Let me begin by thanking the organizers for the invitation to spend some time here and to speak at this conference.

It's been a great pleasure to be here. Okay, 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-Pitievsky. 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.

Okay, 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.

Okay, 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. Okay, and so let me start by recalling some relevant quantities. So these would include, for example, the mass or rather density.

There's a momentum, which I'll write in this way. So I'll consistently abuse notation in this fashion. And so I'll write a dot product here between two complex numbers, indicate the real dot product, so I can identify them as vectors in R2.

And so there's the momentum density, the free energy. And so this is the quantity whose integral gives the conserved Hamiltonian for the Gross-Pityasi 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 omega.

Capital J of u is one half of the curl of the momentum density. And 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 product of, say, the real part,

the gradient of the real part and the gradient of the imaginary part. OK. And so what this tells us, incidentally, 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 the 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, OK, where things are not degenerate, 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. The 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 the j-th derivative. I take the row as 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 that 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 the total energy is constant. When I study the Gross-Petithet's 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, then in particular, this u squared minus 1 is quite small in L2. And so it's convenient to think of this in this way. 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, is 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 0. And so then this tells us what the divergence of the current is, is small. The curl of the current, the vorticity, again, has structure. It's basically 0 in a lot of places. It lives on curves.

And so we have a very good understanding of both the divergence and 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, the simplest, say, what one refers to as a vortex solution. And so a special solution will be one

who depends only on the horizontal x variables. 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

in a polar coordinate. So it'll be e to the i d theta. And then a modulus 2 depends on d and scales like r over epsilon. OK, and so if you make this ansatz, then you get ode, let's say fd of 0 is 0. fd increases to 1 as the radial parameter goes to infinity.

OK, and so if you make this ansatz, you get ode for fd. 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 is basically this, right? So on the scale of order epsilon around the origin of order 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 cutoff on a ball of radius epsilon times d squared over, so the, if you just compute the energy of this, you get d squared over r squared. I think there's a factor of 2. So this is, and so, for example, the energy of,

if I integrate this over a ball of radius epsilon, over a ball of radius 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. Okay, this is the reason, this is, 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? 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 translation in the 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.

Okay, it is a smooth function who will be, okay, 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 one, and we scaled in this fashion.

Okay, 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, I'll always be interested in d equals one. We can see very naively from this that, if I call this a vortex, then a vortex of degree two

costs order four pi log one over epsilon energy. And so two vortices of degree one are much cheaper energetically than one vortex of degree two is worse. So vortices of degree plus and minus one have sort of strong instability properties

which are not possessed by other vortices. Okay, 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 one 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. Okay, 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 what it 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. Okay, and so the picture is... Sorry, are all these, I just got a little smeared,

are all these d equal one? Yeah, these are all d equals one, that's right. So these are all of the same sign in particular. One, yeah. So the picture's on a small scale h epsilon. Here are these vortices, and if I look at a certain height z,

I will see these structures centered at points f one of z up to f sub n of z scaled into this small tube. Okay, so this is what I am interested in. This is the situation I'd like to study. And let me say, so one might imagine

writing a solution as, 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. Okay, 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, for 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, for either 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.

Okay, and so in some way, the story I want to describe begins with work of Delpino. This was mentioned by Larry Banneker yesterday and Kowalczyk from 2008.

It was published. And so they show essentially, I'm going to state this a bit imprecisely, but they showed 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 them in a certain way. So I'll subtract off, it turns out the energy will diverge on a logarithmic 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 one 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.

Okay, so essentially they define this and they said that if we consider G epsilon of the above ansatz, and so built into this rescaling is the fact that I'm considering something with n, n vortex lines, the parameter n is there. G epsilon of the ansatz is equal to, I guess, pi over 2 f i,

okay, something over i, and then logarithmic interaction terms, so some involving i and j different. I think there's a pi here also.

Log of f i minus f j of z, the whole thing integrated with respect to the z variable. And so their theorem was that this is true up to a little over one error terms. And they showed also that these little over one error terms in some way depend smoothly on f,

on the lines f here. And I'll call this G0. Okay, 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 Del Pino and Kvosik was to make a choice of the phase,

which is, I guess they didn't prove upper bounds. This is only lower bounds. They're committing 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.

Okay, and so based on this one can conjecture that, well, I think they either conjectured or came close to conjecturing that there should be solutions of the gross PTSD equation. Well, let's say for the elliptic equation, 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 Schrodinger equation associated with this Hamiltonian. Okay, so this is the, I think the first conjecture was explicit in the paper of Del Pino and collaborators,

and the second one may have been implicit. Sorry, I should say Kvosik 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 Zeamer,

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,

rather a 3D domain of revolution, but the point is that you want a geometry force that's a local minimizer of the arc length functional, connecting two points on the boundary. And so what they showed is that there exist solutions of the Ginzburg Landau equation for small epsilon,

such that the energy density divided by the rescaling, divided by the divergent logarithmic factor, converges to n pi times Hausdorff one-dimensional measure, restricted to the segment, say L here.

