim would be this number d1. OK. And by the way, this number, dk is sort of odd number with pi and e in the denominator and so forth. Well, some of it's natural, like the sigma k is the area of the unit k sphere. So that seems kind of natural in this setting. But where does e come in? It's not so obvious that it should come into this at all. But this dk has a very nice meaning, which you'll see in a minute. OK, so I want to basically show you the proof of this theorem, the square root of 2 theorem. And so it uses just a few facts that I want to tell you about. So one of the key facts is, I've already talked about the density, but an important property of the density is that it's upper semi-continuous. The density of a surface, now we're always talking about minimal surface, not arbitrary surface, but for minimal surface, for things that are critical for the area functional. The density is an upper semi-continuous function of both m and x, the surface and the point. So if you have a sequence of minimal varieties, mn converging to m and xn converging to x, then you have this thing, the density of the limit is greater than or equal to the limit. I guess you'd say a little soup of the, but anyway, yeah. So when you pass to a limit, the density can jump up, but it can never jump down. And just pick your favorite one, weak convergence, like verifolds, you know, but flat. Because it doesn't use any way to go strong convergence by standard regularity here. Well, it doesn't. It doesn't really use, this is sort of pre-regularity. This follows just from monasticity and nothing else. So any sort of notion of weak convergence would work here. Weak convergence of verifolds or flat convergence of integral currents, whatever. So for instance, of course, this is minimal, not area minimizing, but one, if you look at an immersed surface that you can see that, I mean, you can see an example of it, at all the points where it's embedded, the density is 1. So if you take a sequence of points moving to where it's immersed, the density jumps up to 2. So that just illustrates how it can be upper semi-continuous as a function of the point x. But to give an example of it being upper semi-continuous as a functional surface, let's consider two, say, lines. So one-dimensional minimal surfaces in R3, namely straight lines. So if you take two straight lines in R3, that's a minimal variety. And if you move them together, density is 1 everywhere, but if you move them together, suddenly, boom, when the density jumps up in the intersection point to 2. So anyway, those are examples where you see the density can jump up. But as I say, you can prove it's very easy, follows from the Monte-Histi formula, that the density can never jump down. So it's upper semi-continuous. Not continuous, but it's upper semi-continuous. So that's one key ingredient in our proof. Another key fact is a beautiful fact about minimal hypercones proved by Hart and Simon back in 1985. So to take any simple area minimizing hypercone, then that cone is part of a foliation. And it turns out that cone is one leaf of a foliation of the Euclidean space by area minimizing hyper surfaces. Of course, the cone is singular, but all the other leaves are smooth. So I say foliation, but there's exactly one leaf that has exactly one singular point. So schematically, say here's your cone, then remember, it doesn't look like this in the picture, but this should be dividing the ambient space into two components. You should think of this and this as being the same component and this as being the same component. So it says the complement, it's part of a foliation, so the complement, here's the leaf on one side of the cone. And part of the theorem, if I stay here, the leaves are all smooth radial graphs. So if you take a ray from the origin, it intersects this leaf exactly once. So these surfaces are all radial graphs. And furthermore, the ones on one side of the cone are all related to each other by dilation. So once you have one leaf, you get all the others just by dilating that one. So you have a bunch on one side, you have a bunch on one side, and then you have different leaves on the other side. Why are those leaves area-minimized? Well, they were constructed to be, oh well, there are several ways of saying that, they constructed it by, I mean, they constructed leaves by minimizing area, but it's also a true fact that anytime you have a foliation by minimal hyper-surfces, that automatically implies they have a very strong area-minimizing property. So sort of by construction, but even if you didn't know their construction, once you have this foliation, you could conclude, oh, they have to be area-minimizing in a very strong sense. OK, oh well, here I drew a picture of it. It's the same picture I just drew, but the cone, except it's sideways. But anyway, there's the cone, and it's part of this foliation. OK, then the third thing is we use mean curvature flow. So mean curvature flow, you start with some manifold