Thank you for the introduction. And also, I'd like to start by thanking the organizers for the invitation to ITS.

It's my first time here. And also, for this opportunity to give a talk at this wonderful conference. So the subject of my talk would be global regularity and scattering properties of two nonlinear wave equations, which you can think of as generalization

of the Maxwell equation. So we'll start from there. So Maxwell equation is a system of linear equations which concern a real valued one form on a space time, which

is supposed to be the electromagnetic potential. And in my talk, my space time will always be the Minkowski space, d plus one dimensional Minkowski space. And from this, you define what's called the electromagnetic field, which

is nothing but the curl of A. So this is the anti-symmetric part of the gradient of A. And the Maxwell equation reads as follows. It's the space time divergence of the electromagnetic field is equal to the charge current vector. So this is the inogenous term for the Maxwell equation.

The first nonlinear equation that will be the subject of my talk is what's called the Maxwell Klein Gordon equation, which is a nonlinear generalization of the Maxwell equation by coupling it to a scalar field, which evolves via a Klein Gordon equation.

So how this looks is follows. So this is an equation for an electromagnetic field, sorry, potential, and a complex valued scalar field, which I call phi. So the electromagnetic potential acts on phi

as a potential in the Klein Gordon equation. So in order to make sense of that, I introduce what's called the covariant derivative, which is a first order operator, which is in the mu direction. It's just the partial derivative plus the potential term given by the electromagnetic potential.

And then the phi equation, the phi satisfies the Klein Gordon equation with respect to this covariant derivative. A covariant derivative. And then the a evolves by the Maxwell equation, where the source is given in terms of phi

by the following expression. So it's the imaginary part of phi times the complex conjugate of the covariant derivative of phi. So the second nonlinear wave equation

that will be a subject of my talk is the Yang-Mills equation, which can be also thought of as a nonlinear generalization of the Maxwell equation. But it's a generalization of geometric nature. So actually, in order to properly formulate this,

I want to come back to Maxwell equation and Maxwell-Klein Gordon equation and point out an important property of these, which is called the gauge invariance. So I want to get to here, talk about that first before I want to talk about Yang-Mills.

So in both of these equations, there's inherent ambiguity in the description of the solution. So in the case of Maxwell equation, it's clear by what I mean. Or what I mean is, if you look at this f, then this value of f is independent of does not change.

Under the change, if you change a by adding a gradient of some real value function, because the curl kills the gradient. So this is disambiguity.

So it's called the gauge invariance of the system. And the Maxwell-Klein Gordon has a similar invariance, where a transforms in the same way. And a phi transforms by multiplication

by e to the i chi. But then once you write like this, you realize that actually this lens, this phenomenon of gauge invariance lends a different interpretation of these equations, right? So what you can think is, in fact, the Maxwell-Klein Gordon equation, the complex field here,

can be thought of as a section of a complex line bundle, which can be identified with a complex value function by choosing a frame. So by frame, I just mean a choice of basis of the fiber at each point. In this case, it would just be the choice of which vector should be deemed as one in the complex plane.

And then this gauge invariance is nothing but the rotation of the frame at each point. And this derivative is nothing but the covariant derivative on sections of a complex fine bundle. The reason why I set this all up is because Yang-Mills

equation is a generalization of the Maxwell equation. In the case when the vector bundle has a gauge group, which is more general than just the group given by multiplication of this. So in fact, this can be formulated in a more general way. But let me stick to the most concrete yet non-trivial case,

which is when the gauge group is the group SU2. So in this case, the Yang-Mills equation would be an equation for a 1-form, which takes values not in the real numbers, but in the matrix algebra, little SU2, which

is the set of traceless and prime Hermitian 2 by 2 complex matrices.

And then you can formulate a similar connection. So this gives you a connection on vector bundles with SU2 as the gauge group. And then it turns out that the analog of this electromagnetic field is nothing but the curvature

2-form, which is the commutator of the two coherent derivatives. And you can explicitly compute how it looks like. It looks like the following. d mu A mu minus d mu A mu. So so far it's the same as the Maxwell case.

But then you have an extra term, the matrix commutator between A mu and A mu. So you see that the formula from A to F is now nonlinear, and nonlinearity is due to the non-competivity of the matrix algebra. And the Yang-Mills equation now is the analog of Maxwell equation

with zero source, which is this. So it's the covariant derivative of the covariant spacetime divergence of F is equal to 0, whereby this enacts on you to view F as a tensor of its stick value in SU2.

