Non-Uniqueness Of Locally Minimizing Clusters - Anna Skorobogatova

Added: 2026-04-29

[00:00:14]It's a pleasure to have Anna Skorobogatova from ETH and she's going to talk about non-uniqueness of minimizing local minimizers.

[00:00:28]Okay, thank you for the introduction for the introduction and yeah, for the invitation to be here and so great pleasure to be back in Princeton.

[00:00:36]Yeah, so I'll talk about some non-uniqueness results for optimal cluster problems where so you might know the classical multiple bubble problems, you have a bunch of finite volume chambers that represent your bubble cluster and then you have one infinite volume chamber representing the complement of the cluster in Euclidean space. And I'll talk about a variant of this problem where you have more than one infinite volume chamber and the differences that can occur in this framework.

[00:01:03]Um Okay, so uh And this is all based on the joint work with uh uh Lea Bronstein, Robert Neumayer, and Mike Novack.

[00:01:26]So let's Let's start with the the setup. So I'll look at partitioning my Euclidean space into a bunch of chambers which are sets of locally finite perimeter.

[00:01:46]So I'll call this pi and this will be you have a bunch of chambers which are finite volume and a bunch which will be infinite volume.

[00:02:11]These will be And then I want to look at local minimizers of interfacial perimeter subject to the volumes of the um the finite volume chambers fixed.

[00:02:33]So perimeter I got it.

[00:02:43]by two because I was counting everything twice.

[00:02:50]Uh I got it.

[00:02:54]Uh so this is the reduced boundary or Yeah, I think it's probably everyone is familiar with what that is. Um Okay.

[00:03:07]Subject two.

[00:03:19]And so So you're just comparing it on finite sets, right? They agree there. Uh yeah, I mean we'll look at it so like the uh the competitors are always differing uh outside uh sorry, they're they're agreeing with my set outside outside of a compact set. Yeah.

[00:03:38]Okay, and so uh Yeah, already here so I kind of underline one of the wrong equations that you can see. So yeah, I mean maybe I can draw just a picture. So a priori you're not and I mean sort of the hope is that a posteriori you recover things like connectedness for instance for your local minimizers. But yeah, this really isn't clear a priori so that one chamber might have different disconnected components.

[00:04:05]You have some finite volume, some infinite volume.

[00:04:10]So you know, maybe this uh one This could also be a piece of pi one.

[00:04:20]This one.

[00:04:21]So um Okay, and so yeah, already so the underlying criticality conditions here will be that for for the interfaces that bound any finite volume chamber, you're going to have constant mean curvature.

[00:04:39]So it's CMC but because there the volume constraint is sort of coming into play.

[00:04:48]Um And then uh whereas for for any interfaces between pairs of infinite volume chambers, you just have minimality so mean curvature zero because there you're not seeing some kind of volume constraint.

[00:05:15]I'm confused and the in that picture you can draw the this interface this perimeter will be infinite, right? Is that right or what?

[00:05:24]Uh yeah, but when we're doing the comparison of perimeter, we have we're comparing with uh competitors that agree outside of a a compact set. So then you're only computing you'll have something finite. So what do you mean by local minimizer?

[00:05:42]So I mean it's uh the perimeter is going to be better than that of anything that agrees with this guy outside of a compact set.

[00:05:52]So then actually what you're computing in the end is finite.

[00:05:56]Uh >> [laughter] >> what is that? Um Excuse me.

[00:06:09]Okay, and so um Let me Let me start with sort of what has been known already for a while. So let's start with some simple cases. So already when you have just one finite volume chamber and one infinite volume chamber, this is the classical isoperimetric problem.

[00:06:31]It's well known that that's a unique minimizer once you fix the the the volume of your finite volume chamber, that's just a Right, now let's if we have still one infinite volume chamber but more than one finite volume chamber, this is the classical multiple bubble problem.

[00:06:59]So Um Um two one And there's been a lot of attention on this problem and which decades I won't uh Yeah, I won't discuss sort of everything that's been known to date but there's a well-known conjecture by Sullivan from the '90s I think which says that if you don't have too many of the finite volume chambers then there's a unique there should be a unique configuration for your local minimizer which is the so-called standard n-bubble which yeah, I also won't define precisely but you can define it by sort of taking a bunch of equidistributed points on a um on an n-sphere and then taking a Voronoi partition of these points so you're basically yeah, you're decomposing this sphere into connected components that are sort of equi-partitioning these points and then you can stereographically project this onto onto your Euclidean space. But you yeah, so you'll see configurations like for instance this will be like a double bubble where you have It's a terrible picture but you should also you should always have like in low dimensions equal angles of 120 degrees.

