Wave maps on hyperbolic space
CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor.
Lawrie, Andrew
The Cauchy problem for wave maps on hyperbolic space exhibits several features of a different nature than the corresponding problem on flat space. In this talk we'll focus on the question of gauge choice and we'll sketch the proof of small data global well-posedness and scattering in high dimensions using the moving frame approach introduced by Shatah and Struwe. In this setting of a curved domain, the argument will rely crucially on the fact that the main dynamic equations in Tao's caloric gauge are scalar, rather than the tensorial equations that arise in say, the Coulomb gauge. This talk is based on joint work with Sung-Jin Oh and Sohrab Shahshahani.
 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. So everything, the theorem I'll present today is joint work with Sungjin Oh and Sae-Rab 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 Lorentzian metric, eta, taking values in a Romanian manifold, N with metric, G. And in this talk, we'll let M take the product form, so it'll be of the form R cross sigma, where sigma is a Romanian 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. So wave maps are formal critical points of a Lagrangian 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. And here in coordinates, so it's take, for example, Tx coordinates on M and coordinates Uj on the target. And then this can be, this is expressed as follows, so this is equal to eta alpha beta Tx, Gij of U, d alpha Ui, d beta Uj. And then the volume form is square root of determinant of eta dt dx. So this is the action, and critical points, satisfy the following Euler-Grangic equations, d alpha U is equal to zero. Okay, in the talk, I'll use the convention that I'll be 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 Ijk for indices on the target. Okay, 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. Okay, so this is still quite abstract, in the case when N, I'm going to run out of chalk here, N is embedded, isometrically embedded into Rn, then this equation takes the form, the delimbursion on M of U is perpendicular to the tangent space of U, sorry, of N over the point U, and this can then be simplified as follows, so the delimbursion 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, I mean it's quadratic in the derivatives of U. Okay, so we'll study the Cauchy problem, which means that we'll give ourselves initial data, U, dT of U, where initial data, so U0 is a map from sigma into N, so I'll take a smooth map say into 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 and N, so this is an initial data set for the wave map problem. Okay, 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. Okay, so the main theorem for today is that in the case, so sigma is HD, so d dimension like revolvex 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 sub-bluff 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, 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. Okay, 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 in an appropriate gauge. Yeah, this is a constant map, so this is a measurement of 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. What's n? Sorry, what's n? No, n is just a nice Riemannian manifold, say, bounded geometry, so including, say, hyperbolic space, the sphere, n is a nice Riemannian manifold. This is a small data result, and for large data, then the dynamics of the wave map equation depends a lot on the target, but here this is a small data problem, so here this is for any target. Okay, 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? This is critical, I don't have a scaling invariance, but in the case sigma equals rd, then h.d over 2, h.d over 2 minus 1 is invariant, is a unique solvab space, 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. Okay, 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 h.d? 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 consensational curvature geometry. And well, we have a head start in that there's, this is the linear theory on h.d, so the linear theory has been well studied. So for, so solutions this equation we'll study it in particular. So you have nice strict-guards estimates, so there's nice dispersive theory. And so the global strict-guards 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. There's, for the Schrodinger equation, there's work by UNESCO and Stoffelani in Valerio Bonica. And then for the wave equation on hyperbolic space, we rely on the strict-guards estimates of Metcalf and Taylor and Anker and Piafolecci. Okay, so, okay, so I'm still trying to answer this question, why h.d? And then second, why high dimensions? Well, okay, again, this is starting in high dimensions, this is an easier problem because we have better dispersion in higher dimensions. The wave of my problem, it becomes quite difficult outside of symmetry in dimensions two and three. Okay, two is the most interesting, interesting dimension because you see when d equals two, the critical Soblov norm is the energy space, so it's h1 cross l2. So what we're really interested in is studying this problem in the energy space when d equals two. But we're starting with this theory in high dimensions to develop some technique needed to address the low-dimensional case. Okay, so d equals four is easier. It's a starting point. And, okay, 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, so 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 Eel-Sampson, 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, 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. OK, 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 Kleinerman and McAdon and Kleinerman 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. OK, and this was followed then by work of Kleinerman and Ronjanski, who established the same theorem for more general targets, and then Jochen Krieger, as well for the case of hyperbolic target. OK, so let me, and 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, 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, regularity of 2D harmonic maps. OK, but then, so there's this breakthrough work of Tao, so this is the case in SD. And then there was a deep simplification of the Tao theory, the Taro-Tao theory, in high dimensions due to Schatt-Struve and Namhad Stefanoff and Ullinbeck 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 Schatt-Struve as it's their general approach to apply here. So what Schatt-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. So Schatt-Struve worked in the derivative formulation, which means that they work with, the idea is to work with a 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 their idea is to choose, 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. So why does such a global orthonormal frame exists? Because the base is contractable manifold in our case. It's r cross hd, so the base is contractable, so you always have at least one such frame. So choose a global orthonormal frame field, and then the choice of E gives rise to a connection form A. So in particular, the co-variant derivative on u star tn can be expressed in terms of this connection one form A, as follows, A alpha. We're here. This is the LaVe-Schwitte connection on the base manifold m. So this extends naturally to tensor, the tensor bundle, taking tensor-staking 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 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 a wave map translates now in this psi formulation to d alpha 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 like this. Let me write this as AB, AB, F, 