So in fact, what I mean by this is it's in this case,

there is the same. There is a gauge invariance as well, which is given as follows. So if you take A mu and transform it by the following function.

So take any function from the spacetime, which takes value in now the group large SU2. If you make the change like this, then the equation remains invariant.

And the interpretation is, of course, that these are the same objects, but just the frames are rotated by this U. So we'll be concerned about large data theory for our initial value problem for these equations. And of course, the starting point

is the conserved quantities for this equation. So both equations have conserved energy. So in the case of Maxwell-Planck-Gordon, the energy measured at time t of a solution which consists of A and phi has to form spatial interval

over the size constant t of basically the sum of squares of each component of f. Yes?

So I just chose it so that it's the simplest example of a non-abelian group. So it should be the simplest non-trivial example of the equation. So of course, you can formulate the equation for other lead groups. For completeness, I'll just fix it to be as you do.

Any other questions? All right, so the conserved energy for the Maxwell-Planck-Gordon equation is basically just the L2 norm of f and the covariant derivatives of phi.

And the conserved norm energy for Yang-Mills equation is basically just the L2 norm of f. So I can write it in this case as just this,

because now these are matrix valued. And both systems also have, OK, so at least in the Maslow's case here, and always in the Yang-Mills case,

you have scaling invariance. And both systems, so the systems are invariant under the following scaling. So in the Maxwell-Planck-Gordon case, with mass equal to 0, it's invariant in the scaling where you map A and phi

into something like this. So in other words, both A and phi scale like inverse

of length, which is natural, because A is at the level of a connection coefficient. And in the Yang-Mills case, A also scales, has a dimension of inverse length.

And you can see how the conserved energy behaves under the scaling invariance of the system. And you realize that for both equations, the energy critical dimension is when d is equal to 4.

So when the space dimension is equal to 4. So this, with the case that will be concentrated from the rest of the talk. And also I should say that, in fact, our real motivation would be to try to understand the Yang-Mills equation. But then Maxwell-Planck-Gordon equation

has been traditionally sort of, as you will see, shares a lot of analogy with the Yang-Mills equation. So the viewpoint we'll take is that the Maxwell-Planck-Gordon equation would be a simple model problem for the Yang-Mills equation. And in that sense, because the Yang-Mills equation is massless, there's no mass in the Yang-Mills equation, I'll take the mass parameter in the Klein-Gordon to be zero.

So if, more properly speaking, one should call this the Maxwell scalar field equation, but unfortunately the literature, even the massless case is called the Maxwell-Klein-Gordon equation. I'll just stick to the dimension. Is it okay? All right, so I'll erase this here.

Then now I'm not lying about anything here. Okay, so dimensions below four are so-called energy sub-critical dimensions. And in those cases, there were results

about global regularity of these equations from early on in the 80s and the 90s by the work of early Moncrieff and Kleine-Moncadon and so on. But the understanding of the critical dimension and at the energy regularity has been obtained

even in the perturbative regime very recently. So in the case of Maxwell-Klein-Gordon equation, this is the theorem due to Krieger, disturbance, and Tataru in 2012

that small data global existence hold at energy regularity in d equals four. So let me just say, the usual value problem for the Maxwell-Klein-Gordon equation with zero mass

on R one plus four under some additional gauge condition, I'll explain, I'll get back to this some soon,

is global opposed and scattering holds if the energy of data is sufficiently small.

Okay, and I added this condition. So actually, so this condition is important to have in order to actually have even a formidable system. So this is a,

and this has to do with the gauge invariance that I explained earlier here. So by the presence of gauge invariance, if you just write down the equation, then you realize that it's a underdetermined, and you have too few equations to have a well-posed system. So you have to, the way you fix it is, you impose one more condition

that you want your connection to satisfy. And this is one way of doing it, and this is called the Coulomb gauge condition.

And the choice here is crucial because it's in the Coulomb gauge where the fine law structure of the Maxwell-Klein-Gordon equation is apparent. This is a observation that goes back to Kline and McAdon in the 90s.

There's a corresponding theorem for Yang-Mills equation. And this was proven only last year. Let me just say, same result hold.

So with this small data result in hand, our goal is to address the large data problem for these equations. Okay, that's a good question.

So, okay, so it will be a whole story, so let me not get into too much details. But scattering here would just mean that under this gauge, the solution to the problem would converge asymptotically to a simpler equation of, yeah.

