 Thank you very much to the organizers for the invitation to speak, and also to stay here for a while. It's been a lovely month. OK, so everything for the theorem I'll present today is joint work with Sungjin Oh and Serab Shahshani. And it's part of a project that we've been working on for a few years now. And I'll present the latest result that we have. So the topic is wave maps, and in particular, wave maps on hyperbolic space. So I'll start by just introducing the wave map equation. So wave maps concern maps, u from a Lorentzian manifold, m with a Lorentzian metric eta, taking values in a Riemannian manifold, n with metric g. And in this talk, we'll let m take the product form. So it'll be able to form r cross sigma, where sigma is a Riemannian manifold. And as I mentioned, we'll focus on the case sigma equals hd. And we'll also talk a bit about the case sigma, this d-dimensional Euclidean space. OK, so wave maps are formal critical points of a Lorentzian action. So L of u is 1 half integral over m du du d ball on m. And so this is a very natural action. It's nothing other than the Lorentzian analog of the Dirichlet energy for the harmonic map equation, for example. And here in coordinates, so take, for example, tx coordinates on m and coordinates uj on the target. Then this is expressed as follows. So this is equal to eta alpha beta tx g ij of u d alpha ui d beta uj. And then the volume form is square root of determinant of eta dt dx. OK, so this is the action. And critical points satisfy the following Euler-Range equations, d alpha u is equal to 0. OK, in the talk, I'll use a convention that I'm summing over repeated indices. And I'll also stick with the convention that the Greek indices are for our spacetime indices on m. And I'll use Roman letters, so ij k for indices on the target. OK, so here, this is the wave map equation. It's a beautiful equation written like this. It's hard to understand what exactly is going on here. But the nonlinear structure is encoded in this capital D, which is the pullback covariant derivative on the pullback bundle u star tn. OK, so this is still quite abstract. In the case when n is embedded, isometrically embedded into Rn, then this equation takes the form, the delimbersion on m of u is perpendicular to the tangent space of u of n over the point u. And this can then be simplified as follows. So the delimbersion of u is equal to S of u, d alpha u, d alpha u, where S is the second fundamental form of the embedding. So this is a nonlinear wave equation where the nonlinearity arises from the geometry of the target manifold. And it's quadratic in the derivatives of u. OK, so we'll study the Cauchy problem, which means it will give ourselves initial data, u t of u, where initial data, so u0 is a map from sigma n to n. So let's take a smooth map, say n to n. And u1 at each point x and sigma, so u1 is a map into the tangent space, is an element of t u0 of x. So this is an initial data set for the wave map problem. OK, let me just start by just stating our theorem, and then I'll talk a bit about the proof, or rather, more like the formulation of the problem, how to formulate the problem correctly in order to prove it. OK, so the main theorem for today is that in the case, so sigma is hd, so d dimension hyperbolic space, and d is bigger than or equal to 4. So if you have high enough dimensions, then there exists an epsilon bigger than 0, such that for every smooth initial data set, u0, u1, and let's also assume for simplicity that, let's say, u0 is equal to a constant, u0 of x, outside a compact set. So it takes smooth, compactly supported data. In the case of a map, this means it's a constant map outside of a compact set with small critical norm. So u1 measured in a soil-blast space, hd over 2 cross hd over 2 minus 1. It's less than epsilon. So for any such data, there exists a unique, smooth, global in time solution, u of t. And moreover, we can say that the same norm of u of t, so the sup and t in our u of t, this stays around the size epsilon. And moreover, we have the following qualitative estimate on the asymptotic behavior of u. So u of t minus u infinity and measured in L infinity in space converges to 0 as t goes to plus or minus infinity. OK, so we also can prove a bit more. In particular, we have a more precise statement on the asymptotic behavior, namely, that of scattering. But it's difficult to formulate scattering at this point. I need to introduce the notion of a gauge first. So we also prove scattering, but inappropriate gauge. Yeah, this is a constant map. So this is a measure in L infinity. So yes, it's given by the initial data. So the solution converges to a constant, just to a point on the target as t goes to infinity. Sorry, what's n? Ah, and n, the target is no, n is just a nice Riemannian manifold, say, a bounded geometry, so including, say, hyperbolic