 Let me begin by thanking the organizers for the invitation to spend some time here and to speak with the conference. So it's been a great pleasure to be here. OK. So let me first write down the equations I'll be studying. And so there are two equations I'm interested in. The first I'll call the Gross-Bitievsky. It has a parameter epsilon. And it looks like this. So the scaling is here's the epsilon. I'll also consider the elliptic counterpart. So I'll call it Ginsburg-Landau with the epsilon. This is a bit of a misnomer. OK. And then the setting for all this will be, say, in three dimensions in a cylindrical domain. So omega cross 0L, where little omega is a bounded open subset of R2, say, simply connected. In fact, we can take it to be a ball, if we like, for most of the talk. And notation I will use rather consistently is I'll write points in omega in the form x and z, where x is in little omega, and z is the vertical variable. OK. So these are my equations. For concreteness, let's say I'll consider, for example, so this is the more interesting one. I'll consider for here, say, Neumann boundary conditions on the lateral boundary and periodic in the z direction. And here I would want probably some Dirichlet data, for example, somewhere. OK. And so let me start by recalling some relevant quantities. So these would include, for example, the mass or rather density. There is a momentum, which I'll write in this way. So I'll consistently abuse notation in this fashion. So I'll write a dot product here between two complex numbers to indicate the real dot product. So I can identify the mass of vectors between as vectors in r, c. And so there's really the momentum density, the free energy. And so this is the quantity whose integral gives the conserved Hamiltonian for the gross pitiase equation. And let me write this as, I'll write little j to denote the momentum. I'll call this e sub epsilon of u. And then finally, I'll define the vorticity, which I'll write as capital J of u. I can't call it omega because I called the domain little omega. Capital J of u is 1 half of the curl of the momentum density. And it's a short calculation shows that this is, in fact, equal to, again, if I identify u as an r2-valued map with components u1 and u2. Then this will just be the, in three dimensions, this is the cross part of, say, the real part, the grade in the real part and the grade in the imaginary part. OK. And so what this tells us, interestingly, is that, of course, this gradient u1 is orthogonal to level sets of the real part. This is orthogonal to level sets of the imaginary part. And so the vorticity is orthogonal to both of these normals. And hence, it's parallel to level sets of the complex value function. And so if you like, traditionally, one defines a vortex filament as a integral curve of the vorticity vector field. Here, because of this geometric property I've just described, such integral curves will just be level sets of the complex value function, where things are not as generic, and so on. OK. And so there are a couple of conservation laws, a number of conservation laws that I will recall. And so, for example, conservation of mass has the form. OK. So this is the continuity equation. Time derivative of this is the divergence of that. There's a conservation of momentum. And I believe there's a 2 here. And then, OK. So I form the 3 by 3 matrix whose ij entry is the real inner product of the i-th derivative and j-th derivative. I take the row-wise divergence. And then there's also a gradient term, which I won't care about much. The reason I won't care about it is I'm about to take the curl of this thing. And so to understand how the vorticity evolves, I just take the curl of this equation. And I'll find that d dt of the vorticity is, well, this disappears. And I have curl of divergence of grad u tensor, grad u. OK. So there's a nice evolution equation for the vorticity. And finally, the energy is also conserved formally. And it'll be conserved with the boundary conditions I have here. And so for example, this tells me that the total energy is constant. I'll be, when I study the Gross-Petitius equation and actually also when I study the elliptic equation, I'll be interested in situations where the energy is of order log epsilon. This is natural for reasons we'll see in a second. And so if the energy is of order log epsilon, in particular, this u squared minus 1 is quite small in L2. And so it's convenient to think of this in this way. The u squared minus 1 over epsilon is, I guess, logarithmic in L2. And so this tells me that the divergence of j is small in the regime I'm interested in. It's of order epsilon log epsilon, maybe. It's also true, in fact. We'll see that in this situation, in this regime where the energy is of order log epsilon, that the vorticity