But the linear system will be actually not exactly the linear wave equation. But I'll get back, yeah, let me not say too much about this. Okay, so the question for large data,

the large data question for the Maxwell-Klein-Gordon equation is a subject of a result that I proved

with Daniel Tataro in last year. And we can prove, we can show that the result is that the global regularity and scattering, whatever that means, hold for arbitrarily finite energy data.

So for the problem, for arbitrarily finite energy.

So one goal of today's talk will be try to give you an overview of the proof of this theorem. Okay, I should remark that basically at the same time, and independently, there is an alternative proof

of this theorem by Krieger and Luhrmann, who follow the kinetic-Mural constant compactness

and rigidity approach to prove this theorem. And more precisely, they follow the refinement, the development of the scheme due to Krieger and Schlock,

which were done earlier in the wave map case. But as you see in this talk, the approach that we take is different.

And on the one hand, as you'll see, it is a bit more direct. And on the other hand, the proof that we use relies more on the nonlinear structure of the particular nonlinear structure of the Max-e-Kline Gordon equation, I guess, than the other proof. Okay, so this is the picture

for Max-e-Kline Gordon equation. And then you might ask, what would be the corresponding result for the Yang-Mills equation? And of course, in the Yang-Mills equation, you cannot hope for such a unconditional on global post-instrumental result, because there are time-independent solutions

to the Yang-Mills equation, which have finite energy, and therefore it doesn't scatter. So for Yang-Mills equations, these are nothing but instantons,

which are a solution to the elliptic four-dimensional Yang-Mills equation, which had been studied in the 80s in the geometric analysis community. And moreover, it's also known that instantons can lead to,

can serve as a profile for finite time blow-up. So there are explicit constructions of finite blow-up, known due to Rafał and Lianski in this case. So in this case, the proper problem, the large data result, would not be an unconditional result, but rather the question, until which energy

does global post-instrumental scattering hold? So this would be the problem for Yang-Mills equation. And the problem will be as follows. Is the initial value problem for Yang-Mills global post,

and you have scattering for energy up to some threshold,

and the natural threshold, as I'll try to explain, to conjecture would be that of the first, the Brown state, by which I mean the lowest energy,

non-trivial solution, time-like solution, time-independent solution to... So the goal of my talk will be as follows. So in the bulk of my talk,

I'll try to explain the proof, strategy of the proof of this theorem, okay? And it will serve two purposes. On the one hand, of course, it'll be an expression of the proof of this theorem, but on the other hand, the deeper motivation is I wanted, this would be,

this would be a road map to a possible proof of this problem for the Yang-Mills equation. And then if time permits, I'll report on the recent progress on this problem, which is obtained by collaboration with Daniel Tataru.

Okay? Any questions? Okay. Right, so let me describe to you

the strategy of the proof of the Global Poissonness and Scattering Theorem for the maximum Klein-Golomb equation. And this is a, this builds upon the earlier strategy due to Sturben's and Tataru, which was carried out in the case of the wave map equation.

And so for the purpose, for concreteness, I'll focus on just the case of Global Poissonance, okay? I'll try to tell you how Global Poissonance can be proved. So the whole argument is a, of course, contradiction argument, okay? So we'll assume that Global Poissonance fails,

or in other words, finite time will occur, and then we'll try to derive a contradiction. And as is well known, by the combination

of the small data result and finite speed of propagation, you can immediately say that if you have a finite time blow up, then it must happen by energy concentration at finitely many points, okay? So what I mean is the following. So there will be some time before infinity

where your solution breaks down, so you have finite time blow up. You can identify a point on the final slice, such that if you look at the buoyant of influence of the point to the past, so it would be a light cone emanating from this point, and you measure the, so let's call the cross section of, the intersection of the constant t slice and this cone St.

Your energy measured on this St does not go to zero as your St approaches the final time, okay? And of course you should contrast it with a case when this point is a part of the regular space time,

in which case by continuity, this would of course go to zero, okay? But then it turns out that you can say much more, and this goes in line with actually the picture that Piotr described yesterday, which is that you can say that blow up, this constant energy happens in a particular form.

So what we prove is that you can extract a bubble from this finite time blow up with a particular profile. So what I mean by this is the following. So the claim is that you can find a sequence of times

and space coordinates and scales on the end, okay? So it's that, so picture should have mind is somehow there are these space time coordinates which approach this tip of my cone.