space, the sphere, nice, yeah, yeah, n is a nice Riemannian manifold. And this is a small data result. So for large data, then the dynamics of the map equation depends a lot on the target. But here, this is a small data problem. So here, this is for any target. OK, so let me make a few remarks. And so one, this is a small data critical theory for the wave map problem. And why do I say critical? So in what sense is this critical? I don't have a scaling invariance. But in the case sigma equals Rd, then h dot d over 2 plus h dot d over 2 minus 1 is invariant, is unique, so both ways. So the norm here is invariant under the scaling of the equation, under the scaling. Thank you. So if u is a solution, then so is u lambda t of x, which is u of lambda t, lambda x. And so this is the analog of a small data scaling critical global result. OK, and I'll say more about this problem in Euclidean space in a minute. So just a couple of remarks, why are we doing this? So why hd? Well, one, it's just a natural starting point to study a geometric wave equation on a curved background. It's a nice, it's a constant sectional curvature geometry. And well, we have a head start in that there's well, this is the linear theory on hd. So the linear theory has been well studied. So for solutions to this equation, we'll study it in particular. So you have nice Sturckart's estimates. So there's nice dispersive theory. And so their global Sturckart's estimates prove for the free equation on hyperbolic space. And this is, I'm not going to write all the citations on the board just to the sake of time, but I'll mention a few. For the Schrodinger equation, there's work by UNESCO and Stoffelani in Valeria Banica. And then for the wave equation on hyperbolic space, we rely on the Sturckart's estimates of Metcalf and Taylor and Anker and Pierfellicci. So I'm still trying to answer this question, why hd? And then second, why high dimensions? Well, again, this is starting in high dimensions is an easier problem because we have better dispersion in higher dimensions. The way of my problem, it becomes quite difficult outside of symmetry in dimensions 2 and 3. 2 is the most interesting dimension because you see when d equals 2, the critical Soblov norm is the energy space. So it's h1 cross l2. So what we're really interested is in studying this problem in the energy space when d equals 2. But we're starting with this theory in high dimensions to develop some technique needed to address the low dimensional case. OK, so d equals 4 is easier. It's a starting point. And OK, so now maybe I'll get to this question of why. Well, we've taken a look at the energy critical case, but under a symmetry reduction. So with my colleagues, we've studied the energy critical case under co-rotational symmetry. And so what does this mean? So particularly, we looked at maps from r cross h2 taking values in either s2 or h2. So in two different two-dimensional rotationally symmetric targets, and co-rotational symmetry means that the map respects the action of rotation of both the domain and the target. And in this case, we saw qualitatively different behavior than one sees in the Euclidean problem. So I won't talk too much about this today, but many of you have seen one of us talk about some of these works in the past. But what we found interesting is that even in the case, so for the h2 target in the two-dimensional case under co-rotational symmetry, there exists many finite energy harmonic maps. And these play an interesting role in the dynamics. So they're all asymptotically stable. And in particular, so not just asymptotically stable, but they're globally asymptotically stable. So meaning you have stable soliton resolution. In the case of the h2 target, so you can look at the initial data. And there's a geometric property of the initial data that you can identify that tells you which harmonic map you scatter to. OK, so this is in stark contrast to the Euclidean case where it's well known, say, from the 60s, even the work of Il Samson, that there are no finite energy harmonic maps from r2 into a manifold of negative sectional curvature. OK, the case of the h2 target is perhaps even more interesting. There is, again, a continuous family of harmonic maps. And these display some interesting stability properties that I won't get into today. But it relates to the talk of Michael on Monday, meaning that one sees, in fact, anonymously slow decay. So you have stable harmonic maps, one sees anonymously slow decay to these harmonic maps. So arbitrarily slow decay rates. So both of these problems we thought were interesting. And our goal in studying the wave map problem outside of symmetry is to try to then study non-corotational perturbations of these harmonic maps. So to study the full theory. And OK, and then there's, of course, then many more finite energy harmonic maps once we leave