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-Struve 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-Struve framework for the wave map equation. And also the similar framework used by Namhad Ullimbeck and Stefanoff. I should say that this actually goes back to the Chant Chatan Ullimbeck. Chine. The Chant Chatan Ullimbeck on the Schrodinger map. Oh, OK. So yeah, so I was attributing it to Heller and the harmonic map formulation. In the first time that they wrote it like that. Was in the Schrodinger map. OK, sorry. What 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 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 set up. And where do we go from here? Sorry for the. OK, so here we just now will 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 beta equals, and the following nonlinear wave equation for psi. Psi alpha psi 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 components, 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 b 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, Chattano-Struve analyzed this system using LP estimates for, so standard elliptic theory estimates for A, and then strict-arts estimates for this wave equation. And in particular, they prove control over, they prove operator estimates, L2D, 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. So the dimension comes in handling this term. So in high enough dimensions, you have to look where can we analyze this term. So the right-hand side needs to go in L1, T, L2, x in dimension 4, 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 4. 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 in T and L infinity in x. OK, and this is the point is that we have no choice but to always put this in L infinity in T. And then in lower dimensions, we can't control A in L1 in T. We have a failure of intergrability. And so you need to rely on more sophisticated structure on the right-hand side. And, ably, 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, based on strict rate testaments, aren't quite enough to close. So in high dimensions, though, this we barely. At dimension 4 is a threshold for which a simple argument on strict rate testaments work. And it's because of this difficult term here. 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 test that'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 is 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 a 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 bound to 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. So how do we deal with tensor reality and also suitably renormalize? OK. And so tau's caloric age 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 u. So 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 the Gaussian 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, where should I go here? Oh, I can put all this up, I guess. OK, so it converges the heat flow under this nice bootstrap assumptions. It converges to a fixed point as s goes 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 over, 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 caloricate, these are Tau's caloricate 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 caloricate 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 write as before, except we also have ds of u. This 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 that this is manifestly scalar, and 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 can suitably renormalize the non-linearity. And also we deal with tensorality of the main equations. Yes? So we just assume it's the same u infinity? Yes, it's all the same u infinity. Yeah, so this is fixed by the data and by our bootstrap assumptions. So just we bootstrap, 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 little with 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 the Gaussian, right? In the case of this is sigma is rd, for a second. And the Fourier side is e to the minus s xc squared. F naught, 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 little of 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 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 little 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 little Paley resolution just by integrating, by integration. So this is ds over s, and this is our little Paley projection. OK, so here, little 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 little Paley of the little Paley via heat flow. So this is our candidate for main dynamic variable, and it's a nonlinear little 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 d s, so a s is equal to 0, is the same thing as d s 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 is the only one that remains is d s, 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, d s psi alpha, d s prime, which is now using this caloric gauge condition, and we're going to s to infinity, capital D, 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 d s, a minus this integral of f s, sorry, alpha, 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 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 d s psi alpha. And now we can pull the s derivatives out. And remember, this is just d s, but we introduce a curvature term. So this is d s 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 we would actually act on like this. So this becomes the scalar del inversion of psi s is equal to d s 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 a alpha psi s minus lambda alpha a alpha psi s. OK, so this becomes the main dynamic equation that we'll then estimate using strict. This is now a scalar equation. We'll then estimate this using strict. It's 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 heat flow and wave map equation don't commute. So w restricted to s equals 0 since we've 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 is at heat time. Oh, 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 s, which are the low frequency part of our solution, 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 improve 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 Navman-Steffanov-Ulimbeck 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-wastemates allow us to 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 gauge 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 perturbed of in this small data setting, but they become very problematic in the large data setting. So the cloricage 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 cloricage setting. So in fact, this appears in our analysis in a subtle way. But it becomes very important in the low data setting, and it was very important in particular to work on Schrodinger maps by Bejanaro, UNESCO, Kenig, and Tatar. I hope I get the names. We're all right. OK. Oh. I'll stop there. Thanks. Thank you. Thank you. Thank you. Thank you. Thank you. Any questions? So I didn't understand what depends on the source being the hyperbolic space, in what 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 Stryker's estimates for the Dalen-Version hyperbolic space. And so if you have a different geometry, so none of this relied in particular on hyperbolic geometry what I talked about today, but then if you want to start to improving estimates, you need something like this. So is it true that if you have good Stryker's estimates? Then this works, yeah. So it means that if, for instance, take a sigma to be the Euclidean space with asymptotically flat metric, you should get the simulator? Yes. Yeah, in particular. This is part of my thesis. I did something like this. Yeah. No, it's all right. It's a long time. OK. Yeah. So some other questions for you, Max? OK, so let's thank you. Thank you.