[00:08:37]And you always have pieces of spherical caps for the for the finite volume chambers. So and this is as long as so when uh the number of your number of finite volume chambers is at most n plus one because after this it's sort of not clear. So for instance if you have So for up to three bubbles in the plane it's sort of clear that then somehow you have the should be a uh It should be something like this. Again, this is a local picture.

[00:09:14]>> [laughter] >> Um But then sort of if you try to have four in the plane already it's yeah, it's kind of not so clear where you'd want to put the fourth one.

[00:09:22]Um I think it's local minimizer one And yeah, there's been there was a lot of results towards this in low dimensions so in the plane and also in 3D uh at least for sort of low numbers of uh uh of chambers already like a couple of decades ago, but there's the one result I want to mention which is was really a ground breaking ground breaking result towards resolving this problem is by Milman and Neeman in 2022.

[00:10:20]Where they show that this conjecture is true whenever the number of finite volume chambers is at most the minimum of So this covers the triple bubble and all three, the quadruple bubble and all four.

[00:10:40]And yeah, so it's kind of almost reaching the the optimal boundary exactly on the number of the number of finite volume chambers. And so the so they also have a result that about a year after this I think where they also show that in in R5 the quintuple bubble is a minimizer.

[00:11:08]But they don't have uniqueness.

[00:11:11]That's okay.

[00:11:15]Okay, so yeah, I guess at least for this when you have one infinite volume chamber yeah, it's sort of expected that when you don't have too many finite volume ones you really have this unique unique minimizing configuration. And so then um the natural thing to ask next is okay, what if you have more than one more than one finite volume chamber.

[00:11:39]And um now uh and I'll start so from now on I'll mostly restrict myself to the case where I have the simplest case where I just have one finite volume chamber and these two infinite volume chambers and that one.

[00:12:02]Because we'll see that already in this case things start to become far from trivial when the dimension becomes high enough.

[00:12:09]And so yeah, so I should say so and aside from this being a natural a natural generalization of this classical multiple bubble problem it's there's also some kind of physical motivation for considering this problem which comes from the structures formed by copolymer chains. So inside I should mention is right.

[00:12:52]There arises kind of suitable uh sharp interface and blow up limits of so called triblock copolymer melts.

[00:13:09]Uh where these are This is a model that represents uh Yeah, structures that can form by copolymers where you have three different phases two that uh occupy much more space than the third where you see structures a little bit like So it's uh Here there's kind of an energy that you're minimizing which is it's a bit like for for instance for the liquid drop model which people might be more familiar with which is also a model for diblock copolymer melts so the two phase version of this where so these are kind of like two different uh two different phases occupying a lot more space than this small one.

[00:14:04]And yeah, here you're the kind of energy that you're minimizing is you have a um perimeter which is like a local attractive energy, but then you also have um some kind of long range repulsive energy that's not local and there's a kind of balance between these two for minimizers. And yeah, it turns out that somehow if you start sort of suitably blowing up around places like this you kind of in the end hope that you see structures like this where you have two infinite volume chambers and one finite volume one. But yeah, that's kind of just something I'll say that there is some sort of motivation for for studying this problem.

[00:14:53]And the yeah, maybe the the starting work on this was from a few years back by Almgren and Taylor Greene Greene who studied this problem initially in the plane and they showed that in R2 there is just like for the classical multiple bubble problem there exists a unique local minimizer which is so called standard lens and it's unique up to the up to rigid motions.

[00:15:43]And the yeah, I mean the picture is exactly like this one. So you have a planar interface between the two infinite volume chambers and this is your finite volume chamber which is so here you have equal angles of 120 degrees.

[00:16:01]And uh yeah, and the the the finite volume chamber is bounded by two circle arcs symmetrically about this this hyperplane well this line interface.

[00:16:13]Okay, and then so this is in the plane and then the natural thing to ask is what happens in higher dimensions.

[00:16:25]And yeah, what we've seen so far is that we have we have what we expect uniqueness for local minimizers in general it seems.

[00:16:49]What's the natural upper bound that you would look over here? So Well [clears throat] >> Like like because you have V and V greater than or equal than two. Now here you're taking them exactly equal to four.

[00:16:59]>> Ah, sorry.

[00:17:01]>> [laughter] >> Sorry, sorry.

[00:17:03]I should say that from now on I'm considering just this case of three.