with boundary. People usually do it with manifolds without boundary, but you can do it with boundary. And for us, we need it to do it with boundary. So let's start with a minimal surface boundary. And well, if we do it without boundary, you just take a surface, and then you let it evolve so that at each point, its velocity is equal to the mean curvature. So if you take a curve at each point, like I've drawn here, each point where the curve curves one way, it starts moving the other way. So it moves for the curve, it just moves with velocity equal to its curvature, so it starts moving along. Now, as I say, people, when they study this problem, they usually do it for closed curves or closed surfaces. But you can just as well do surfaces with boundary. And in that case, if you have a boundary, well, you could just fix the boundary. You fix the boundary and let the surface move. Or you can also prescribe the motion of the boundary, just like when you saw the heat equation, you subscribe on the bar, you prescribe the temperatures, the endpoints, and the heat equation tells you what happens in between. Same thing here. You can move the endpoints however you like, and then the equation will make the curve move accordingly. So you can prescribe the motion of the boundary, and then the equation, velocity equals mean curvature, gives you a flow of the surface as away from the boundary. OK, now those turns out, again, even if you start with a smooth surface, as it flows, it's likely to develop singularities. So there's a regularity theory for mean curvature flow, just as there's for minimal surface. There's also a monotonicity formula due to Gerhardt Hueskin, and there's associated notion of density. So at a spacetime point, so this flow, you should really think of it as happening in spacetime. There's a point. If a singularity happens, it happens at a certain point x at a point t. So you have a singularity in spacetime, and there's an associated density of that singularity. The formula is more complicated, so I won't write it down. The monotonicity is more complicated, but it has the same general properties as the density we talked about for minimal surfaces. So in particular, the density for mean curvature flow, it's more complicated. I say the formula looks a lot more complicated, but it has the same properties. In particular, it depends upper semi-continuously on both arguments, on the spacetime point, where the singularity happens also on the flow. It's upper semi-continuous. And another thing that's important, so it's a little confusing here. You have these two different notions of very different looking definitions of density, but fortunately they're compatible in the following way. Let's take a minimal variety, minimal cone or anything else. Take any minimal variety. Well, if you let it flow by mean curvature, it just sits there, because it's mean curvature is zero everywhere. So if you take any minimal variety and let it flow by mean curvature, it doesn't move. So any minimal variety you can think of as a solution of mean curvature flow. It's a solution that just a minimal variety is the same as a static solution of mean curvature flow. So a priori, that's problematic because now we have two different definitions of density. The minimal variety, we could give the definition I wrote down, or you could think of it as a static flow, and there's the more complicated definition I didn't write down. But fortunately, in that case, where both definitions apply, they give the same number. Again, that would be important to us. So now let me give you the outline of the proof of the main theorem I stated about square root of 2. So we start with a simple topologically non-trivial area minimizing hypercone. And now I want to intersect it with the unit ball. Of course, the cone goes on forever. It has no boundary. But now I want to intersect the cone with the unit ball. So I only look at the part of the cone in the unit ball. So now I have a minimal variety in the unit ball, and its boundary is this manifold in the sphere. So now I have a minimal con, but with boundary. And I think of that as a static mean curvature flow. So this is the way our proof goes. What we do is we approximate that static flow by mean curvature flows that are really moving by dynamic mean curvature flows. So this static flow, let me just call it lm. That's where the cone just sits there and doesn't move. But now we approximate it by a sequence of mean curvature flows where the surfaces are really moving. So we find a sequence of dynamic mean curvature flows that converge to this static one. OK. Next, we show that each of these guys has a singularity. So each of these guys has a singularity, space time singularity x in, at which the density is greater than square root of 2. And now, furthermore, we show that, well, so the singular is happening somewhere or another. But by passing to a subsequence, we can assume that these converge to some space time point x. But