And similarly, the vorticity converges to a world of vector 0, 0, 1, if you will forgive a bit of experimentation, times the same thing. And so one understands this as saying that 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?

Okay, and let me say also a, I won't write this down, but a main tool in the Sternberg, in the Montero-Sternberg Theorem, were earlier results of myself and Soner and Alberto, Alberto Baldo-Orlandi. Okay, so then, okay, and 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, 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, and h1 omega complex valued functions with the following two properties. So first we'll 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 one half length scale. And so I'd like to see, let's write it this way. So at each, if I look at the,

I'll write 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 u one. If you like, it's the determinant of the horizontal gradient of u, viewed as a real value, as an arbitrary value map.

So I'd like this to look like a bunch of delta functions to leading order concentrated near the origin, so w minus one, one in omega dz. Let me ask this to be, okay,

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. We assume this, and I'll also assume that this g epsilon of u epsilon,

that the energy rescaled in the above fashion by subtracting off the correct diversion part, that this is bounded by some constant c two. 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 horizontal directions. So let's let v epsilon of x and z

be u epsilon of h epsilon x and z. 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 one to a measure who is concentrated on the graph

over the vertical axis of n h one 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 domain, the rescaled domain will contain that ball. And so I mean, for example, on w minus one,

one of a big ball across zero l for every r. OK, so after passing the subgenings, there exists an f such that this holds. And moreover, whenever this holds,

then the limit from the energies g epsilon is greater than g zero 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 Kvoljic. This shows that the upper bound of Del Pino and Kvoljic is sharp.

In other words, for any f, I can construct a sequence of 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 theoremsic corollary. Let me state this not too precisely. So the corollary would be that in our geometry that there exist solutions of the Ginzburg-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 Jacobians 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 the z direction. OK, and so this tells us for these solutions the vorticity is indeed, sorry, and fi is a critical point of the limiting function and 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 Kowalczyk and Delpino 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.J. 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, 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 Ginzburg-Landau equation in 3D. And so in some, I guess I have to,

so a lot of this follows from, say, seminal work of Beth Wahl-Bosie in the early 90s on the Ginzburg-Landau equation in 2D.

And so the general picture in 3D is that one, is that when epsilon is small, energy and vorticity concentrate around lines. And more generally, in n dimensions, where n greater than or equal to three, energy and vorticity concentrate around co-dimension two minimal surfaces. This is the general picture.

And so this has been studied, for example, a major convergent in food, in more or less chronological order, Beth Wahl and Riviere, Lin and Riviere, Beth Wahl-Bosie-Orlandi, Beth Wahl-Bourguin-Bosie-Orlandi, Beth Wahl-Bosie-Orlandi-Smitz, et cetera. So these all made important contributions to this analysis. And so, of course, in 3D, minimal surfaces is a 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-Landau 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 minimal surfaces. And one can have multiple interfaces clustering around a single minimal surface. There's a lot of rather recent work on this

by people including Delpino-Kowalczyk-Wey, well, let's say Delpino-Kowalczyk-Wey-Packard, and basically various subsets of, in fact, subsets of those permutations. A few other authors have contributed as well,

as well as a paper of Kowalczyk-Wey and Packard, or a paper of Delpino-Kowalczyk-Wey and Yang. And so a lot of recent work on interface clustering. This is the first parallel result in higher dimensions showing vortex clustering. OK, so theorem two then would be, and so henceforth I'm going to always state things

in terms of rescaled variable v rather than u. And so let me write down what the equation, I'm going to start at the Gross-Pityffs equation. Let me write down what it looks like after this rescaling. And so I'll have, say, I guess I want,

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 models and rescaled versions to conserve quantities.

So I'd like to assume this. I'd like to assume that the vorticity initially concentrates. Well, it has these properties. I guess it's not written here.

So this converges. We can make a conversion. The preciseness doesn't matter, but let's say w minus 1, 1, or this topology over here. This will converge, too.

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 condition, 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 g naught of this f0.

And here, f naught denotes a smooth solution. And say h4 is enough, continuous into h4.

Exactly the Hamiltonian system associated to g naught,

which is exactly the model that Valeria Banica spoke about yesterday in her talk.

So f naught solves. t f naught 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 to the same thing, in fact.

Well, I guess this is, 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,

so in other words, what this is, this is the derivation of the equation studied by Valeria Banica yesterday. And so let's remember from Valeria's talk, this was first formally written down by guys in the fluids community, by Klein, Maida, and Damodaran in the mid-'90s,

following earlier work of Zakharov in the late-'80s. And basically, the way these guys argue is by matched formal asymptotics. And so this is, as far as we know, the first, 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 problems. I should have said, the Gross-Pitigapsy equation, in principle, describes superfluids, and so quantum mechanical fluids.