[00:17:07]>> [laughter] >> But yeah, one thing I would not mention that is known I said yeah, I was meaning to mention this actually. So yeah, basically in above 2D we basically know nothing as of yet.

[00:17:22]Um but and and this is kind of in general so for uh so also not two there's another there's a result by Novaga and Milani and Tortorelli also from 2023 who fully characterize the possible local minimizers when the sum of the finite and infinite volume chambers is at most four.

[00:17:58]And they show that the local minimizers are either so there are What are the possibilities? Well, firstly you have the the classical I mean you have this standard and bubble I guess when you have two This was already known by which are all I think already before them. So you have this standard Well, that was for the when you have three three finite volume one infinite volume.

[00:18:33]And then there's yeah, also when you have two this was yeah, basically the this case was already known before then anyway.

[00:18:43]and bubble Or what are the other possibilities? You have the lens.

[00:18:51]And then you have the so called peanut where you have two finite two infinite.

[00:18:58]So this is something like this. Okay, this is a terrible picture.

[00:19:03]These should be circle arcs.

[00:19:09]So here again you have equal angles of 120 degrees.

[00:19:14]Also here.

[00:19:16]Or there's a case where you have one finite volume chamber three infinite volume chambers which is the right for love.

[00:19:27]Uh where this is something like like some kind of compact body.

[00:19:45]But yeah, other other than that we yeah, other than in the plane we basically don't know and that's something we want to investigate, but yeah, it seems there are some difficulties in higher dimensions of these book what I'm about to mention that we know in the case of exactly two infinite volume tubes.

[00:20:04]Okay, so All right.

[00:20:11]So now let me stick to when there's yeah, really one infinite one finite volume chamber two infinite volume chambers. So in this case in higher dimensions so there was first there was a result by Front side and Novak a couple of years ago that shows that up to ambient dimension seven the the standard lens is still the unique local minimizer up to rigid motions. Now you're I mean instead of circle also taking spherical caps that again meet symmetrically about the hyperplanes interface.

[00:21:03]Still the minimizer.

[00:21:14]Okay, and then right.

[00:21:17]The natural thing to ask is why why stop at seven? Well, uh yeah, essentially it turns out that unlike everything I mentioned so far from starting from ambient dimension eight uniqueness is no longer true and we stop being able to sort of take we stop being able to produce other examples and this is what we were able to show last year.

[00:21:43]So we are Robin and Mike.

[00:21:55]Uh and we can now do this in uh yeah, so uh we can do this in a large number of dimensions starting from eight uh now I'll explain this second so when n is between uh eight and seven numbers starting from uh there exist local minimizers that are not the lens.

[00:22:35]And so so firstly uh the reason for this is there's a computer assisted aspect to our proof which we don't expect that this is a general this is a real upper bound to the dimension we can go up to. Our hope is to remove the computer assistance, but yeah, in the end we things come down to a competitor comparison where we have to compute energies explicitly and although the things we're computing these energies for have a lot of symmetries uh nevertheless I mean it comes down to computing integrals that boil down to some special hypergeometric functions which the hope is we can use good enough estimates for these functions to to actually show that we have this for all dimensions starting from eight. Uh but yeah, for now uh this is as far as we can go and I'll say a bit more on the computer assistance uh later um and I should say that it's specifically the the eight dimensional case pretty much simultaneously treated by Navarro Collignon and Tralli.

[00:23:44]So eight eight And so Okay, so what is what is the point here?

[00:24:05]Well, the the key underlying observation is that if you take any one of these uh uh locally minimizing on two clusters and you blow it down then the interface uh the the blow down of the interface is going to be an area minimizing hyperplane.

[00:24:24]And there's a well known result by Simons from uh 68 which says that all area minimizing uh hypercones in Rn when n is at most seven are planes.

[00:24:52]And the the characterizing property of the lens is that it blows down to a plane.

[00:25:00]Uh but then um but then starting from um starting from dimension eight it's no longer true.

[00:25:15]And there's the well known example of Simons cone which is part of a larger class of cones that I'll come back to later known as the quadratic or Lawson cones.

[00:25:28]And eight there exist area minimizing uh house hypercones that are singular and not plane.

[00:25:47]And the um the the class of examples is these what's called the graphic cones which are you take uh you seek it out.

[00:26:05]So these will be so I split so this will be n the hypercone and Rk plus l plus two.

[00:26:18]k plus one plus l plus one such that so they're given by quadratic varieties where x squared is k over l y squared.

[00:26:33]Um yeah, the particular case so k is l is three, this is Simons cone and it's due to Bombieri and Giusti that this thing is minimizing.