has a huge amount of structure. And there are actually two ways of thinking of this. One is that when I study the evolution equation, I'll put this structure in my initial data. But more generally, it's true that merely under the assumption of small energy, the vorticity has a huge amount of structure. And so in particular, it'll be concentrated along one dimensional curves in 3D. And away from those one dimensional curves, it'll be close to zero. And so then this tells us what the divergence of the current is small. The curl of the current, the vorticity, again, has structure. It's basically zero in a lot of places. It lives on curves and so on. And so we have a very good understanding of both the diversions in the curl of the current density. Or we will when I tell you more about what this is doing. OK, so then let me describe a special solution of both of these equations. It's simplest to say what one refers to as a vortex solution. And so a special solution will be one who depends only on the horizontal x variable. So this is independent of z and of t. And so for every integer d, let's write u upper d. Depending only on x, depending on the parameter epsilon, we'll have the form of a modulus writing it in polar coordinates. So it'll be e to the i d theta. And then a modulus here depends on d and scales like r over epsilon. OK, and so if you make this ansatz, then you get an ode. Let's say f d of 0 is 0 f d increases to 1 as the radial parameter goes to infinity. OK, and so if you make this ansatz, you get ode for f d. You can easily solve it. You can find this in a number of ways. Let's then remark some properties. And so the energy density of this, so this away from the origin, this looks like it's homogeneous of degree away from the origin. It's basically this. So on a scale of order epsilon around the origin of a ball over epsilon, the modulus will be less than 1. Away from the origin is basically e to the i d theta. And so the energy of that will be like, say, a cut off on a ball of various epsilon times d squared over, so if you just compute the energy of this, you get d squared over r squared. I think there's a factor of 2. And so for example, the energy, if I integrate this over a ball of various epsilon, over ball of various r, excuse me, over a macroscopic ball, I'll get something, I guess, d squared times log r over epsilon plus big O of 1. OK, this is the reason, if you like, the basic reason I'll be interested in this logarithmic energy scale. This is the scale on which one sees these. So one thing, so this is being a basic vortex. And this logarithmic scale is the one on which vortices appear. Any questions? No? Excuse me, there's no z dependence on your special solutions? Here, no. That's right. And so this is really a 2D solution. It's really 2D. Yeah. And so you can think of it as being a cylinder, as being a translation variant in z direction, and in the t direction. Another attribute of these solutions is if I look at the vorticity, well, this is really going to be, I guess, pi times d. And then what's left will be an approximate identity. It'll have the form. And so this is really an approximate identity. OK, it is a smooth function who will be, OK, so this is determined by the exact form of this f. And maybe it'll depend on d. But this is a non-negative function with integral 1. And we scaled in this fashion. OK, and so we can see here that for this special solution, the vorticity is basically a smeared out delta function around the origin with multiple d pi times the integer d. And the energy scales like d squared. And what follows will always be interesting d equals 1. And we can see very naively from this that if I call this a vortex, then a vortex of degree 2 costs order 4 pi log 1 epsilon energy. And so 2 vortices of degree 1 are much cheaper energetically than 1 vortices of degree 2 is worse. So vortices of degree plus minus 1 have strong stability properties, which are not possessed by other vortices. OK, and so the ansatz I'm interested in is the following. So I want to look for solutions of these two equations. What I would like is I want to, my ansatz is, I want to look for u epsilon depending on who looks something like, well, a product of, I take the basic, say, degree 1 vortex on a scale epsilon on each horizontal slice. And then I translate it by an amount depending on the vertical and time variables, f, i of z and t. OK, and so then if I look at a single product, this is just the basic vortex translated. And I multiply them together, I'll get a. And so the picture is this should look like, and sorry, I also want to put a, I want it to be a small translation, so it'll be a small scale H epsilon. OK, and so the picture is. Sorry, all these are all the D equals 1. Yeah, these are all D equals 1, that's right. So these will