now by the upper semi-continuity of density, here we have a sequence of flows with singularities with density greater than square root of 2. That means by upper semi-continuity, the density of the limit flow at the limit point has to be greater than or equal to the lm soup of the density of the nth flow at the nth point. And this is, well, OK, I'm not going to quite get the greater than. I'll only get greater than or equal to. But each of these is greater than square root of 2, so OK. So and now, but then by the compatibility, this density is, in fact, the same as the density of the cone. So what we proved is there's some point on this cone where the density is greater than or equal to the square root of 2. But of course, all these points are regular, so they have density 1. So in fact, it's this point. So we end up showing that point of the minimal variety has density square root of 2. The point x must be the vertex because the density is one away from the vertex. OK, so in a nutshell, the basic strategy that's proof is you take the problem and you reduce it to a harder problem because the original problem is, you know, try to understand singularities of just static mean curvature flows. And that already seems quite hard. Now instead of reducing the problem, considering dynamic, general mean curvature flows seem much harder than minimal surfaces. But anyway, here, fortunately, it actually helped us instead of making things worse. So I'll show you that. So let me just elaborate a little bit. So the two questions, one, where do those approximating flows come from? And then in those approximating flows, how do you get this square root of 2 bound? So I think I have a slide with the picture anyway. So what we do is, let me help draw. So what we do is we start with this, as I said, we look at our cone, the portion of our cone in the unit ball. And now remember, it's part of that foliation by minimal area minimizing hyper surfaces. So what we do is we take the leaf on one side that comes very close to the origin, maybe distance. So we'll start, I'll say, with this. This is a leaf on one side that comes maybe distance 1 over n to the origin. And now what I want to do is I want to describe this approximating mean curvature flow to you. So what we do is up to time 0, like from time minus infinity to time 0, this is our flow. It's just a minimal surface. It just sits there. It doesn't do anything. It's mean curvature is 0, so it has no reason to move. It would just sit there forever if we let it. But remember, we're allowed to move the boundary. So now at time 0, we start moving the boundary. And we move the boundary until, so we actually move the boundary through where those leaves intersect the sphere. So we start moving the boundary. And as we move the boundary, that forces the surface to start moving, too, by mean curvature flow. And we move our boundary until we end up on the boundary of the leaf that's on the other side. Let me just draw dotted lines here. So I'm not moving the surfaces directly. I'm only moving the boundaries. And then mean curvature is deciding what mean curvature flow decides what happens to the surfaces. But I start here, and I move the boundary until I get to here, and I stop when I get my boundaries on this leaf that say it's distance 1 over n on the other side. OK. OK, now by the way, so I'm moving the boundaries just through where the leaves intersect the sphere. But then I let the surface move by mean curvature flow. The surface definitely will not, that flow will not be along those leaves, because those leaves have mean curvature 0. They wouldn't move at all. So as soon as I start. Why is the Ln plus on the picture? Sorry? Oh, sorry. Let me see. Ln plus is where we. Ln minus is, let's see. Yeah, we start on Ln minus. So the way I was drawing it, this is my starting point, Ln minus. And then I move, and it'll flow. Where is Ln plus? OK, so yeah, OK, so sorry. Ln minus is the leaf on one side that's distance 1 over n from the origin. Ln plus is the leaf on the other side that's distance 1 over n from the origin. So what I want, what I claim we can do is find a mean curvature flow that starts with this one, and as t goes to infinity, goes to that one. And the way we do that, remember, what we have control over is what the boundary does. So I'll just take this, I'll take the boundary of my starting leaf, and I'll move it till it's at the boundary of what I want my limit leaf to be. But again, I don't get to control what happens inside. Mean curvature flow does that. But it turns out, if you do that, if you do what I just said, you start moving here, and then you stop when you get here, then as t goes to infinity, you can prove the flow does converge to the leaf Ln plus. So here, the colors probably don't match up at all, but anyway, the initial surface is the leaf on one side, and the final surface is t goes to infinity. It flows to the corresponding leaf on the other side. OK, so it turns out it's not