So tn approaches t plus these xn remain in the cone. And there is sequence of scales

which actually go to zero faster than the radius of the cone. So it's little o of this. Such that if you blow up this picture and make this scale lambda one to be unit, so which means that we use the scaling to think about the following rescaled objects.

So we rescale everything by lambda n centered at tn and xn. Then the sequence converge to something

to another solution to the Maxwell-Kline-Gordon equation in a strong local H1 sense. So to be more precise on a sort of unit time interval.

Where this object solves the Maxwell-Kline-Gordon equation and what's more, it is a stationary solution to the Maxwell-Kline-Gordon equation. And in fact, I can even give you

in which direction it is stationary, okay? So by passing to some subsequence, you can assume without loss of generality that these points are also told to some time-like curve, which I'll denote like this. And its generator I'll denote as y, okay? And the solution turns out to be stationary

in respect to that y variable, direction. Why does it have to be time-like and not null? Well, the short answer is because nothing can travel, nothing with mass can travel at the speed of light because of finite energy.

But this is an important point. Actually, it'll actually come up, so yeah. And by stationary, I just mean the following. So the contraction of y with f formed by b is zero and the covariant derivative of phi

with respect to the y variable, okay? And then you say the following. Okay, well, we can always apply a Lorentz transform to y to make it a dt, right? And then the claim is this. The claim, the second claim, this is one claim,

the second claim is that there does not exist any time-independent or actually stationary non-trivial finite energy solution to the Maxwell-Planck-Gordon equation, okay?

And this is very simple. You apply the, I can even give you a proof, you apply the Lorentz transform to make it dt, then you realize these will obey the following equation,

okay, where it looks formally like the Maxwell-Planck-Gordon equation, but this is the equation on R4, and you just realize that you can multiply this equation by psi integrated parts.

This tells you that the covariant derivative of psi is equal to zero if you can multiply the parts, and this tells you that the equations decouple, and therefore, f must be trivial, so you can take b to be zero

and then psi also has to be zero. And of course, the reason why you can do this integration by parts is because you have finite energy assumption. Okay. So it all boils down to showing this bubble extraction on a clang.

Actually, let me, I saved this board for this purpose. So, and of fundamental importance to this bubble extraction is a very beautiful monotonicity formula for the Maxwell-Planck-Gordon

equation, which I'll now try to explain. So the key monotonicity formula, which is responsible for this, is something like this. Okay, so but I need to first fix some patients, okay? So let me, for convenience, translate the,

let me for convenience translate the singularity to the origin of my space-time, okay? And moreover, let me flip the time direction so that I look at something that blows up to the past, okay?

I just wanna work with sort of the, this half space. And let me also assume, for simplicity, that I've cut off everything that happens outside this region, so that essentially the energy outside is very small.

Okay, so that we only concentrate on what happens here. Okay, this can be done by some gluing of initial, excision and gluing of initial data. And in order to state the monotonicity formula, let me set up some notation, okay? So let me introduce the vector field, which I call x, which is the normalized

scaling vector field, okay? So this is the scaling vector field, but then multiplied by a factor so that its Minkowski norm is always equal to minus one. Okay, so this rho is square root of p squared minus multiply by x squared.

And if you plot this vector field here, it'll just be something like this, right? So it'll always point in the scaling direction, right? But then you'll see some degeneration as you approach the boundary of the cone,

because there this vector field becomes null, okay? Let me also introduce some null variables, u, which is, according to my convention, t minus f sub i of x, and v, which is t plus f sub i of x, okay? So in particular, it's chosen so that this boundary

of the cone is the u equals to zero hypersurface. Okay, and with this, I'm ready to tell you the key monotonicity formula for bubble extraction. This is actually, okay, so let me just first state it. So let's take two time slices,

which I'll call s t one and s t two, okay? So the monotonicity formula is something like this. So morally, the following monotonicity formula holds. So there exists some weighted non-negative energy quantity p

such that its integral on this slice bounds from below the integral of the same density on this slice, on the other slice, closer to the similarity.

And in fact, the difference is given by a space-time integral of certain quantity density, which I'll call q, which is also non-negative,

up to some error, which goes to zero as you approach the singular time, okay? And here, this density of the space-time integral

is given as follows, is one over rho times the interior, the contraction of this expected field to the Maxwell field squared plus this self-similar derivative of phi squared.