all have the same sign in particular. One, yeah. So the pictures on the small scale H epsilon, you hear there are these vortices. And if I look at the certain height z, I will see these structures centered at points f1 of z up to epsilon of z scaled into the small tube. OK, so this is what I am interested in. This is the situation I'd like to study. And let me say at the, so one might imagine writing a solution and also let me add, if I have boundary conditions, I may want to multiply this by a global phase function in order to satisfy those boundary conditions. OK, so one might imagine writing a solution of either of those equations as something of this form and then an error term and then doing a linear analysis. For whatever reasons, well it's just an empirical fact that this argument has not been carried out successfully for any of the problems, for essentially any of the problems I'm considering here, especially for any three-dimensional question for any of these problems or any similar problem. And so somehow linearizing has not been successful for these questions. But this is the picture to have in mind. So this ansatz will not actually appear in any proofs. OK, and so in some way, the story I want to describe begins with work of Del Pino. This was mentioned by Valerio Banica yesterday and Kowalczyk from 2008, it was published. And so they show essentially I'm going to state this a bit imprecisely. But they show that, well let's look at the following quantity. I'll define this G sub E of. So let this be the integral over the domain of the total energy density. And I want to rescale it in a certain way. So I'll subtract off, it turns out the energy will divert on a logarithm of the scale, pi n minus, and I guess I need l here as well, l being the length, being the vertical height of the domain, and pi l log epsilon minus n, n minus 1, pi l log of the smaller scale h epsilon. So in fact they show that in some sense the correct scale to take will be h epsilon is log epsilon to the minus 1 half. I'll tell you why this is an interesting scale in a moment. Then there's some constant that depends on n and the domain omega. OK, so essentially they define this and they said that if we consider G epsilon of the above ansets, and so built into this re-scaling is the fact that I'm considering something with n and vortex lines, the parameter n is there. G epsilon of the ansets is equal to, I guess, pi over 2 fi. OK, something over i. And then a logarithmic interaction term. So some involving i and j different. I think there's a pi here also, log of fi minus fj of z, the whole thing in red with the z variable. And so their theorem was that this is true up to a little over 1 error terms. And they showed also that these little over 1 error terms in some way depend smoothly on f on the lines f. And I'll call this G0. This is the. So OK, and just one or two remarks about this. Of course, the choice I make for the phase here will affect the constant I get there. And part of the point of the paper of Dill-Peyton-Covacic was to make choice of the phase, which is, I guess, they didn't prove upper bounds. This is only lower bounds. They're computing the energy of a test function. But to make a choice which is presumably a good one and close to optimal, that makes this term as small as possible. And so in some way, they identified the right constant here, which is connected to the right phase there. OK, and so based on this one can conjecture that, well, I think they either conjectured or became close to conjecturing, that there should be solutions of the gross PTSD equation. Well, I will say for the elliptic equation. One would expect one might hope, at least, to see solutions to the elliptic equation, which have roughly this form for f, a map from 0L into n fold copy of R2, for f being a critical point of this energy function, or for f a critical point here. And one might further hope to have solutions of the gross PTSD equation of this form, where f solves some kind of shorting or equation associated with this Hamiltonian. OK, so this is the, I think, the first conjecture was explicit in the paper of D'Alpino and collaborators. And the second one may have been implicit. Sorry, I should say, Kovacic and collaborators. So, and let me remark also, there's an earlier, I think they were, in some way, motivated by an earlier paper of Montero Sternberg and Ziemler, who showed that for suitable geometries, so they were considering the elliptic problem in certain 3D domains. The domain looked like this. Imagine a surface of revolution generated by this figure. And so it doesn't have to be a surface of revolution, or a 3D domain of revolution. But the point is that you want a geometry force. There's a local minimizer of the arc lane functional, connecting two points on the boundary. And