hard to show, by the way, that it does flow to this, because all those leaves kind of act as barriers. That's the only thing that it could possibly flow to, basically, and so it flows to that. OK, but now, this is where the topologically non-triviality, you remember our hypothesis was that the cone was topologically non-trivial. That implies easily that this leaf is not isotopic to this one. So that means you have this deformation, and some, they're not isotopic, so when you flow from one to the other, there had to have been a singularity somewhere in between. So there has to be a singularity. OK, that's just saying the same thing. That's repetition of the earlier thing. OK, so y squared of 2. Well, so as I said, here, at first, it seems like we've made things worse. Now we need to know, we just wanted to prove a bound of square root of 2 for minimal varieties. Now we need to prove, we want to say the same bound for general singularities of mean curvature flows. But these are very, this is a very special kind of mean curvature flow. It's called a mean convex flow. So notice, when we started moving the boundary, we moved it in one direction, towards one side of the surface. When we kept on moving it always in one direction, sort of positive multiple unit normal. And that implies by a maximum principle, actually, if these surfaces, as they flow, they're always, the maximum principle tells you that the mean curvature for these moving surfaces, once they start moving, will always be positive. The surface always move in one direction. And that's a very, very strong, nice condition. So in general, we know much less about singularities of mean curvature flow than we know about minimal singularities of minimal surfaces. But the one case when you have a pretty good understanding of singularities of mean curvature flow is that mean convex case. For mean convex flows, we roughly have a classification of singularities. Singularities are all self-similar shrinking spheres or cylinders. So what that means, if you take a surface flowing by mean curvature, if it's always moving in one direction, if the surface are mean convex in a singularity forms, if you look at it, look at the surface under a microscope just before the singularity, what you either see is essentially a sphere or a cylinder. So the sphere case, for instance, there's a celebrated theorem of Fuskin, one of the first things theorems big breakthroughs. If you take any, say, convex body, convex body in Arjen and let it flow by mean curvature, he proved it shrinks to a point. And as it shrinks, it becomes rounder and rounder. So it's asymptotically spherical. So again, as I say, if you look at it under a microscope right before it disappears, what you see is essentially a sphere. And another kind of singularity would be if you have, well, suppose you have a very, very thin torus of revolution. That's also mean convex. The mean curvature is always pointing inward. So as it flows, it always moves in one direction. And of course, by symmetry, it shrinks until it becomes a circle. And again, if you look at it under a microscope right before it shrinks away, you see basically a cylinder. Maybe it can be more common. Wouldn't have to be if it's a asymmetrical torus. It's sort of thin, or it's a torus, mean convex, but it's thinner on point. This part will shrink away before that does. So you get a kind of neck pinch. But again, we know that when the singularity forms, you sort of have a cusp. And if you look under a microscope just before the singularity, what you see is essentially a cylinder. So that's what I mean when I say they've been classified. For mean convex flow, the singularities all look like either shrinking spheres or shrinking cylinders. And you know what those Houston densities are at those things, namely the density at one of these. If you have a shrinking cylinder, so that's an sk cross r something another, then its density is given by this formula. And in particular, these numbers are all greater than square root of 2. So again, we had our flow. It flowed around. We knew just topologically that a singularity had to happen because it's a mean convex flow. We know roughly speaking what that singularity looks like, we have a classification. And that singularity had density square root of 2. Sigma k is the area of the unit k sphere. Well, that's a purely local thing. So it's the same. If you're away from the boundary, which we are, it's the same. So anyway, so we got that the density is greater than dk, but dk is greater than square root of 2. So that's the theorem. So one interesting thing. So of course, this theorem, I said it's sharp. In a sense, it is sharp. But in some sense, I mean, it's sharp only because we're taking infimum over all dimensions. If we looked in a particular dimension, if you looked at minimal hypercones in R10 or something, our result would not be sharp. So in some sense, it's asymptotically sharp. So if you take the infimum over all dimensions, all