Thank you so much and thanks to the organizers for giving me the opportunity to give a talk and actually my

original plan was as said in the title to talk about critical half-wave problems, which would mean two the L2 critical problem that Patrick Girard was talking about on Monday and another new problem, which I will now try to focus on because Patrick did such a great job It's hard to really add something to this. So let me

let me Let me explain what I want to discuss is what I phrase is a half-wave maps problem Okay, so what's that? So it's an equation that is extremely easy to write down

so, let's see and I will say a few words so the wedge means the cross product in R3 and U is our unknown function Depending on time on an interval say from 0 to capital T and say it's defined on

Rd With values in S2, which we always of course think being embedded in R3 Yeah, so this is it's a quite I mean Simple minded at first glance generalization of the Schrodinger maps problem

Yeah, so you just take the say Landau-Lifshitz equation This would be the Laplacian or it would be minus the Laplacian, but minus signs can be easily so to say Absorbed by redefining direction of time. This is an Hamiltonian equation, by the ways I will show to you soon So I might get some various sign mistakes in my formulas, but they don't matter. So that's only where it matters

I think I'm right. So, let's see okay, so this is the half-wave maps equation and And so there are two reasons why I recently started to work on that first of all I've seen this equation before but so only recently I realized that this has a really somewhat very nice physical

interpretation that's the one bonus thing about it, which I try to make Somewhat clear to you here And the second thing is which I was always also afraid about is that you cannot do much with that That is rather quite Unexplicit what you can do compared to what like Schrodinger maps or wave maps equation where you have a nice

explicit structure say of Equivariant wave maps and so on it studies stability and blowup problems like that But it turns out to be wrong this statement that I will show to you that you can do a lot of it Explicitly because I will as my main

Result today discuss that in the one-dimensional case, which is the energy critical case I will show to you how to completely classify all traveling solitary wave solutions in an explicit form in terms of so-called Blaschka products. Okay, so this is something I Recently found and something that amazed me and gives me hope to continue this

Study of this energy critical problem. Okay, so as I said, this is an Hamiltonian equation. So It's an energy functional attached to it Which of course

Looks like this Somewhat guess what this is going to be. So maybe there's a half here if you like that and For later purposes, I will just Do this here for record that this is of course

Expressible as a double integral which will as you see make a connection to a very interesting physical application of this model and The Poisson bracket attached to that so that you can see that you can write this

Equation of as a Hamiltonian equation of motion is the one that is given by Functions with values in s2 say the components at x and y satisfy

Poisson bracket structure, which is given by this expression So this is the antisymmetric Levitivita symbol and the Delta function So if you if you use this structure at least formally you see that the half wave maps equation is formally equivalent to this evolution equation given by

Poisson bracket given with H, okay So in particular H is at least a formally conserved quantity So let me now start and discuss the motivation behind this equation. So first of all

It is clear now that you get the original equation. Oh, yeah, if you are clear I mean you have to plug in what this Poisson structure is and You work out what this is by plugging this in and then you finally see that this is the equation that you get I mean if you take for instance Here the Dirichlet integral you get the Landau-Lifshitz equation with that Poisson structure. The Poisson structure is some universal thing

It's just the energy function of it determines whether you have the half wave maps equation as I call it or the Landau-Lifshitz equation The Landau-Lifshitz is the Schrödinger maps equation. I

I don't have to write it down. Yeah Schrödinger map or Landau-Lifshitz coming from physics like the continuous version of the Heisenberg equation for a ferromagnetic system Okay Yeah, so d equal to 1 by the way would correspond to the energy

critical case Everything above d equal to 2 and higher is of course an energy subcritical a supercritical So by the way and the energy critical case, it's also Physically most relevant

as I will explain in a second and There's also another thing attached to it it has a conformal invariance property, so this is the case I will focus on so the motivation

So one motivation comes from differential geometry, I mean recently there was some

Growing interest in what is called a half harmonic map with target s2, okay So that would correspond of course obviously to static solutions to my flow equation

So in particular these static solutions will play also an important role for this time dependent equation Of course, you might also think about something like a half harmonic map heat flow Which would be a parabolic analog to this equation, which I don't discuss here. So this is something that was recently introduced

Riviere and Dalio in particular the regularity theory for these equations like say if you are the critical dimension that an h1 half map