symmetry as well. OK, so let me leave this aside. This is some motivation. And now talk a bit about how to prove this theorem. So first, why is it difficult? This is a semi-linear problem. And I have small, smooth, compactly supported data. Well, the main difficulty lies in the quadratic derivative non-linearity. And it's well known that this is non-perturbative, meaning that you can't close an iteration argument based solely on estimates for box. So in fact, one can't just close using semi-linear techniques. And this was a difficult, for in the case when sigma is Rd, this small data problem attracted quite a lot of research over the interest over the 90s and early 2000s. There was fundamental work by Kleinermann and Macadam and Kleinermann and Selberg for the sharp subcritical local opposeness theory. And then there was breakthrough work of Tataru and then followed by Tao in around 2001 who proved the analog of this theorem in the case when sigma is Rd and when the target manifold is the sphere. And this was followed then by work of Kleinermann and Ronjanski who established the same theorem for more general targets and then Jochen Krieger as well for the case of the hyperbolic target. So we'll get to a bit more of the history in a minute. But let me just say what was Tao's key insight. So the key insight of Tao was that, maybe I'll just say this, was that because I'll get into it in a second on the board, was that the gauge structure of the wave map equation can be used to renormalize away the non-perturbative part of the non-linearity. So he performed a gauge transform which rendered this quadratic, or at least the worst part of this quadratic derivative interaction, perturbative, amenable to perturbative analysis. OK. And this idea of using gauge structure arose in the harmonic map literature in the work of Hiller on 2D harmonic maps, the regularity of 2D harmonic maps. OK. But then, so there's this breakthrough work of Tao. So this is the case in this SD. And then there was a deep simplification of the Tao theory, the Taro-Tao theory, in high dimensions due to Shata Struve and Namhad Stefanoff and Ulymbek in around 2000. 2002. There was this flurry of work in the early 2000s. And in particular, I'm going to focus on the work of Shata and Struve as it's their general approach to apply here. So what Shata and Struve realized was that you could perform a, so this is some deep simplification, one can perform a global gauge transform. And so they found a nice gauge transform, which in particular, they formulated the derivative formulation in the Coulomb gauge. So I'll explain this in a second. But what they saw is that if you put yourself in the Coulomb gauge, the whole non-linearity is essentially becomes cubic or higher, and hence a minimal to perturbative analysis. So this is roughly the approach we'll use. However, we'll see that the Coulomb gauge is in fact very poorly suited to the case of a curved background domain, and we'll have to resort to a different technique. But we'll roughly follow their approach. So let me outline how this goes. OK, so Shata and Struve work in the derivative formulation, which means that the idea is to work with du instead of u. And du is a one form over m with values and u star taking values in the pullback bundle. And it has the nice advantage of being linear. It's a one form. It's a section of this bundle. So this is a linear object. And the idea is to formulate the problem in terms of du. But the first step is to choose a global frame field, e, collection of n tangent vectors, varying smoothly, on u star tn, which is a vector bundle over m. OK, so why does such an object, why does such a global orthonormal frame exists because, OK, so the base is contractible manifold in our case. It's r cross hd, so the base is contractible. So you always have at least one such frame. OK, so choose a global orthonormal frame field. And then the choice of e gives rise to a connection form a. So in particular, the connection, the covariant derivative on u star tn can be expressed in terms of this connection in one form a, as follows, a alpha. We're here. This is the Leve Chavita connection on the base manifold m. So this extends naturally to tensor, the tensor bundle. I'm taking tensors taking values in this bundle. OK, and so this again appears in the formulation of the wave map problem. And now we'll express du in this frame. So write, so find one form psi such that du is e psi. Sorry, maybe I'll write this e j psi j. And the component will write the components of du as with lowercase psi. So d alpha u is e psi alpha. OK, so now the wave map problem becomes the following div curl system. So u of wave map translates now in this psi formulation to psi alpha is equal to 0. And then we have the following torsion free property of our connection. So this is our curl relation, d alpha psi beta minus