minimal source and all dimensions, then it's square root of 2. But it's kind of surprising that bounds are very good, even in low dimensions. So for example, let's take the one that says, it gives you a bound, if one component of the complement is not simply connected, then the density is greater than this number, d1. I guess I erased a bit. If one component of the complement is not simply connected, then the density of the cone has to be bigger than d1, this number is a 1. Well, the smallest known example of an area minimizing hypercone that has that property is the cone over s1 cross s6. So I guess that's a eight-dimensional cone in R9. Anyway, that's the smallest dimensional example we know that satisfies the hypothesis. Its density is 1.523, and our bound is 1.520. So the bound is rather good, even though we only prove its bound is optimal when you go off to infinity. But it seems rather good, even in low dimensions. And everything I said is for area minimizing hyper-sources. But actually, by modifications, we can say some things about cones that are not area minimizing. And for instance, the method's implied that the density of a simple singular minimal hypercone in R4 is greater than or equal to 1.52. And of course, by Marquez, I said codo, but it's really Marquez. Marquez and Neves, the best constant is 1.57. The constants are not bad, even in very low dimensions. OK, so let me just mention some interesting up-and-questions. So conjectures or whatever. So one is that it seems reasonable to guess that these Simon's cones achieve the minimum in any particular dimension. We only know that in the case of n equals 1. That's a recent result, again. It's part of the solution of the Wilmore problem. Well, if you're in an odd dimension, that cone doesn't exist, but Sn cross Sn minus 1 the cone over, that seems like a good candidate. Again, now if you look at cones, if you again look in a fixed dimension and look at the cones where one of the components, say, has non-trivial k-dimensional homotopy, again, these conjectures are all basically that the minimum should be attained by a sub-manifold that's the product of two spheres. Just because those are kind of the simplest ones. But anyway, in our theorem says that asymptotically, those really are the best. But we don't know their best in any particular dimension, except in the case of two dimension in S3. Another problem is higher co-dimension. Our techniques, I can't imagine any technique is very strongly for hyperserves in many ways. I have no idea how to begin to attack this kind of thing in higher co-dimension. Of course, Jean's result was for higher co-dimension. But if you talk about higher dimension and higher co-dimension, I have no idea. And then one thing that says in our theorem is, explain why the proof gives good bounds. So this is something that I find completely baffling. We had this proof. And we were very happy to get any bound whatsoever. OK, you do the proof and you get some bound. But the proof said nothing about Clifford Torai or anything. It almost didn't seem to have much to do with any particular minimal surface. So you get some bound. And then it seems like just sort of by a miracle it happens that that bound is sharp. But I have released asymptotically sharp. But it doesn't seem like they could just be a coincidence. But as they have no, from the proof, it just seems like a miracle that happens to happen. The proof gives you lower bounds. And then there happen to be these examples that give it lower bounds. Usually when you prove something as sharp, you know, it proves a sharp inequality, sharp isoparametric inequality or something, then you can kind of see how this critical case, all the inequalities, become equalities or something. But here it's not like that. Anyway, OK, I'll stop there. Thanks. I'm a little bit confused about the k-dimensional homotopy and the bounds. You're saying that if one of the components has non-trivial k-dimensional homotopy, then the bound is the function. The lower bound is the function of k, which goes up. Yeah. Which goes up with k. Sorry, it goes down with k. But does it go down with k? Yeah, so for instance, d1, and you know, there's numbers, the formula is kind of complicated. So if you just look at the formula, it's not very obvious. But d1 was, I think, about 1.52 roughly. And sort of d infinity, the limit of these d's was square root of 2. So yeah, if you restrict your classic cones, yeah, you should have a bigger infeemum rate. Questions? Do you think there are trivial, airy minimizing cones? Topologic trivial? You know, I would kind of guess, just because why not. But I mean, I have no idea. I have no reason to think one way or the other. Your guess is good to mine. What do you think? Yes, I agree. There are, you know, were you shown, and some other people, made examples of minimal hypercones, where in fact, the link is minimal, topologically trivial. In fact, the link is topologically a sphere. But those