And this p takes the following form. P is some weighted energy quantity, where up to constant factors that I'll probably mess up, it's this weight times the r inverse dv,

r phi squared plus the reverse weight times du, the same quantity, r inverse du r phi squared

plus these weights times the angular derivative

plus a hardy term, okay? And furthermore, okay, and actually, in fact, there is also a contribution of some terms which involve a, which is non-negative, so I'll now talk about this, okay?

Okay, so a few remarks before, okay, so I don't have much time. Right, so yes, so that was the first mark I'm gonna say. So of course, you might be puzzled how, okay, it's curious that you see a monotonicity formula

in a reversible, time-reversible wave equation, right? And of course, the magic lies in this error term, okay, which is actually the contribution of a flux. And in fact, the reason why you have this decay, so that you have this sort of modulo, this error,

is that your priori have decay of flux, right? And decay of flux just comes from the fact that the flux density, energy flux density is non-negative, so you have finite integrals, so if you take your, yeah, small, then it's gonna go to zero. So let me just say that this error

is due to flux decay, energy flux decay, which is a nice feature about this equation. And second, let me also say that I lied a little bit.

So it actually goes to zero, yeah, okay, sure. Yeah, yeah, yeah, yeah. Okay, so I should also say that, let me not write it out because we don't have much time. I should say that this formula is actually only morally true because the way I wrote it, the weight you see actually goes to infinity at the boundary, okay?

So you need some regularization procedure, but let me stick with the prettier lie rather than the messier truth, okay? You hope you believe me that that's a technical thing that can be worked out, okay? But very nice. So what you get out of it, first of all, is the following,

that you have, in fact, by independent means, you can prove that this integral, these integrals, is bounded from below above by some absolute constant time, the energy, okay? So this is something that's always bounded,

at least as you approach the similarity. And what this tells you is that this space-time integral is finite. Let me write it out.

And this signifies the following. So look, there is a weight here which goes to zero basically linearly as you approach the tip, right?

So this signifies that the fact that this integral is finite tells you that there is some logarithmic decay of this integrand as you go to zero, right? Otherwise, this cannot be finite. So this is actually the key crucial ingredient for the bubble extraction, as I'll try to explain now.

So what this tells you is that, morally, the derivative, this sort of time-like derivative, decays in some integrated sense as you approach the tip, right? So you wanna make use of that. You wanna say that you can extract a solution,

extract a limit of a sequence of solutions, and say that that must be stationary like that due to the decay of the, integrated decay which comes from this, right? But then, you have to divide into two cases.

One, okay, but then, what you also realize is that there is degeneracy of this vector field as you go to the boundary of the cone, right? So this is only effective in a fixed time-like cone, not in the whole cone.

So you have to divide into cases. One, when there is, when there exists a fixed time-like cone, which I call C tilde, such that some non-trivial amount of energy

remains in the time-like cone. So this would be our case one.

And you also have to think of the case when this doesn't happen. And by time-like cone, I just mean a cone with opening which is strictly smaller than this null cone. You've drawn it inside S t,

but it's not the case because of your intersection? Sorry? So, wait, wait, so I'm only concerning energy in this region. So what this is trying to say is that this is the case when there's some energy remaining in a fixed time-like cone.

And this is the case when the energy exits all fixed time-like cones. And let me just say that, so what happens is that in the case of one, the decay of such a time-like vector field is effective.

Okay, and this allows you to find the desired sequence of space, time, and scales, which should give you the bubble extraction.

You see, this explains why you have this particular direction of stationarity, right? Because this is nothing, so you imagine that you found these sequences, right? And imagine that you can do it in such a way that the same integrant decays. In these boxes, and then you realize that

that x vector field asymptotically becomes this y vector field, right? The direction, unit time-like vector in the direction of this asymptotic time-like ray that these approach. But then, so is this okay?

So the smaller cone is at a given distance from the other one, right? Are you sure that you actually get something time-like? Right, right, but then the point is, in fact, you can actually extract such a sequence.

I can't go, I don't want to go into details, but you can extract such a sequence if this holds for any fixed time-like cone. So the case where you cannot use that monotonicity formula is exactly when you cannot find such a cone. So in other words, the energy exits every time-like cone

as you go toward the singularity. And in fact, this is precisely the case where you have to think about, this is precisely the case when everything sort of is traveling at the speed of light, right? And this is the case where your priori would think

that there cannot be any nonlinear bubbles, right? Because if there were any bubbles, then because of finite energy assumption, you cannot travel at the speed of light. And in fact, what you can do is you can now look at this statement, that this weighted quantity is always bounded,

and realize that it has a very good weight on certain components of the field phi, right? So in particular, the dV derivative of phi decays, and the angular derivative of phi decays. And this tells you that, in fact,

there cannot be any nonlinear bubbles. And more quantitatively, you can say the following. In fact, one way to quantify this will be to find.