Is actually a smooth map like an analog of Elland's regularity theory for harmonic maps in two dimensions yeah, but it's a different thing because Riviere used some kind of commutator type arguments to get this regularity result and also from a more geometric point of view

in a very recent work by Fraser and Shane It is seen that this equation Corresponds to a free boundary minimal disk problem. I Will come to this later

That is to say it is also naturally connected that you can look for something which is a minimal surface say inside the ball The unit ball with a certain condition how the boundary of this minimal surface hits the boundary of the ball Which is s2 and in a in this setting it has to hit perpendicular and you can actually see that this problem is

equivalent to this to study the free boundary minimal disk Yes, this is a boundary equation I will come to this point where you see that it's a completely natural connection to the theory of minimal surfaces Okay, I come to this and

B this is the physical motivation behind that Of course, I will later look for traveling solitary waves, which will be Generalizing these half harmonic maps in the kit in the sense that there is another non-zero right-hand side Okay, and I will classify also all these solutions and B is physics

And this is a typical thing you have to know the right names to look for because physicists like these names you see week ordering Gibbs state if you don't know what who week was who Gibbs was it's hard to find out in the literature What's this about once you have the right code words it works. So the first

two names that somewhat play a role in these business a Haldane and chest ring to physicists Who wrote down a Hamiltonian? for Spin chain, and I will be a little bit formal here, of course, which is some

Object like that Okay, these are spin operators. I won't go into any details and There is a so to say interaction among spins. This is on a discrete

Lattice say in one dimension So you have fixed lattice sides and XK XK plus one and so forth are the lattice sides and to each lattice sides you attach a quantum spin I don't want it to go into details what this is about and they interact

With certain what's called long-range potentials Oh It might be a sine squared of the difference of the lattice sides Or there's also the so to say algebraic model where this is a square of this and so on There's a whole zoo of this and the point about this These are exactly solvable models in the sense

You can calculate the spectrum and so forth and you can also find lax pairs and so on. These are long-range quantum chains, which have a very nice Particular structure from the physical point of view So it's hard to draw what these quantum operators do actually

But I installed Planck's constant here just to indicate that in a certain limiting regime Which you can think of the semi-classical limit or also a large spin limit You are at least formally led to what is called a system of classical spins So then this Hamiltonian becomes something which you could phrase as a calogero

Moser type Hamiltonian with spin which would look like this that formally at least the spin operators get replaced By something of this form where the s case are now

Unit vectors in 3-space. Okay, so in the classical regime the spins are given by By an arrow which lies on the unit sphere in three-dimensional space That's at least for a physicist a typical procedure to go from a quantum spin system to a classical spin system

So if you just had just nearest neighbor interaction so this wasn't this function, but just only nearest neighbors couple you get the Heisenberg chain and the continuum limit of the Heisenberg chain in the sense that this Parameter a shrinks to zero would be the classical Landau-Lischitz-Schrodinger maps equation. However here you can of course somewhat

Anticipate that in the next Step of taking this limit you get that this sum Leads to a double integral of that form that you have a continuous spin variable, which I call s

Which is of this form And we are here on one space dimension Making this rigorous makes it of course if one step is to study actually that the dynamics converge in a certain sense And of course in higher dimensions might be even more difficult in the critical case. I think it's still

Doable, but it's it's it's not so simple and on account of the fact that this is of unit lengths You can of course rewrite it As this which is the Hamiltonian I wrote above in one

Space dimension, okay Again, this Calogero-Mosa type Hamiltonian Also shares the magic property that you have lags pairs You have a very explicit structure here and you might wonder if this survives actually for this limiting PDE There is a strong indication for that because I will show to you an explicit

Classification of all solitary waves. It might be a hint that this is a completely integrable system Which is yet energy critical so it shows a criticality which in principle can also say that you have Finite time blow up and so on. So, let's see

So now I come to the more Yes, exactly in Dimension 2 and larger super critical and in one dimension is energy critical and in principle you can think like for Energy critical Schrodinger maps you have also blob solutions

That in this case you might have because of an integrable structure too many conservation laws which forbids such a thing but it could also be like for the cubic Chigo equation that they only live up to certain level of regularity and you still see some kind of Sobolev type growth phenomena Which are a little bit counterintuitive first if you think of a completely integral if I think of a completely integrable