d beta psi alpha is equal to 0. So the wave map problem becomes this div curl system. And what about this connection form a? Well, this satisfies, so a satisfies a curl type formula. So let me write this as a b alpha. So where f is the curvature two form on the pullback bundle. And since this is the pullback curvature, this is nothing other than the Riemannian curvature tensor on the target n at psi b psi alpha. So this is the fact that this is the pullback on the curvature bundle on n. So a satisfies this equation, but is otherwise undetermined. OK, so we have some freedom here. And our freedom amounts to the choice of e, which then determines a. And so Chateau-Schruve make a particular choice. They require a to be divergence free. So db, spatially divergence free. Equals 0, and this is called the Coulomb condition. OK, so this is the Chateau-Schruve framework for the wave map equation. And also the similar framework used by Namhad Ullimbeck and Steffenhoff. As you say, that this actually goes back to the Chan-Chateau-Ullimbeck, the Chan-Chateau-Ullimbeck on the Schrodinger map. Oh, OK. So yeah, so I was attributing it to Hiller and the harmonic map formulation. Was in the Schrodinger map. OK, sorry. What, to Ullimbeck's paper? Sure, sure. No, of course. Yeah, so I was going to mention the Ullimbeck paper from the early 80s, which shows you can always, in a small data setting, you can always, the question, we have to solve this divergence equation or define such a certain frame. And this is always be done in a small data setting with a fundamental paper of Ullimbeck from 82, I think. So thank you, Andrew. OK. So this is the column gauge setup. And where do we go from here? I'm sorry for the, OK. So here we just, now we'll just differentiate these equations. And we obtain an elliptic equation for A and a wave equation for psi. So we differentiate, and we obtain the following elliptic equation for A, A alpha. We have a curvature term, A beta, equals in the following linear wave equation for psi. Psi alpha beta is equal to, so excuse me for a second while I write all these indices down, psi alpha psi beta. OK. All right, this is a complicated looking equation. But in the case sigma is Rd, it simplifies quite nicely because these are curvature terms. So these are 0 in the case of the domain manifold is flat. And I won't write this all out again. But the left-hand side simply becomes the Laplacian of the components of A. So A is a one form. But for each component, we have the scalar Laplacian of A is equal to the right-hand side. I won't write this all out, R of U psi alpha. And the scalar del inversion of psi alpha is, so we have the following coupled system of elliptic equation for A and a nonlinear wave equation for psi. And now, Chateau-Struve, analyze this system using LP estimates for the standard elliptic theory estimates for A, and then strict arts estimates for this wave equation. And in particular, they prove control over, they prove opera estimates of a controlling norm for psi, and then they're able to close their small data argument. OK, so this is very nice. It's a very simple argument to prove the small data theory for wave maps. At least simple with today's at this point. Ah, so the dimension comes in handling this term. So in high enough dimensions, you have to look where can we, how do we analyze this term? So the right-hand side, let's say in, the right-hand side needs to go in L1T, L2X, say, in dimension four, for example. And we have no choice but to put this term now in energy type norm. So in L infinity in time and L2 in X, because we control one, this is in dimension four, this is H2. And so we control one, two derivatives of U, which is one derivative of psi. So this should go in L2. But that means we have to put A in L1 and T and L infinity in X. OK, and this So the point is that we have no choice but to always put this in L infinity and T. And then in lower dimensions, we can't control A in L1 and T. We have a failure of integrability. And so you need to rely on more sophisticated structure on the right-hand side. And there's no structure apparent here, which one relies on in lower dimensions. It's a very delicate analysis. And these simple spaces as well aren't, based on strict-rate systems, aren't quite enough to close. So in high dimensions, though, this we barely, at dimension four is a threshold for which a simple argument on strict-rate systems work. And it's because of this difficult term here. OK, sorry. All right. OK, so how about extending this now to the setting of a curved background? Well, OK, this now we're in trouble because we have these curvature terms here. And so the left-hand side doesn't reduce to a nice scalar elliptic equation or wave equation. And so what are these left-hand sides? Again, they're nice objects. This you can see is the Hodge-Laplacian of the one-form