[00:26:48]Um but it turns out that Giusti and Um yeah, but it turns out that that these start so starting from when the ambient dimension is nine these are always area minimizing. Whereas specifically in dimension eight you have Simons cone and you have the cone uh you have k is two l is four or vice versa. Those are also minimizing uh and then you have the one other singular possibility which is k is one or l is five or vice versa. This one uh I believe is not minimizing uh it's minimizing to one side. It's stationary stable, but not minimizing to That somehow doesn't matter so much because we basically need just one such cone in every dimension in which we want to prove our result in the end and hopefully that'll become apparent soon enough.

[00:27:46]Um okay, um and what I should say is that so and uh this will become clearer when I discuss the ideas of the argument, but in improving this theorem other than saying that we can find something that's not the lens, we can basically say nothing else about these minimizers we produce.

[00:28:06]So the way we'll yeah, the way we'll show that we have something that's not the lens will be precisely because of this uh this fact that as I said the lens blows down to a plane and we'll produce something that blows down to a a singular area minimizing cone that's not a plane uh in each of these dimensions.

[00:28:23]Um but yeah, that'll hopefully become clear from the argument that uh sort of at least at this stage that's all we can say, but uh I'll hopefully have time to sit uh to discuss what we're now able to say in work in preparation with Mike and Robin uh where we can we can show some more precise properties. For instance, we can prescribe we can prescribe the cones that we blow down to uh and for these local minimizers we produce and we can also produce things for instance that at least to our kind of surprise they don't necessarily stick to the cone that they blow down to.

[00:29:07]Okay, so um and now let me discuss uh ideas.

[00:29:13]The proof Okay, so so the idea is that we'll we'll start we'll do this through a uh a kind of partial concentration compactness procedure where sort of classically in concentration compactness arguments you'd like to basically fully characterize the way in which you have loss of compactness which in our case will appear as uh loss of mass at infinity which is really kind of the thing that we have to care about which in classical multiple bubble problems is not an issue um because there there's we don't they don't have this infinite interface which we now really have to uh take care of.

[00:29:57]Um but yeah, we don't need to so it seems like we can't really easily fully characterize the ways in which which mass can be lost at infinity, but all we actually need to do is rule out the one enemy situation which will prevent us from producing the local minimizer that we want that blows down to uh one of these singular uh cones and this one enemy situation will be that all the mass flies off to the to infinity looking like a lens because yeah, this will be in the basically there's nothing we can extract uh in this compactness procedure that will give us a local minimizer with the structure we want. So how do we do this? Well, we start we take uh some very large scales RK to RK going to infinity infinity um then in this very large annulus between scale RK 2 RK we prescribe some conical data which is for now it's just going to be an area minimizing hyper cone which uh yeah, these always for for hyper cones these always dis- disconnect the ambient Euclidean space into two connected components which yeah, is perfect for us to start modeling minimizers off of these uh and so then um and yeah, for now it's just going to be arbitrary in the end we'll take it to be something very specific in each dimension and then we solve so we solve a penalized minimization problem so rather we so we look at locally minimizing the perim- uh perimeter plus a penalization term which I won't uh write down rigorously, but the point of this penalization is basically that since we're fixing this sort of hard boundary data we don't want like we don't want pieces of our finite volume chamber to really attach somewhere here because this would be bad because then we can only yeah, basically we can only take one-sided variations here and this will be a problem.

[00:32:11]So we've prescribed this penalization such that basically uh the the the local minimizers which are called chi K in these large balls that are going out to infinity uh we want this to >> [clears throat] >> to not attach the boundary of of the previous one.

[00:32:39]Uh and nevertheless we want it such that in the limit in we extract sort of a limiting limiting local minimizer um will this will this will at the same time go to zero so in the end we get just a local minimizer from to and it's a plane.

[00:33:04]Which yeah, one can do. And so yeah, in the end we're getting some kind of minimizers that are doing something inside and they have some pieces of the the finite volume chamber doing something that we don't yet know.