System So what I'm here interested in is to consider the traveling solitary waves For this model and this is interesting in one respect first that for the Landau-Lifshitz equation

Such a thing does not exist in dimension. There are only static finite energy solutions. You don't cannot create a Traveling solitary waves for the Landau-Lifshitz equation You might think you can apply a Galilean boost, but this is not working

It's not but of course for the wave maps equation. You can do a Lorentz boost to get a traveling Wave map if you wish. No, but here this is not a Lorentz invariant system, so we cannot apply something But you will see that we get a rather explicit way of constructing these things and I'll show to you that this is the only

way to get them Okay, so you make an ansatz Of course that you say I have a velocity given a real number v And you say I look for solutions of the form u t x is equal to say some profile u v of

x minus v t and This is simple to see that this now satisfies an equation Of this form. Okay. There's a minus here, but it's Inessential, okay, and I have to be a little bit more precise

What I mean here, I mean a solution which is an H dot One half going from the real line To the unit sphere. Okay, and there's a certain boundary condition

at infinity fixed Say the South Pole if you wish maybe huh? Okay. This is always understood implicitly One thing which I will completely skip is that you might ask for a regularity theory for this equation Of course We want to ultimately that any solution is a smooth solution because the arguments I will present you will in some way need this

Kind of property, but I will completely skip this and just mention that Joint work with my former postdoc Armin Chikora you get this smooth Okay

Naturally, of course It's not obvious at all because the perturbation so to say that you add is also a first-order Operator and this is it's a massive so to say change of the equation We know already smoothness for the v is equal to zero case Okay, but let's leave this issue aside for a moment. So

What you can do first is of course you can try to cook up a Solution okay, how could you maybe try to do that first? There's one way of course, you can think you are check this variationally this is critical points of the at least formerly of

the energy and Then you have to find another functional which is a side condition. Maybe it's such that the left-hand side comes From that side condition in V is a Lagrange multiplier. Okay

Call this functional UV Constant which in a sense would correspond to a linear momentum, but this is a tricky business in this setting I Will not write down you can in fact find a function at least formally so that you get this the problem is as you will see is that both functionals show a conformal invariance property and hence

Discussing minimizing say sequences is not so simple. It's a little bit like the plateau problem So it might be a little bit painful to attack this problem variationally, but in principle it should be doable Okay, one thing so we won't follow this road

Second road is of course an implicit function theorem at least for small V. You might be able to construct in the neighborhood of these Whatever are the half harmonic maps at least slow moving traveling waves, okay, but

We won't do that. So there's a lucky punch here Which does the job? Okay, at least you get solutions you make an ansatz so you start With a half harmonic map which corresponds to a static solution and

You can find such things like just living say in the Equator plane. Oh, so the third component is zero, of course then I

Have to make sure that this point also lies in that plane, but it's fine Okay, and now you make an ansatz that for UV and that's sort of a magic thing here But you will see a geometric reason for that later

Of course, I made a much more general ansatz first because I was thought I was thinking about kind of mimicking Lorentz boosts, however alpha and beta are just constants and if and only if alpha

And that it's not true. There's there's a sign. So if alpha is 1 minus V squared So this of course assumes that V is less than unity Which means the solitary waves cannot propagate faster than the speed 1 in my units. Okay Well, let's assume that for a moment and B is equal to V

You get a solution it's quite Fascinating that you get an explicit cheap trick to construct so to say boosted solutions So it's it's even I don't think so. I don't think so. There is no Lorentz invariance. There is no Galilean invariance and

The funny thing is also that this Transform which comes from the static to a traveling one is just acting so to say from the outside on the target I mean you don't have to do anything in the arguments of X

Yeah, yeah, but still I mean you could I mean Okay, let me draw a picture The picture is this Okay, I

changed the boundary conditions But I can rotate back so I can always that's everything I say is modular rotation on the target. That's true I violate the boundary condition, but I could rotate back and then I would get the same boundary condition, but I'm gentle about that. So so what I did is

That this half harmonic map and of course F and G are certain special functions But it's it's a certain kind of parameterization of the gray of a great circle, huh? Gets in a sense boosted to some other circle Which lives here and in the extreme case when we say tends to one

We're just on the get the constant which just lifts on the tip of the sphere. Okay Okay, so the theorem that I'm going to talk about now says this is the only way to get the solutions Okay, the boosted solutions and gives an explicit characterization of all the possible functions F and G you can have here

So let you

Let me be a real number And UV solve this equation makes perfect sense if it's an H dot

one half function and the distributional sense solve that Then you have two cases if V Is bigger and modulus than one or equal to one the only way Is that you have a trivial solution? It's a constant. So it's

Nothing else can happen and if V Satisfies this then What rotations of course

On the sphere on the target You have to have that UV of X Is of that form which I wrote down and now what is F and G?