A. And this is the Hodge-Delimberschen of the one-form Psi. But these are tensor equations. And there are no good estimates known for these objects. So in particular, there's no global strict-rate system. It's known for the Hodge-Delimberschen acting on one-forms. And in fact, in the case of hyperbolic space, it's known that the Hodge-Laplacian behaves poorly on one-forms. In fact, there's a failure of boundiness of the restransform. OK, so there's bad elliptic theory and just unknown dispersive theory in this case. OK, so how the main question becomes, or one initial question becomes, is how do we formulate this problem correctly in the case of a curved background? In particular, how do we deal with the issue of tensor reality on the left-hand side for the main dynamic variable? So the main dynamic variable in this case are A alpha and Psi alpha. And the answer comes, there's a nice way to get around this if we use Tau's caloric gauge, which was introduced in a different context, but miraculously deals with this issue of tensor reality beautifully. So how does Tau's caloric gauge work? So let me actually formulate the question. So the issue is find framework that deals with one, the tensor reality of main dynamic equations, and two, still the point of this whole thing was to renormalize the non-linearity to make this amenable to perturbative analysis here. Sorry, I didn't say enough about this. But you see that in this case, A, we saw this elliptic equation in A, meaning we basically write A as gradient inverse. So it's still a Gaussian inverse of gradient. This is a nice curvature term, or target is a bound that has bounded geometry. So this is like Laplacian of Psi squared, for example. So A can be expressed nicely in terms of inverse derivative of Psi squared. And so we see here in the non-linearity that the non-linearity becomes essentially cubic in Psi. OK, so how do we deal with tensor reality and also suitably renormalize? OK, and so Tau's caloric gauge will do both of these things for us. So how does this work? It's a really beautiful idea, and it's kind of miraculous. Miraculously simple. So we choose the wrong slide. So it's based on the harmonic map heat flow for our wave map view. So we start with our goal is to prove operator estimates like these. So start with a smooth wave map u of tx. And we'll use this as initial data, so on a time interval i. And we'll use this as initial data for the harmonic map heat flow. So we'll introduce a new time variable s and solve ds minus Laplacian u is s of u dk u, dk u. And our initial data at time s equals 0 is u tx. So for each time t, we solve the harmonic map heat flow. So under mild bootstrap assumptions, the heat flow is well behaved in this small data setting. And in particular, it converges to the same constant map for each t and x as s goes to infinity. OK, so now we have, oh, I can put all this up, I guess. OK, so it converges, the heat flow under this nice bootstrap assumptions converges to a fixed point as s goes to infinity for each t and x. And now we're going to choose a, so now the next step is to choose a dynamic frame, e on u star tn, which is now sits over, let's put hd here to be precise, sits so we have introduced a new time variable s. So it sits over here. And there's a canonical choice for this frame at s equals infinity. Since we converge to a constant map, we simply choose the same set of orthonormal vectors over each point t and x in the domain at s equals infinity. So yeah, so we choose a dynamic frame, e, and a connection form a. So e is a function of s, t and x, and a is a function of s, t and x. And we require that e sdx converges to the same fixed frame as s goes to infinity. And then our second requirement, and this is tau's caloric, these are tau's caloric gauge conditions, is that the frame be parallel in s. So we require that ds of v is equal to 0, which is the same as requiring that as equals 0. So these are the caloric gauge conditions. And I'll try to motivate both of these choices in a second. But first, I'll write down our candidate for our new dynamic variable. So we'll express our map u in this frame. So right as before, except we also have ds of u is e psi s. So we'll find such a psi s. And then again, still d alpha of u is e psi alpha. And our candidate for the main dynamic variable will be psi s, which is now, this sits over, so we'll see if this is manifestly scalar, or it satisfies a scalar question. So we want to show now that if we just work with psi s, we satisfy both of those two questions. So we considerably renormalize the non-linearity. And also, we deal with tensorality of the main equations. Yes, it's all the same u infinity. So this is fixed by the data and by our bootstrap assumptions, yeah. So we bootstrap, so we assume that the map converges to infinity for all t next on this interval. OK. All right, so