so what they showed is that there exist solutions of the Ginsburg-Lando equation for small epsilon, such that the energy density divided by the rescaling, divided by the diversion log factor, converges to n pi times house dwarf one-dimensional measure, restricted to the segment, say L here. And similarly, the vorticity converges to a welded vector 0, 0, 1, if you forgive abuse limitation times the same thing. And so one understands this as saying for these solutions, one has n quanta of vorticity concentrating around this curve. And so I think part of the motivation of these authors was to ask, well, on smaller scales, what's the fine scale structure of these solutions? OK, and let me say also a main tool in the Sternberg and in the Monterrestor Museum or theorem were earlier results of myself and Sonar and Alberti Baldo Orlandi. OK, so then let me state the main results of this talk, which are the following. So the first is joint work with Andres Contreras, which addresses this. So really, let's say, as a corollary, it will address the elliptic problem. And let's say, so assume that I have a sequence U epsilon in H1 omega complex valued functions with the following two properties. So the first will say that I want to see, say, n vortex filaments concentrating on this. So from now on, H sub epsilon always denotes this log epsilon to the minus 1 half length scale. And so I'd like to see, let's write it this way. So at each, if I look at the, all right, over here, I guess J sub x of U is just the Jacobian with respect to the x components. And so this is the horizontal gradient of U1. Well, if you like, it's the determinant of the horizontal gradient of U. U is a real value as R divided by n. So I want to, I'd like this to look like a bunch of delta functions to leading order concentrated near the origin. So W minus 1, 1 in omega d z. Let me ask this to be, OK. And so this tells me that if I look at, so theoretically, if I look at a given slice, then on the average, I'll see n vortices within this H epsilon scale of the vertical axis. And we assume this, and I'll also assume that this G epsilon of U epsilon of the energy rescaled in the above fashion by subtracting off the correct diversion part, that this is bounded by some constant c2. So I want these to hold uniformly in epsilon. So the conclusions are that after passing to a subsequence, we have the, let me do the following. So I'd like to rescale them in the horizontal directions. So let's let v epsilon of x and z be U epsilon of H epsilon x and z. And so v, and so v is, I've expanded the domain. I've moved the H epsilon scale out to order one. This satisfies that the horizontal vorticity of v converges, say, in W minus 1, to a measure who is concentrated on the graph over the vertical axis of n H1 curves. And so this converges to, I guess, pi times the sum. And so what I mean here, I guess this is an expanding domain. I mean, for every, if I fix a big ball, then for epsilon small enough, the rescale domain will contain that ball. And so I mean, for example, on W minus 1, one of a big ball across 0L for every r. OK, so after passing to some things, there exists f such that this holds. And moreover, whenever this holds, then the limit for the energies, g epsilon, is greater than g0 of f. OK, and so this is, so one can view this as basically a lower bound to complement the upper bound of del Pino and Kovacic. This shows that the upper bound of del Pino and Kovacic is sharp. In other words, for any f I can construct a sequence u epsilon such that this holds in such a heavy quality here. And a few other conclusions, which I won't write down at the moment. OK, and so this is the first theorem as a corollary. Let me state this not too precisely. So the corollary would be that in our geometry that there exist solutions of the Ginsburg-Landau epsilon equation. OK, so here to be precise, I'm considering Dirichlet data on the top and bottom and Neumann data on the sides. And the point of the Dirichlet data will be to force the presence of n vortex lines near the origin. OK, so there's just solutions of this equation such that, if you like, these such that after rescaling in this fashion, the rescaled jacogons converge to, and I mean, so I guess I mean, right? And so this measure has this form of a sum of delta functions on every z, and then I integrate in z direction. OK, and so this tells us for these solutions, the vorticity is indeed, and sorry, and f I is a critical point of the limiting function, indeed a minimizer, subject to suitable boundary conditions. OK, and so this says that indeed, while not giving the kind of sort of precise point-wise description in terms of the ansatz that Kovacic and Del Pino may have had in mind, this does say that for these sequences of solutions, the vorticity is concentrating