So one way to find a concentration, a point-like concentration, would be to look at phi, and test this, well, integrate this on some ball of radius lambda, okay? But then await it so that it's a dimensionless quantity, so you wanna take something like, I think, this, okay?

And say that this is very small. Well, this is a wave equation, so I need to take a time derivative as well, so I also take something like this. And this, the heuristics that I told you basically translates to the statement

that such a quantity, uniformly for all lambda and all x, goes to zero as you approach the singular time, okay? And of course, what's happening is that outside, we cut off anything that's happening. Inside, there's no energy in any timeline cone,

and where the energy lives, there is no nonlinear bubbles. You have smallness of dv derivative and the angular derivatives, which give you this. And the key hyperbolic, the wave regularity theorem that we proved precisely enters at this point.

So the result is that if this is small enough, then we can continue the solution on past zero. So there's a contradiction. If this holds, so this is a contradiction.

And this ends the summary of the proof. So I think I ran out of time, so I can't say anything about Yang-Mills. I'll just stop here. Thank you.

Central. So if I understand correctly, this monotonicity formula, which is not a monotonicity formula. Yeah, well, yeah, it's hidden, yeah. It doesn't give you any prior estimate on any solution.

You have to take different cases and use it differently in different cases. Is that correct? Yeah, I mean, well, it's a- Because by itself, it doesn't tell you much, right? Yeah, it's a very weak decay statement that it tells you, right? All it tells you is that there's some, this integral quantity is bounded, right?

So this tells you that sort of the integrand actually decays in some logarithmic sense. I don't understand, because you don't have information about this t2, right? Oh, no, so what you can do is, because you know that this is, so I was saying that by separate means, you can show that this is always bounded.

And this means that you can take t2 to zero, right? That's all, it's uniformly bounded. So that will tell you that this base-time integral is bounded. But this base-time integral comes on the Q, I think, it's part of the Q. Yeah, yeah, but it's on the left-hand side.

Right, but, okay, by the way, I'm trying to ask you about how exactly I came up with this. This is uniformly bounded for any t2. That's something that you can prove, you can prove. Okay. So you can take just t2, yeah, yeah, yeah, yeah.

Two questions, but short. This, that's the line, you can use three cards? Oh, that's a, so of course, but we also have to use, sort of, by linear estimates and so on.

So, why is it complicated? Because for the way we have the same thing. Yeah, that's a very good point. So in fact, Because we just, we use Sobolev and we mix the Sobolev and three cards. So this is more of a, more dedicated. Yeah, so, yeah, right, right, right, exactly.

So you see, the phi equation, if you expand, has a term which is a times derivative of phi. And okay, you're right. So this actually statement is a complete triviality for the semi-linear, for the power type non-linearity. Because in there, what you can say is that

in the non-linearity, you can use three cards to say that you have, you have very good off-diagonal decay if you decompose into little pili pieces, right? And the point is that this translates to a smallness of some Bessel floor. Yeah, and that's it. But then here, you realize that because of the presence of the derivative,

and the wave equation only gains one derivative, there's no exponential decay between the low-frequency part of this and the high-frequency part of this. So this is actually the key difficulty that one has to face. And the second question. Okay, this is, I guess, the natural extension of the formula I have with the idea for the critical wave equation.

Yeah, yeah, okay. So you don't go back, as in the proof I have with Carlos, to zero outside, when you have a ring monotonicity formula when you are done, in some sense. You don't have to go in your basket. Right, right, but then we,

right, but then that's when you construct a minimal kind of example, right, right. So, right, I mean, I guess the point is this is more a more direct approach. So we are now assuming that we have a minimal kind of example. We are assuming that, give us any kind of example. But then, okay, so the thing I don't understand

is we lose approximation of current inside and not outside. Outside is not easy? Outside is, I mean, outside is actually, I mean, because I find this propagating out and flux decay outside is very easy. Yes.

You control this term, so it's not, okay. Okay, anyway I can discuss with you this. We continue, and thank you.