And so rather than looking at, say, classical ideal fluids 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-Pitigapsy case. Okay, and so let me note here,

in this setting, all the vortices have the same sign, and so one would like to be able to do this for vortices of opposite sign. That's what one sees in these trailing vortices of airplanes, for example. However, I mean, vortices 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 the history, and so this includes work of Walchlein, Maeder, Damodaran, who derived it,

then Kenig, Ponce, and Vega. Lots of work of Banneker and Milhaud. Also recent work of Walter Craig and Carlos Garcia as Pietja. And I think there's also work of Garcia, Pietja, and Ise. And so one can mention essentially everyone who's worked on this in a sentence or two.

Okay, 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 sort of crude knowledge about the vorticity. We say that we have n vortices concentrating on a certain scale with the origin about this line. And we assume that and energy bounds.

And then we deduce various conclusions. And so in the dynamical setting, we will choose initialators such that these hypotheses hold. And if we can show that 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, will be in the paper of myself in DDA, 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.

So if we consider, let's call this the interacting vortex filament system. So here's the point. So suppose f and f0 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'd like to do the same thing, if possible,

for if f0 solving this and f obtained from the Gross-Pity-FC equation in the limit as epsilon goes to zero. And so let me suppose that I've somehow arranged it

at every time these conditions hold, at every time in some interval. And so at every time, I can pass limits along some subsequence and get some limiting f that will depend on time. If I have some kind of equicontinuity in time, I can do this. I can find a signal subsequence for which it can converge along a 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 hypotheses, under my hypotheses of rather sharp energy bounds. What were the data? Okay, 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 films can have at most this much energy. They may lose a little energy. Energy may leak from the vortices into the rest of the fluid

and that's kind of an enemy that has 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. Okay, but then I'm still left with 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 the point is then, 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 Grunwald inequality and proceed. Of course, all our 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 what I want to do then is to use this equation.

The equation tells me how the vorticity moves. Remember, the vorticity is telling me where the curves are. And so let me multiply this rescaler, I think it's in terms of v instead of u. I multiply the vorticity by test function phi. And after the rescaling, I'll find,

what will be the term, phi sub t. There will be a term, epsilon. So here i and j are sort of like this. I have the horizontal derivatives 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 of phi, and the dot product with.

Okay, the anisotropic rescaling gives rise to log epsilon 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 naught of z t squared.

Okay, and so then I better sum over the components and multiply by cut-off functions. Okay, and so then formally, at least, we'll actually, then it's really true that the phi times the Jacobian converges too, as long as the limiting curves are supported in the region where these cut-off functions are 1.

This will really converge to the L2 norm, the L2 norm squared. And similarly, this will converge to, we know what this is doing. This will give us different pieces of that computation. And so we have to, in particular, we have to know what this is doing.

And so it's the 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 a measure supported along the curve, along the vortex curves, involving the vertical derivative of the limiting curve.

Okay, so these come from the limit. And whereas this involves, if you look at it, this involves, this is just, where the cut-off function is equal to 1, this is just equal to minus dz of f naught of the solution. Okay, and so here we have,

so this will give rise to a term involving the orthogonal gradient of dz of f naught and dz of f, right, the thing I obtained from the limit. This is the hardest term by far, and I don't have time to talk with it. So let me stop here, thank you.

Is there another geometric context of this? Maybe some domain which, not cylindrical, maybe some other geometric,

interesting geometric context in nature. Well, right, so, I'd say for the, for Gross-Patygevsky cylindrical or an unbounded cylinder or all of R3 are the natural context. And so the, as I think Valeria said, this limiting system has been studied most

on the real line, and so that we'd need a domain who's unbounded in the z direction. That would be possible, and for the elliptic equation, yeah, 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 and filaments concentrating on small scales around the geodesic. Well, that's, that would be a bit harder.

I actually have the same question, but just slightly different for my own purposes. You put Dirichlet-Avada conditions at zero and L. Yeah. Can you put periodic? You can. I mean, the,

right, so we have a, I mean, 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, you know, let me mark also, this is, So the Dirichlet conditions make you, allow you to fix the curves at the end. Yeah, that's right. In principle, they could also be fixed by,

by knotting. If I, so the limiting equation in the elliptic case is just this without the time derivative. And if you look at this, 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 knotted solutions who are local minimizers of some energy, which is what we would need to do to relate them to these. But the time dynamics, I mean, the time dynamics allows you to give initial data and follow them for a period of time.

And that wouldn't cause the, you wouldn't need a minimizer. No, that's right.