around curves who minimize this limiting function. OK, that's theorem one. And the second theorem is a dynamic theorem. And so, and this is joint work with D. Day-Smiths, which, so these are different 16s. That 16 means it was submitted, I mean, it was posted recently. And the 16 means it's quite underway at the moment. So this is, we've, and so I have to thank the organizers for this, I mean, so this stay in Paris has given us the opportunity to make progress on this problem. OK, and so theorem two says, let me consider, oh, I guess I should make remarks about some remarks about this. And so, OK, without writing down names, let me just mention there's a huge amount of related work on the Ginsburg-Landau equation in 3D. And so in some, I guess I have to, so a lot of this follows from, say, seminal work of Bethwell-Brasil and the early 90s on the Ginsburg-Landau equation in 2D. People who can, and so the general picture in 3D is that one, is that one epsilon is small, energy and vorticity concentrate around lines. And more generally, in n dimensions, where n greater than or to 3, energy and vorticity concentrate around co-dimension two minimal surfaces. This is the general picture. And so this has been studied, for example, major conversion food in more or less chronological order, Bethwell and Riviera, Linn and Riviera, Bethwell-Brasil-Orlandy, Bethwell-Borghain-Brasil-Orlandy, Bethwell-Orlandy-Smiths, et cetera. So these all made important contributions to this analysis. And so, of course, in 3D, minimal surfaces line. And what this is doing is giving a description of the fine-scale structure in the case of higher degree concentration of vorticity around a line. This is also related in a way to, so another parallel is with the scalar analog of the Ginzburg-Lauhn equation, so the Allen-Kahn equation. And so there's a phenomenon of what one calls interface clustering. And so there are two interfaces are connected in some way to a minimal surface. And one can have multiple interfaces clustering around a single minimal surface. There's a lot of rather reasonable work on this by people including Del Pino-Covaltic Way. Well, let's say Del Pino-Covaltic Way, Peckhardt, and basically various subsets of, in fact, subsets of those permutations that a few other authors have contributed also, as well as a paper of Covaltic Liu and Peckhardt, or a paper of Del Pino-Covaltic Way and Yang. And so a lot of recent work on interface clustering. This is the first parallel result in hard dimensions, showing Vortex clustering. So theorem two then would be, and so henceforth, I'm going to always state things in terms of a rescaled variable v rather than u. And so let me write down what the equation. I'm going to start at the gross pity of the equation. Let me write down what it looks like after it's rescaling. And so I'll have, say, I guess I want to. So the rescaling is anisotropic with respect to x and z, giving rise to certain logarithmic factors in certain places. OK. And so this will, for example, give rise to rescaled versions of these conservation laws and rescaled versions to conserve quantities. So I'd like to assume this. I'd like to assume that the vorticity initially concentrates while it has these properties. I guess it's not written here. So this converges. We can make a conversion. It's a precise essence. Doesn't matter, but let's say w minus 1, 1, or this topology over here. This will converge to, OK, so we have something like this. And again, I keep on forgetting this. OK, so at every height, I have n vortices. And then I integrate in the z direction. And also I want to then assume that I have the minimal energy possible given this conservation addition. And the minimal energy is given by this. So g epsilon, after rescaling. When I rescale, I get some logs in certain places that this should converge to g0 of this f0. And here, f0 denotes is a smooth solution. And say h4 is enough, continuous into h4 of exactly the Hamiltonian system associated to g0, which is exactly the model that Valerio Banica spoke about yesterday when we were talking. So f0 solves t, f0 minus the second. And there are components, j. OK, so I assume I have a smooth solution of this limiting system. And I assume that at time 0, the energy and vorticity are concentrating around the curves who give the initial data for this solution. The conclusion is that then for t here, again, the vorticity converges to the vorticity associated to the solution here, f0 at time t tensor dz. This is true. And similarly, the energy also converges the same thing, in fact. OK, let's leave it at that. OK, we have more conclusions, which I've written down. And so what this says is that, right, it says this. So in other words, this is a derivation of the equation studied by Valerio Banica yesterday. And so let's