V sorry this where F is the real part

say of a function capital F G is the imaginary part. It's the real part of a

holomorphic function in fact on the boundary living on the upper half plane and on the boundary you get the little F and G is the imaginary part Of that function and F is of the form F of Z

Is equal to what is called a finite Blasch product. So there is a K

running from 1 to say D and There's maybe a lambda K Z minus a K. There's a shift of course That is maybe possible plus I and these Lambda Ks are real numbers

Not 0 and the a K's are arbitrary points in R and D is a Natural number of course Okay

Yes for V equal 0 you can characterize that but the point is and this is the Point that you can really reduce it to the case V is equal to 0 so I will explain that the proof is Really an advancement of understanding the case when V is equal to 0, okay? So can this be understood directly on the stationary social and the half harmonic map or unique?

Okay When I look at the stationary solution the half harmonic maps this is not completely easy to see but Doable okay with what's known in the literate in the recent literature say

The point is that you can sort of say also undo the boost transformation. I will explain that okay Yeah, yeah, it gives you all in particular characterization of all half harmonic maps because it's a case of V is equal to 0 Okay, and there's of course a saying there is no traveling wave with speed larger than 1 okay

This is also one point Okay, and the interesting fact is that the energy of these guys is of this form There's a quantization of course involved like for harmonic maps the D the index of the Blasch product

However by tuning the V you can make this arbitrarily small so it's like the L2 critical half wave equation with Patrick Girard where you have L2 criticality, but you have solitary waves which have arbitrarily small L2 mass and now you have

Solitary waves with arbitrary small energy energy is now the critical thing, so it's completely different say from the Landau Lipschitz where you have Don't have that okay Alright, so I have

roughly 20 minutes 17 clock counts Okay, let me explain to you about the proof

Okay, this is actually two statements a and B. Okay. I will along say something about a and But I will of course focus on the more interesting part B So what's the Initial situation

So think first that we are given a solution UV and think it's smooth okay to what any issues here so that the image Of course it raises us a closed curve on the unit sphere Okay, and

Actually what we are posteriorly see that these are just great circles or in the case when V is equal to zero or When V is not equal to zero, it's it's a circle on the sphere. Okay, but first we don't know anything about this More than this. So if you take

So rewrite for simplicity Little u Is equal to uv. I want to skip that index little v because writing is a bit difficult Okay, first thing of course that we do it's also to see that you have a conformal invariance here Of course You consider this as a two-dimensional problem in a sense because this arises as a boundary of something

Yeah, so you take the u to be the harmonic extension of little u That is capital U is a map from R2 plus Which you can also identify with the complex of a half plane

Into R3. It's a component-wise harmonic function Yeah, and the boundary condition star is a boundary condition then because as you know, it's a classical fact that

operator just becomes the Normal derivative with respect to this extension variable or minus the normal derivative So effectively you study the problem minus v of dx capital U and There's a minus so it will change here, but it's

Inessential of dy. This is my new variable that I add to my space Like this on the boundary Okay Fine so far so good. So by the maximum principle

Actually, it's a strong maximum principle that inside it's strictly less than one Well, you see out that you the image of capital U is something inside the sphere Of course because of this and the boundary is so to say given by this little u the boundary curve

profile of the solitary wave Okay now Comes the first step That says, okay

Capital U is a Parameterization of a surface module and there might be branching points and these things It's maybe not embedded, but we just say it's a minimal surface provides a minimal surface

That's the first step. So how to see that the first idea is to use something which is called the Hopf differential So it's a function That you cook up depending on z say so z is just a