[00:33:24]Okay, so what do we do now? Well, yeah, because as I said a priori we don't necessarily know that we have connected this we really yeah, we really have to consider this part this really bad possibility where everything is just drifting off to infinity and so what we can then do is basically decompose we can decompose the pieces of our finite volume chamber into kind of regions that we call concentrations but sort of by how we define them they're going to have they're going to be um so maybe I'll call these concentrations chi K I and so we decompose one and two equations I and it turns out it's possible to do this in a way that basically you can always ensure a uniform uniform bound on the number of these uh and all of them have non-degenerate volume so such that the meaning the yeah, the volumes NK are at most some V0 so um and we have that the distance between pairs of these K is defined by how we set this up going to be going to when I is not equal to J and also we can ensure that they have uniform diameter bounds. So these are all each one is contained in a some uniform ball around some point XK I so and uh I K >> [clears throat] >> I I Okay, so we've kind of decomposed into our finite volume chamber into these pieces and yeah, they might be so the one situation we have to we have to rule out is that each one of these pieces is is looking lens-shaped and they're all disappearing off to infinity because what we'd like to do is basically extract a local a local limit of at least one of these uh subsequentially that blows down to a singular cone and yeah, the the the problematic situation is if we can't do this for any of them because all of them blow down to to planes which would is what would happen in that case. But you got a uniform lower bound. You I mean you're able to do that.

[00:36:36]Yes, uh because hm? Uh [clears throat] yeah, because we have density estimates on there are lower density estimates. So we yeah, you can't have like infinite fragmenting into lots of tiny pieces.

[00:36:48]Is that obvious that you don't I mean like these uh Uh I guess this is kind of a yeah, I mean it's a essentially the standard like uh nucleation argument where you take uh some uh volume fixing variations. I think this was already done like the the classical um >> Okay. like multiple bubble problems.

[00:37:11]Um yeah, I guess because yeah, I mean everything we're doing is in codeimension one and we have boundary data as well.

[00:37:25]Uh Okay, so what do we do?

[00:37:32]Yeah, so we need to rule out the situation as I said it's uh all these separation pieces uh pieces uh are all lens-shaped Okay. So Okay, so in this case uh what we can first do is if this happens we can say well then it's actually first cheaper to combine all of them into just a single concentration. So all of these are flying off to infinity looking like lenses then it's actually cheaper to throw away this and this kind of replace them basically by by planes and then put everything into so instead of this we have just one larger lens-shaped piece. So this becomes a much better something like this. So we're pasting this into here.

[00:38:46]Uh so we put like a single volume one so instead separate and three we say so step one cheaper to have a single one say I think one And this is just by a a concavity argument basically you're exploiting the isoperimetry to say that it's cheaper to do this.

[00:39:27]Okay, so then once we have this, that's where the that's then where the compactness comes in. So until now we didn't use anything about the I mean we were taking arbitrary conical boundary data now finally we need to prescribe it to be something very specific because now we'd like to say, okay, well, if we now just have a single lens-shaped piece flying off to infinity, actually this isn't optimal either because we can find a competitor we can find a better competitor to this specific situation where all of the masses now combine into one piece.

[00:40:01]So, uh there exists a cheaper lens.

[00:40:06]And this we can we need to model this on a specific cone.

[00:40:12]Because this is a cheaper competitor than dimension eight.

[00:40:21]Plus um modeled precisely on these uh quadratic cones. So, we we take a quadratic cone and each one of these dimensions I'll draw what the competitor looks like.

[00:40:38]A second. So, Okay, so what is the What is the question? Well, So, in even dimensions we can just take really the like generalized Simons cones and then yeah, for simplicity in all dimensions we took we differ the these K and L in these quadratic cones just by one. But because yeah, it makes the computations easier, but I think the hope is that um Yeah, I mean Yeah, the hope is that we can really generalize this in dimension and I think probably yeah, also come up with cheaper competitors modeled on at least the other quadratic cones.

[00:41:18]Uh in the yeah, um in these dimensions. So, um what does this competitor look like?

[00:41:29]So, these cones they have a lot of symmetries. They have SO K uh uh they have SO K plus one cross S alpha plus one symmetry. So, if you put you do this in the you can basically boil things down to looking in the plane where the axes are these two uh yeah, these two components in which you're defining this quadratic cone.

[00:41:57]And let me just draw this for the Simons cone for simplicity. So, you can also just do this in one quadrant, in which case so, this is going to be uh C K alpha uh C take K.

[00:42:10]Oops. And in this case the cone forms a really 45° angle, but yeah, for when you when K and L are different, they just have a different angle with the uh with the axes.

[00:42:23]And we basically construct this competitor to be circle arcs that meet at 90° here and here and at 120° on the cone.

[00:42:35]So, something like this.

[00:42:38]And then you can Yeah, so you repeat this in all the in all the the quadrants.

[00:42:50]Um and yeah, you can also I mean, the point is that for for the Simons cones you really have the same the same circle arc because it's also symmetric between these two axes, but otherwise you're going to have two different circle arcs that you're doing for the other quadratic cones.