remember from Valerio's talk, this was first formally written down by guys in the fluids community by Klein, Maida, and Damodar in the mid-90s following earlier work of Zakharov in the late 80s. And basically, the way these guys argue is why I've matched formal asymptotics. And so this is, as far as we know, the first, or I mean, it's not even the first, because it's not finished yet, but we hope this will be the first rigorous derivation in any setting of this model for the dynamics of thin, nearly parallel, vortex filaments, starting from an equation with fluid dynamical rodentite. I should have said, the gross piti-fc equation in principle describes superfluids, and so quantum mechanical fluids. And so rather than looking at, let's say, a classical ideal fluid as described by the older equations, we're considering their quantum counterparts, which in some ways, the analysis is much easier in the quantum case in the gross piti-fc case. OK. And so let me note here, in this setting, all the vertices have the same sign. And so I would like to be able to do this for vertices of opposite sign. That's what one sees in these trailing vertices of airplanes, for example. I mean, vertices at the same time, actually, appears to be more relevant in the quantum context, but who knows. And then, so concerning this equation, Valeria summarized yesterday a lot of history, and so this includes work of, well, Klein, Mara Damodar, who derived it, then, Kenick, Ponson, Vega, lots of work of Banek and Mio, also recent work of Walter Craig and Carlos Garcia as Piatia. And I think there's also work of Garcia, Piatia, and Ize. And so one can sort of mention all the, essentially, everyone who's worked on this in a sentence or two. OK, and so in the remaining eight minutes, I'll say a bit about the proof. And so here's how this goes. And so the overall point is that the proof relies incredibly strongly on, well, say, on things in the spirit of theorem one, which is here, I think. So in theorem one, we assume only some kind of crude knowledge about the vorticity. We have n vortices concentrating on a certain scale about the origin, about this line. And we assume that in energy bounds. And then we deduce various conclusions. And so in the dynamical setting, we will choose initially such that these hypotheses hold. And if we can show these persist in time, then we have access to these conclusions. And further conclusions in the same spirit. And so a large part of this, there's a theorem three, so to speak, which is, again, we'll be in the paper of myself in DGA, which is sort of a refinement of these results under the same or similar hypotheses. We extract still further information about the behavior of the vorticity in these solutions. And so about the proof. If we consider, let's call this the interacting vortex filament system, so here's the point. So suppose f and f naught solve this. Then it's an easy computation to show that this is bounded by a constant times this. OK, so what I want to do, I like to do the same thing if possible for if f naught solving this. And f obtained from the gross pity FG equation in the limit as epsilon goes to 0. And so let me suppose that I've somehow arranged that at every time these things, these conditions hold at every time in some interval. And so at every time I can pass the limits along some subsequence and get some limiting f. That will depend on time. If I have some kind of aquacontinuity in time, I can do this, I can find a signal subsequence and I can take it into versions as long as time interval. And so if we try to mimic this computation, we'll find that, well, for reasons that will become apparent in a moment, I get this. But then I get errors arising from the fact that this is not the right equation. I have to do some approximations. And these errors look like, and when I say f comes from this, I mean under my hypothesis, under my hypothesis of rather sharp energy bounds of welfare data. OK, there's some extra energy here. And so the idea is that if it's possible that, so I have a lower bound here, the limiting vortex elements can have at most this much energy. They may lose their energy. Energy may leak from the vortexes into the rest of the fluid. And that's kind of an enemy you have to be controlled. One thing we can do however is we can replace this by the first variation by a linear term. Let's write it like this. And then I get quadratic terms, which turn out to involve only the L2 norm, not the derivatives. And so I can absorb the quadratic terms into here. And I get a linear term. OK, but then I'm still left controlling something who's linear in f by something who's quadratic. That doesn't look very good. However, it's also true if I go back to this system that for the linear term in question, one has this estimate. And