why is this a natural choice for the dynamic variable? Let me try to motivate this with a really simple example, namely that of the linear heat flow on Euclidean space used as Littlewood-Paley theory. So in analogy, there's a linear heat flow. So given a function f naught, we can solve the heat equation with this data. And the solution to the heat equation, so f of s is nothing other than convolution with a Gaussian. In the case of this is sigma is rd for a second. On the Fourier side is e to the minus s c squared f naught. And this is a Gaussian adapted to the ball of frequency c less than s to the minus 1 half. Gaussian adapted to this ball in frequency space. And so this damps the high frequencies of the function f. OK, to damp the low frequencies, so low frequencies, so damp the low frequencies, we can multiply by s times Laplacian. So we'll define Littlewood-Paley projection onto frequencies comparable to s to the minus 1 half by damping the low frequencies with this operator and then e to the s Laplacian s f naught, which is s times Laplacian of f of s and which is nothing other than, since f solves the heat equation, this is just s ds of f. So this damps low frequencies, this damps high frequencies, and this is roughly like projection onto frequencies like s to the minus 1 half. And one can, in fact, prove all the nice things about Littlewood-Paley theory, including the square function estimate in this framework on Euclidean space. So in particular, and what's nice is that we can recover f from this Littlewood-Paley resolution just by integrating, by integration. So this is ds over s, and this is our Littlewood-Paley projection. OK, so here, Littlewood-Paley projection onto frequencies s to the minus 1 half is given by the s derivative of our initial function. And so we'll make this analogy here. So we'll use the harmonic map heat flow resolution of our map u as geometric analog of Littlewood-Paley via heat flow. So this is our candidate for main dynamic variable, and it's a nonlinear Littlewood-Paley decomposition of our map u. OK, so let's just briefly show how this resolves the remaining issues. And we, of course, need this analog of this reconstruction formula, which is where precisely where tau's caloric gauge conditions enter. So u s tx converges to a fixed u infinity, and e s tx converges to a fixed frame e infinity, as s goes to infinity. And so one can see that all the other derivative components, so psi alpha, s tx, and along with the connection form, these then are forced to converge to 0, as s goes to infinity. And moreover, we have this caloric gauge condition, so ds, so as is equal to 0, is the same thing as ds is just partial s. And again, now this means for a that the curvature, if we have s in any one of the components here, sf alpha, this, remember, involves derivatives of a in the commutator of a with itself. So this, the only one that remains is ds a alpha. OK, so we can write psi alpha of s is the integral minus the integral from s to infinity, just by the fundamental theorem of calculus, ds psi alpha, ds prime, which is now using this caloric gauge condition, remember from s to infinity, capital D, so this is the, which is then, and now we use torsion free property, so this is nothing other than d alpha psi s. So we can recover all these derivatives, the psi alphas from psi s, derivatives of psi s, and the same with the a's. So a alpha s is ds a, minus this integral of fs, which is now the pullback curvature psi s, psi alpha. And so we see again, a is now quadratic in psi s and psi alpha, and we can, since we can recover psi alpha from psi s, we can now recover all the a alphas from just psi s alone as well. And then finally, the equation for psi s is we'll see a scalar nonlinear wave equation. So if we just take d alpha, d alpha psi s, we can commute these two, this is d alpha, ds psi alpha, and now we commute, we pull the s derivatives out. Remember this is just ds, but we introduce a curvature term, so this is ds of d alpha psi alpha, and then plus the curvature arising from these two, this is f alpha s psi alpha, and I'll call this w. So this is a scalar now, d alpha psi alpha is a scalar. This, let me just, OK, and we can expand this left-hand side. So this is just the component psi s, so this first guy acts like, this is like d alpha plus a alpha times, now this is now a tensor, but now it will actually act on like this. So this becomes the scalar del inversion of psi s is equal to ds w, so this term, plus f alpha s psi alpha, and what we get by expanding this product here. So this is minus 2 a alpha d alpha psi s minus a alpha psi s minus alpha a alpha psi s. OK, so this becomes the main dynamic equation that we'll then estimate using, this is now a scalar equation, we can estimate this using strict-rates estimates for the scalar del inversion hyperbolic space. And let me