[00:43:07]Uh and I should say, yeah, and then this is where the So, then uh the computer assistance comes into really comparing basically the energy it costs you to uh to paste this competitor in sort of you take the cone and you paste this in around the origin of the cone uh in place of a little lens that's drifting off to infinity with the same volume.

[00:43:35]Um Um computer assistance.

[00:43:40]So, looking at the path energy with lens.

[00:43:52]Um yeah, I should say so the the what we use here is the um the Flint arc library implemented in C.

[00:44:01]So, I think it's also at least one of one of the ways in which uh one of the uh possible things used by people in fluids I think also when they compute when they do these computations for solutions that blow up, I think.

[00:44:17]Uh but so the point is that it it uses interval arithmetic, so it treats it treats numbers as intervals.

[00:44:25]So, with you can be sure um to a high level of precision you can compute things like for instance we use it to compute integrals. It computes things like this.

[00:44:35]Well, um Uh and yeah, to to a very high level of precision you can be sure that uh the two things will So, it's paired in our case uh um yeah, you really have a an accurate comparison between the two.

[00:45:08]Um And yeah, I should say that we don't think this Yeah, I mean, we're basically sure that this competitor is not actually the the optimal one because I mean, it's something that's uh constant in curvature in these coordinates, but you're changing the metric when you're because you're um in these So, when you when you change to these coordinates, you're then computing a weighted a weighted surface area.

[00:45:39]And so, I think in the end this doesn't actually give you something constant in curvature in your original in your original coordinates. Uh and uh a post-doc a former post-doc Dominik Stadler of uh Leah did a computation that's a I think it should be possible to find at least a critical point um where you solve an ODE to find which curve you need. Yeah, but yeah, so it seems like we have something specifically for Simons cone in dimension eight towards this, but yeah, we haven't really taken this much further. Um but I think it's doable.

[00:46:19]Um Yeah, partially because it's not clear that this critical point gives you something minimizing.

[00:46:25]Uh I think I'm out of time.

[00:46:27]Okay, so maybe now so uh maybe now I have a little bit of time to say what people do. It would be better than yours though, right? Sorry.

[00:46:40]It's probably doing better than that competitor though. Yeah, it's doing it will be doing better than It would be doing better though, right? Yeah. Yeah, for sure.

[00:46:48]Um Yeah.

[00:46:52]But I Yeah, it's not clear that it's really something minimizing, but maybe it is.

[00:47:00]Um okay, so Right, so from this argument basically all we got, so remember all of this was basically an argument by contradiction. We we assumed that the one enemy enemy situation, which was our one obstruction to producing uh um something that's not the lens as a local minimizer.

[00:47:21]Uh yeah, so we we supposed for a contradiction that this did happen, which in our case meant all the mass was drifting off to infinity, and then we showed that actually this cannot be possible. So, in the end we really we really keep we really show that at least one of these non-degenerate pieces of mass uh from our um our finite volume chamber will in the end produce something in the limit that that blows down to a singular cone, which is what we wanted because then this thing is not going to this minimizer this local minimizer we produce is not going to be a lens.

[00:47:53]But this gives us essentially no other information because uh one thing I I didn't mention is an additional piece of information we get from here, which is a consequence of how we we choose this um how we choose this penalization is that actually we get that these all of these pieces of the finite volume chamber are contained in a ball uh of radius square root a constant times square root of K about the origin.

[00:48:27]Which basically means that that all the pieces of the finite volume chamber are existing at infinitesimal scales relative to the scale at which we prescribe the cone. But this is bad because this at least this is bad for telling us anything about this limiting local minimizer we produce because we basically can't compare anything from the cone we prescribe at this much larger scale to the behavior of the the chamber in this infinite interface at this relatively infinitesimal scale.

[00:49:00]Um Nevertheless, what we can And so yeah, in particular I mean, so the one there's the issue that if the if the cone uh so, it could be that the cone we end the the this limiting local minimizer blows down to does not have an isolated singularity, and then already you have the issue of non-uniqueness tangent cones, but aside from this, yeah, there's also just this issue of that you you can't Yeah, you can't compare the information at these these incomparable schedules.

[00:49:40]Um but despite this, what we cannot say but for each one here in a couple of weeks.

[00:49:59]is so we can show for once uh in particular, we don't know that the cone we prescribed at this this large-scale uh comparable to our K is the same cone that we get uh in the limit, but actually now we can say this. So there exists local minimizers at least for for certain cones.

[00:50:25]Okay.

[00:50:26]One.

[00:50:27]Slow down.