are not known to be airy minimized. So I mean, there are some counters. There are already things that exist that might be examples of topologically trivial hypercones, but who knows. OK, well, I didn't really tell you where a k came in. I told you the square root of 2. OK, so let me just go back a minute and say, well, what if you started with a topological cone that was topologically trivial? Well, you could do the same argument. You could still make this flow. But if it's topologically trivial, these two leaves are isotopic. So for all you know, the flow, there still exists this flow that takes you from one leaf to the other. But maybe it's smooth for all time. So that's where we have to use the non-triviality. And for the square root of 2 thing, that's all we need to know. We just need to know there's some singularity. And since it's mean convex, that density is bigger than square root of 2. But if you know something about the topology, not just that the topology changed, but you know something about the way the topology changed, then you can say more about the singularity. So I mean, yeah, so it's kind of like the Morse function. It's something about the manifold that tells you something about what kind of critical points you have. So it's the same idea here. Not only do you have a singularity, but certain. You know these singularities are all like shrinking spheres and cylinders. But for instance, if you took a shrinking sphere, that wouldn't change the fundamental group. So knowing something about the topology puts a restriction on which of these tells you you have to see certain kinds of singularities and then you get a better bound for that reason. Yeah? You get the things for lower dimensions by just knowing which dk could possibly become involved in your construction. Well. Is that all it is? I mean, is that for dimension 3 or 4 or whatever? Well, OK. First of all, unfortunately, we don't even have area-minimizing guys until you're up to seven dimensional and r8. So this whole program sort of doesn't make sense until you're in high dimensions. Because just the smallest dimension in which you get a singular hypercone for area-minimizing when it's seven dimensional and r8. Yeah? How do you extend the flow through the singularity? For those kind of things, I mean, how do you have a singularity if you have some problems? Yeah, that's a great question. So you repeat the question. OK, the question is, how do you extend the flow past the singularity? So mean curvature flow has been studied a lot in recent years by different people. And there's sort of two groups of people that answer that question in different ways. One group says we don't. So you can solve it by PDE. You can just look at it as sort of looking at PDE. And so people who are mainly using PDE generally study the flow up to the singular time. The first time a singularity happens, then they stop. But fortunately for us, there's a good week. There's several different week notions of mean curvature flow that lets you talk about something being a flow even after the singularities happen. Now, there are several different definitions. And unfortunately, they're not equivalent definitions. But fortunately, when you're dealing with mean convex surface, those different definitions all agree in that special case. Anyway, there are ways to talk about weak solutions. And then you can flow right past the singularity. And of course, let's see. Yeah, I think for us, it's really you have to be able to do that. Actually, if you just want the square root of 2 result, it turns out you really wouldn't have to go all the way. You just have to know a singularity happens. And that's good enough. You don't need to keep going. You say that, well, I can't flow forever without a singularity because that's a topological contradiction. So singularity happened. For the argument I showed you, it would be enough to just work up to the time of the first singularity. But if you want to get that sharper thing about the bound depending on k, depending on the topology, then you have to consider these weak flows. Because the problem is, let's say we flow around. So if this region is not simply connected, pi 1 is not trivial, then what you need to know, this is a true fact, that at some point, you may have many singularities. But you need to know at some point, this is a true fact, that if pi 1 is not trivial, then at some point, you had to have an s1 cross r, whatever type singularity. And that means you had to have a singularity with density d1 in that flow. There's no reason that has to be the only singularity, the first singularity. So as you're flowing along, maybe the first singularities are s2 cross something or s3 cross something. So you have to keep on, you can't stop there. You have to keep on flowing until you get to the d1 singularity. So for the first result, you don't need to flow past the singularity, but for the second one, you do. Further questions? Let's thank the speaker again.