speak a bit about this nonlinear structure here. So what's w? w is d alpha psi alpha, and this is at 0, so remember this is the wave map equation, so this should be 0, but it's only 0 at s equals 0, because otherwise we're commuting the wave map, the harmonic map people and wave map equation don't commute. So w restricted to s equals 0, since we come from a wave map. This is equal to 0, and it satisfies, w can show satisfies a covariant heat equation, which we solve from s equals 0 to s. So this equation occurs at heat time s, so we have an equation for each s. So solve on 0 to s. And remember with our analogy to the linear heat flow, this is the s time between 0 and s. This is like the high frequency part of the function f. So w knows the high frequencies of the initial map u, so it arises from the high frequencies. And then the rest of the nonlinearity, these other four terms involve just the frequencies at s and lower, so in particular, so this involves frequencies s prime bigger than equal to s, which are the low frequency part of our initial map u. And we see this in these equations here. So a, when we see psi alpha and a alpha, this involves integrals of psi s from s to infinity. So this is the nonlinear structure that arises from the low frequency component of the map itself. OK, so this is kind of where the analysis now starts. So we need to study this equation and prove estimates. Let me just remark that in the initial Chateau-Struve approach, I said, where is this? The crucial bit about a is that we use elliptic theory to estimate a. And this is the same in the Naman-Stefanov-Ullembeck approach, use elliptic theory to estimate a. What's the replacement for that? Well, it's nothing other than the regularity theory for the harmonic map heat flow in this case. So we recover a and psi alpha just by integration, just by fundamental theorem here. But we use regularity theory for the parabolic equation, which is a scalar parabolic equation, to prove estimates that pseudo-wesemates and also estimate a in terms of psi s and psi alpha in terms of psi s. A couple other last remarks. So one, just some advertisement for the caloric age is that this is, we can see this is nicely, so all of this will work in the 2D setting as well. In fact, the 2D setting that I laid out earlier, because it just relies on understanding the harmonic map heat flow, which is a well-studied equation. So in particular, it works nicely in large data settings as long as you have a handle on the harmonic map heat flow. And in the cases where you want to study the wave map equation, you do have such a nice understanding. So it works well in large data setting where the column gauge is a little bit of a disaster once you leave the small data setting. You have a lot of difficulty with this elliptic equation here. In particular, there's quadratic interactions of A, which then become relevant, which I didn't even write down here, which they're perturbative in this small data setting, but they become very problematic in the large data setting. So the caloric age is nice in large data settings. And the last remark in my last minute is that the worst part of the non-linearity, so it also is better in the worst interactions in the non-linearity. So here I said that A was roughly, in the column gauge setting, A is roughly like inverse derivative of psi squared, at least in a small data setting. And here the worst interaction, you have a dangerous high-high to low interaction. So you have high-high inputs with a low output, and then they're hit with an inverse derivative. So this becomes dangerous in the column gauge, and this interaction just doesn't appear in the caloric gauge setting. So in fact, this appears in our analysis in a subtle way. But it's very important, becomes very important in the low data setting. And it was very important in particular to work on Schrodinger maps by Bejanaro, UNESCO, Kennegg, and Tataru. I hope I get the names. We're to write. OK. I'll stop there. Any questions? So I didn't understand what depends on the source being the hyperbolic space in which you said? So not much in what I said here. But the crucial fact is that at some point we're now going to start analyzing this equation, and we use Stryckart's estimates for the delimburson hyperbolic space. And so if you have a different geometry, so none of this relied in particular on the hyperbolic geometry, what I talked about today. But then if you want to start proving estimates, you need something like this. So is it true that if you have good Stryckart's estimates? Then this works, yeah. So it means that if, for instance, take sigma to be the Euclidean space with asymptotically flat metric, or Schrodinger's estimator? Yes, yes. Yeah, in particular. This is part of my thesis. I did something like this. Yeah. It's OK. No, it's all right. It's a long time ago. OK. Yeah.