[00:50:33]Prescribed.

[00:50:36]Okay.

[00:50:38]So this we can do uh um so we can do this for any uh any cone uh any area minimizing.

[00:50:53]Um isolated singularity as long as as long as it satisfies this uh this energy comparison. So like we verified this specifically for these quadratic cones, but as long as you can verify this for any other cone with an isolated singularity, you can actually make sure that if you prescribe this cone at this huge scale, actually the thing you get will will blow down to the same cone.

[00:51:20]Uh but that's fine.

[00:51:25]energy comparison.

[00:51:30]lens So you mean you only uh okay, you only need to be the lens. Yes. Yeah, but the lens. Uh the second thing we can do, which is I think a bit more surprising, at least uh it was to us, is that uh despite even being able to make sure that you can blow down to some prescribed cone, as I briefly mentioned, uh it doesn't need to be the case that the infinite interface actually attaches to the cone itself. So bonus.

[00:51:59]Um uh using this this uh we'll say this agree with You mean it's asymptotically it? Yes. So it's asymptotic to the cone and >> But it's not it. Yeah, and this we can do for Yeah, so you need area minimizing. We also need So we need uh so-called uh strictly area minimizing, which basically allows us to I mean essentially the information we need is we need to know exactly the the asymptotic behavior, and this we know for these so-called strictly minimizing cones, where you're basically looking at like the the Jacobi field of this cone, and you need to know you have like some positive ones that you're basically your this infinite interface you know exactly the asymptotics of this at infinity. We need We need this.

[00:53:07]Um So wait, you're claiming that it does not it really does not. So it is different.

[00:53:14]That's what you're proving. Yes.

[00:53:16]Okay.

[00:53:17]Uh so for uh agree I think cones again with the energy comparison.

[00:53:34]So I don't know, maybe I'll say a couple of words. Uh yeah, maybe let me just say the third thing we can show, which uh so and one of the cones which we can prescribe are also cones with with non-isolated singularities. So I guess what I described so far was this competitor comparison for these quadratic cones, which have isolated singularities.

[00:53:54]Um so uh the cone prescribed um this uh blow down and you can do this specifically uh uh which I think even having one example is already interesting.

[00:54:10]Uh it seems like the specific construction we have doesn't easily generalize to give something very general. We kind of have to do it by hand.

[00:54:19]Uh but we can do this for Simon's cones.

[00:54:22]Uh C33 cross R.

[00:54:24]So I think yeah.

[00:54:26]At least to us it seems Yeah, we think having at least one example with a non-isolated singularity is already somehow interesting in itself. Um But yeah, maybe let me say a few words about this.

[00:54:41]So uh So why can we Why can we do something like this?

[00:54:46]Well the cones for area minimizing cones with isolated singularities there is this so-called Hopf-Simon foliation think of in Simon '84.

[00:55:04]Uh so they show uh that there exists a Okay, I guess I'm basically running out of time. So maybe I just draw a picture of what's happening. So if you have this cone with an isolated singularity, there exists a unique uh foliation by smooth area minimizing surfaces either side of this cone, which is decomposing the ambient space into two connected components, where this foliation is given by rescalings of a single surface, and they're all asymptotic to this cone.

[00:55:37]And so basically what we do is we essentially show um that you Yeah, that it's somehow more actually more optimal for So if you prescribe at these super large scales, so to I mean we can run our construction, we prescribe conical boundary data, what I described, but you can also You can do this actually for general uh area minimizing hypersurfaces, you can prescribe that as the data. So if you prescribe the data being a leaf of this foliation at these super large scales, then actually it's more optimal essentially for the for this finite volume chamber to be I mean we don't show something as uh as precise as this, but it's more optimal for it to basically be looking like some lens type thing that's attaching to the leaf instead of somehow things sticking to the cone in the end. And so yeah, that's kind of what we're using.

[00:56:34]Um And that's also we also use this Hopf-Simon foliation in the the first case to to show that you can blow down to a prescribed to prescribed cones as well. But in particular, you can make an example which sticks to the cone, right?