again, this is straightforward. And so the point is then, well, when I go over here and again do the same approximation arguments and so on, I'll find that for f coming from the gross PTSD limit, this is bounded by the same expressions. And so at this point, I'm able to do a grung wall inequality and proceed. The, of course, all of work has been obtained in this assertion that I can do this. And so let me say a little bit about that in the remaining three minutes, if I can. So then, what I want to do then is to use this equation. The equation tells me how the vorticity. Remember, the vorticity is telling me where the curves are. And so let me multiply this rescaler, I think, in terms of v instead of u. I multiply the vorticity by test function phi. And after the rescaling, I'll find, well, there'll be a term phi sub t. There'll be a term epsilon. So here i and j are, well, let's write it like this. I have the horizontal derivative of phi. And then the Hilbert-Schmidt inner product with something, the tensor product of v perp with v. And then I have dz, the horizontal derivative phi, and the dot product with, OK, the anisotropic rescaling gives rise to log ups on here. I'd like to pass the limits in this expression. And what I'll do is I'll take a phi, who, for example, looks like 1 half x minus f not of z t squared, OK. And so then I better sum over the components and multiply by cutoff functions, OK. And so then formally, at least, well, actually, then it's really true that the phi times the Jacobian converges to as long as the limiting curves are supported in the region where these cutoff functions are 1. This will really converge to the L2 norm, the L2 norm squared. And similarly, this will converge to what we know what this is doing. This will give us some different pieces of that computation. And so we have to, in particular, we have to know what this is doing. And so it's a fact that this is converging. So this is one of the things proved in the theorem 3. I didn't write down. This is converging to dz. So if I fix a time t, then this will converge to, so it'll converge to measure supported along the curve, along the vortex curves, involving the vertical derivative of the limiting curve. OK, so these come from the limit. And whereas this involves, if you look at it, this involves where the cutoff function is equal to 1, this is just equal to minus dz of f0 of the solution. And so here we have, so this will give rise to a term involving the orthogonal gradient of dz of f0 and dz of f, the thing I obtained from the limit. This is the hardest term by far. And I don't have time to talk with that. So let me stop here. Thank you. So is there another geometric context of this? Maybe some domain which is not cylindrical. Maybe some other geometric, interesting geometric context. Well, all right. So I'd say for Grosby-Tegesky cylindrical or an unbounded cylinder, or all of R3, are the natural contexts. And so as I think Valerius said, this limiting system has been studied most on the real line. And so with that, we'd need a domain which is unbounded in the z direction. That would be possible. For the elliptic equation, so there what's much more interesting is exactly the interplay between the geometry of the domain and the geometry in the limiting solutions. And so for example, one could presumably do the same thing, but it would be harder. And so one could look for, if I have a three-dimensional manifold with a closed geodesic, one could look for solutions with n filaments conserving on small scales around the geodesic. But that's going to be a bit harder. I actually have the same question, but just slightly different from my own purposes. You put derrically bounded conditions at 0 and L. Can you put periodic? You can. So this corollary says, really, if we have a local minimizer of the limiting functional, then we get local minimizing solutions of the epsilon functional. And so if I just have n curves, if they're not linked in some way, then they'll always want to spread apart and lower their energy. So there presumably are, let me mark also, this is. So the derrick conditions allow you to fix the curves at the end? Yeah, that's right. In principle, they could also be fixed by nodding. So the limiting equation in the elliptic case is just this without the time derivative. And if you look at it, this is really a planar n body problem, where z plays the role of the time variable. And so people have constructed solutions. We didn't, however, the thing we didn't do was to identify nodded solutions who are local minimizers of some energy, which is what we need to do to relate them to these. But the time dynamics allows you to give initial data and to follow them for a period of time. And that wouldn't cost you. You wouldn't need a minimizer. That's right. We wouldn't.