Notation of course for this and dz is the Wiltinger derivative That's that's a fairly standard step step here what I do, but I just want to explain it. Ah So I remember there should be maybe a 4 to have everything nice here so now you

take the derivative of Capital U with respect to the Z. So it's actually just this you combine them. So that's a C 3 valued function It's a 3 vector with complex entries and you to take the scalar product not the Hermitian product. So

Take this product. This is a standard thing and minimal surface theory So what you get is this? warm here That's nothing to do really with the questions it's the climate here, okay

Okay, so the the upshot is that this is a Because u is a harmonic function. This is a holomorphic function. Okay, so you either check the dz bar the anti-Wiltinger derivative of this Zero hence, it's a holomorphic function. It's easy to see because the

Z bar times dz is just times the constant the Laplace operator. Okay, so far so good and now comes the point that You look at that function and you consider the imaginary part of this hope differential

on the boundary Okay, so then you use the boundary equation, okay And now you have to explicitly work out what this means actually

It means in terms of real parts That you have this Minus maybe 2i. I'm not 100% sure whether there is a plus or minus, but it doesn't matter here This okay. So the imaginary part on the real axis of that

Holomorphic function in the upper half plane is the scalar product of d x u times scalar product d y u but now look at the Equation here. This immediately tells you that at least when v is not equal to 0 this has to be perpendicular to that vector and

You see that the imaginary part of that holomorphic function on the real line is 0 You can extend it to the whole Complex plane plane the imaginary part by odd reflection. However, the imaginary part of Phi

Because on the boundary it's h 1 half so this is actually an h dot 1 function. So this is This times this is an l1 function. It's integrable So then you can easily conclude that the imaginary part Is a constant identically 0. Okay, so because it's a holomorphic function

The real part has to be a constant a real constant. However, it's also an l1. So it is also identically 0 So this hopf differential is actually identically 0 fine

So that tells you this is always equal to this and this is always 0 so it's Right angle so U capital U is a conformal map and this is just saying It's an harmonic function with which is conformal. Hence. It's a minimal surface what you get

Now as a next step Yes as a next step But there could be lots of minimal surfaces inside the unit ball Of course, they are very funny ones, but they have to meet the boundary in a certain specific way They have to respect of course this boundary condition here. Yeah, and

When V is equal to 0 this is something which I first referred as a free minimal disk Which was studied by Fraser and Shane and they classified that this can only be a plain disk and then you are in the situation That the boundary is a great circle and then you are in good shape to get to get your classification theorem

However their proof I Could not see how to make it really work in the case when V is not equal to 0 because you have a more complicated boundary condition, okay But I show you an argument which includes the Fraser and Shane result by considering the following thing

So what you learn from that first step, that's a conformal map You also learn that you can rewrite The boundary equation, now I write little u here in a very interesting way

People from harmonic maps know such a thing That you would maybe expect here a square, but this is not true in the fractional case. It's it's not a square It's a one and there's a deeper reason for that because if you test this equation Against u then this term of course drops out and u times u scalar wise is one so you get this

Okay, what is this geometrically the right-hand side corresponds to the length of the curve and this if you use the harmonic extension

The lift corresponds to the area you span Okay, and you will see that it will actually satisfy an isoparametric inequality Set a rate in isoparametric. It has satisfies of course the isoparametric inequality otherwise would be bad Saturates and from that you could also conclude that you have to have plain discs

But you don't know yet what the value of the left-hand side is, but this is just as a side remark So the next step Is and this is then the little innovation maybe is to consider another Hopf differential Differential well, what is that take now this?

You can now I leave the details because this is maybe too much on the blackboard write down what that is again It's a holomorphic function you consider you can study. What's on the boundary you want to make?

May maybe run a trick like that, but of course you have to work a little bit harder But using this formulation of the Euler-Lagrange equation you also see That this is a holomorphic function, which is identically zero But that's too much a minimal surface which of course

Satisfies the first thing there are lots of them however it also has to satisfy this okay, and then You can use what's called the Weierstrass representation of a minimal surface

just want to Say these things here So that you can deduce So what's that so it means that you can this is a classical thing parameterize the

Wilting a derivative of this u in terms of a holomorphic function f and a meromorphic function which satisfies some Comparatively condition, but if you plug this into this equation you work a little bit You will see ultimately that u the image of u can only lie in a fixed plane in R3