[00:56:49]Actually this we can't seem to do. We thought this should be what we do. So you can actually make it a stick to one of the of the foliations, but not necessarily to uh So uh we don't have something Yeah, as I was saying, we don't have something as precise as it really sticks to the foliation. All we show is that it doesn't Yeah, so we show We don't give a contradiction to say suppose we produce something that is the same, and this cannot happen, yeah. There is maybe trapped between two levels of the foliation. That Is that what you It might Yeah, with this we don't know, but I I don't know. Naively I would hope that actually I mean I guess super naively I thought maybe we can even hope for cuz there's this result of uh Simon and Solomon that at least for quadratic cones for area minimizing hypersurfaces that are So from Hopf-Simon you get that if you have an area minimizing hypersurface lying to one side of uh any cone with isolated singularity, um then it has If it lies to one side, it's either the cone or it's a leaf of the foliation. But for quadratic cones, you can drop the one-sidedness, and you still get this rigidity, and this is the Simon-Solomon thing. And I kind of naively thought, "Oh, maybe at least for those cones we can get some similar rigidity, but I the infinite interface is either a a leaf of the foliation or the cone itself. But I guess the trouble is we do have this finite volume chamber, and so it seems not so easy to use these uh Yeah, it seems like this could as precisely be the reason we lose this rigidity, cuz maybe you can glue in something weird where you Yeah, I don't know. It seems At least for critical points I think it's possible to produce some weird critical point where Yeah, um but for minimizers maybe the second one, no.

[00:58:40]But yeah, I would really like to say that actually the only things we have for the infinite interface for these Yeah, but Yeah, so far all we all we have is essentially starting from dimension eight, it seems we have Yeah, not just non-uniqueness, but in principle the whole zoo of possibilities of what can happen. Um and so I guess yeah, that's what I can say. Thanks.

[00:59:11]Questions?

[00:59:16]So just a curiosity, can you tell us what is the personalization, especially the property that they're more interested in is the fact that the classes lie in the ball of square root?

[00:59:28]So Right. So I guess we're assuming the nation that's just acting on the finite on the one finite volume chamber and it's yeah, I mean basically we're taking some integral over this.

[00:59:43]What am I going to say?

[00:59:45]Yeah, but this thing of some function GK where we need we essentially need that the we just need that the Lipschitz constant of GK is something like uh uh one over square root of K and that GK is only kicking in starting from scale square root square root of K. So it's living yeah, it's kicking in above this.

[01:00:17]You integrate the weight which is actually kind of so it seems like yeah, these are kind of the conditions we needed to force both of these simultaneous. So you can also yeah, I'm particular sort of forces.

[01:00:32]Is there any specific reason for choosing like square root of RK? I mean you could choose other powers, right?

[01:00:38]Uh it seemed to us that we couldn't choose so we can this was somehow the we couldn't choose a a higher power. So this seemed like the largest power of RK but yeah, I don't know where Probably because it I guess the density estimate because the density estimate is sort of uniform in this perturbation, right?

[01:01:10]Yeah, but yeah, now I yeah, I mean at the time when we did this computation it seemed like this was kind of the the optimal thing that we could choose essentially.

[01:01:26]Yeah, so basically we couldn't yeah, we couldn't choose something that kicked in at any larger scale it seemed.

[01:01:34]So at least.

[01:01:40]Yeah.

[01:01:46]And the non-cylindrical example the one you have is specifically with a line. I mean if I asked you or like make a plane We tried so we tried both this and the yeah, >> [laughter] >> it's so basically we because we did this by yeah, I mean we taken the cross-section the competitor we have for Simon's cone and then we we somehow extended in the transverse direction. Okay, I guess like it's just yeah, we we didn't try super hard.

[01:02:15]Yeah, it might be something that we try at some point but just the yeah, the naive thing that seemed to work for Simon's cone cross R Yeah, I mean we really just took like a mod X type thing in the but yeah, it seems like doing something a little more complicated yeah, then we have to yeah, I mean I guess it's just that we then need to really implement this into again into this flint thing to do the computation whereas that one we could do with Mathematica even and it was fine but but we yeah, we didn't spend a huge amount of time on this but somehow yeah, it was not clear to us yeah, how yeah, whether it was really interesting to produce a huge generality of these kind of examples or yeah, this kind but I yeah, I think it's it's probably possible to produce something with that's flat more than just a line. I agree.

[01:03:09]And you don't recover out of your methods that um it's convenient to be connected. So this is not something you This is not generally known yeah, uh so yeah, I mean I think it's definitely expected but yeah, I think in other than in the plane we don't have this.

[01:03:35]So not even in R3?

[01:03:37]Uh as far as I'm aware, no.

[01:03:40]Um yeah, I guess I don't know maybe it's Yeah, I mean I don't know I it feels to me like it should be with like somehow with some isoperimetry one should be able to show that's true but I suspect it's not as I suspect people have tried this but yeah, somehow we didn't try to show specifically this but yeah, maybe it's not that obvious that Okay, I guess we can get tea and cookie, right?