Okay, so so what you get from that finally leave the details of course that

the image lies in the plane In R3 so it's a flat really flat minimal surface So it's a disk so you will ultimately see that this is a disk Huh, and of course the boundary of the disk is a circle on the sphere now

You can undo this transformation of course and go back to this case Where's the unboosted case okay, and then you're almost done because now you can invoke some complex analysis Because in the unboosted case

Say you Transform it like this that it's in the XY plane Then you will see with this analysis that F and G have to be say the real and the imaginary part of a complex function f

I G on the upper half complex plane with complex values and Because u lies on the unit sphere the modulus squared of f is 1 on the boundary

So That's a very special holomorphic function on the upper half plane such that the modulus is 1 identically on the boundary and there's a classification for that in terms of Blaschka products and because also f is In H dot 1 it's not hardy. It's a Sobolev space here

You Can conclude that this is actually a finite Blaschka product which I wrote down so they only have finitely many factors otherwise you would have Infinite energy, which is a bit formal, okay?

Hence you get the complete classification to you, and I finally should say that The last idea of using these Blaschka products for this kind of problem is also something that already Mironesco

and Pisante Used for something which is related where you look for what's called maybe a half harmonic map from s1 Into s1 okay, I should take lots of credit to this worker somewhat, okay? So what I want to say is now

Okay, if you work out these real imagining parts of these finite Blaschka products It's just kind of rational functions you have so it's very explicit You can now study the stability or instability you have very explicit Formulas for your solitary waves, okay, and this is also the point. I like to end because it's freaking warm here

So

Oh No, this is what I this is a byproduct of the proof so if you have a stationary solution in my terminology v is equal to 0 and It has should have finite energy other Okay, modular rotations on the sphere of course, but they all have to

Okay, yeah, yeah, yes this is Yeah, that's a byproduct actually so and I should maybe say about a I should be looking for things Yeah, but I cannot okay what I show finally is that I'm

Allowed to do this because they're always equatorial discs so I can always rotate it that the last is identically zero Okay, and I should say a is is a funny proof you test the equation against the Hilbert transform of you and do some magic In the in the l2 critical half-wave case we cannot actually rule out

Solitary waves which have speed more than one. We don't know how to do this in this model It's a magic thing to really get the sharp limit for the V

Oh, okay Yes, I have not talked about existence of the flow yes, of course, yeah, that's a lot of things to do Stability I would expect

I would expect that but Also because Formally you see if you think of the Schrodinger map blow up you have an unstable one Because you have a slow decay of a resonance I mean if a slow decay in the linearization a zero mode, which is very slow decay and I think here

It does not happen that you have I don't I don't think so But I would not bet much my I have to look into after the euro Hahaha look into the things yeah, maybe yes, maybe next week. I have time

So what is your momentum functional oh, yeah, that is a good question It's a bit like this. I draw a picture so the momentum of a configuration you closed curve is the

Solid angle That this curve so to say in the unit sphere Gives you but this is of course not well defined up to four pies because you have ambiguities so so in the physical literature, there's a kind of a

Dispute about what is whether there is a true momentum or not for such a model, but you have a you have a conserved quantity Modulo a value of four pi you could consider an exponential of that, but it's not it's a it's a It's a it's a funny functional and the momentum is it's really not that straightforward

Say for these models, of course Also, you might wonder whether the traveling solitary waves you get are also existing in the discrete models And this is always a delicate question for NLS There are might some Traveling waves in the discrete lattice model or not and so on and here it's also not so clear What survives or what gets created by this continuum limit?

There's so much structure it's actually very beautiful there's a lot of structure and it does come from Not just one interval system, but a class Yes, so what would you speculate I would speculate it's also completely integrable, but I don't

Mean I try to cook up legs past but it's not my prime education, so Take some time maybe but I don't know maybe there's maybe I'm wrong. Maybe you would you see really Serious blow up. Yeah

Yeah, yeah that maybe in some weird way you see like for the Chego equation you might have infinitely many conservation laws In evolution up to a certain kind of regularity in higher regularity you might as have at least grow up For infant, I mean some small kind of infinite type blow up or something like that. I don't know

Conservation laws needn't be Needn't be coercive. So for example blues and esk is nicely integral that solutions just don't exist You will teach me more than I know that's good Yeah, it depends on how you define this