 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.  So everything, the theorem I'll present today is joint work with Sungjin Oh and Sae-Rab 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 Lorentzian metric, eta,  taking values in a Romanian manifold, N with metric, G.  And in this talk, we'll let M take the product form, so it'll be of the form R cross sigma,  where sigma is a Romanian 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.  So wave maps are formal critical points of a Lagrangian 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.  And here in coordinates, so it's take, for example, Tx coordinates on M and coordinates Uj on the target.  And then this can be, this is expressed as follows, so this is equal to eta alpha beta Tx, Gij of U, d alpha Ui, d beta Uj.  And then the volume form is square root of determinant of eta dt dx.  So this is the action, and critical points, satisfy the following Euler-Grangic equations, d alpha U is equal to zero.  Okay, in the talk, I'll use the convention that I'll be 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 Ijk for indices on the target.  Okay, 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.  Okay, so this is still quite abstract, in the case when N, I'm going to run out of chalk here, N is embedded,  isometrically embedded into Rn, then this equation takes the form, the delimbursion on M of U is perpendicular to the tangent space of U,  sorry, of N over the point U, and this can then be simplified as follows,  so the delimbursion 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,  I mean it's quadratic in the derivatives of U.  Okay, so we'll study the Cauchy problem, which means that we'll give ourselves initial data, U, dT of U,  where initial data, so U0 is a map from sigma into N, so I'll take a smooth map say into 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 and N, so this is an initial data set for the wave map problem.  Okay, 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.  Okay, so the main theorem for today is that in the case, so sigma is HD,  so d dimension like revolvex 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 sub-bluff 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, 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.  Okay, 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 in an appropriate gauge.  Yeah, this is a constant map, so this is a measurement of 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.  What's n?  Sorry, what's n?  No, n is just a nice Riemannian manifold, say, bounded geometry,  so including, say, hyperbolic space, the sphere,  n is a nice Riemannian manifold.  This is a small data result,  and for large data, then the dynamics of the wave map equation depends a lot on the target,  but here this is a small data problem, so here this is for any target.  Okay, 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?  This is critical, I don't have a scaling invariance,  but in the case sigma equals rd,  then h.d over 2,  h.d over 2 minus 1 is invariant,  is a unique solvab space,  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.  Okay, 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 h.d?  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 consensational curvature geometry.  And well, we have a head start in that there's,  this is the linear theory on h.d,  so the linear theory has been well studied.  So for,  so solutions this equation we'll study it in particular.  So you have nice strict-guards estimates,  so there's nice dispersive theory.  And so the global strict-guards 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.  There's, for the Schrodinger equation,  there's work by UNESCO and Stoffelani in Valerio Bonica.  And then for the wave equation on hyperbolic space,  we rely on the strict-guards estimates of Metcalf and Taylor  and Anker and Piafolecci.  Okay, so, okay, so I'm still trying to answer this question, why h.d?  And then second, why high dimensions?  Well, okay, again, this is starting in high dimensions,  this is an easier problem because we have better dispersion  in higher dimensions.  The wave of my problem, it becomes quite difficult  outside of symmetry in dimensions two and three.  Okay, two is the most interesting,  interesting dimension because you see when d equals two,  the critical Soblov norm is the energy space,  so it's h1 cross l2.  So what we're really interested in is studying this problem  in the energy space when d equals two.  But we're starting with this theory in high dimensions  to develop some technique needed to address the low-dimensional  case.  Okay, so d equals four is easier.  It's a starting point.  And, okay, 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,  so 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 Eel-Sampson, 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, 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.  OK, 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 Kleinerman and McAdon  and Kleinerman 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.  OK, and this was followed then by work of Kleinerman  and Ronjanski, who established the same theorem  for more general targets, and then Jochen Krieger,  as well for the case of hyperbolic target.  OK, so let me, and 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,  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, regularity of 2D harmonic maps.  OK, but then, so there's this breakthrough work of Tao,  so this is the case in SD.  And then there was a deep simplification of the Tao  theory, the Taro-Tao theory, in high dimensions due to  Schatt-Struve and Namhad Stefanoff and Ullinbeck  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  Schatt-Struve as it's their general approach to apply here.  So what Schatt-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.  So Schatt-Struve worked in the derivative formulation,  which means that they work with, the idea is to work with  a 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 their idea is to  choose, 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.  So why does such a global orthonormal frame  exists?  Because the base is contractable manifold in our case.  It's r cross hd, so the base is contractable,  so you always have at least one such frame.  So choose a global orthonormal frame field,  and then the choice of E gives rise to a connection form A.  So in particular, the co-variant derivative on u star tn  can be expressed in terms of this connection one form A,  as follows, A alpha.  We're here.  This is the LaVe-Schwitte connection on the base manifold m.  So this extends naturally to tensor, the tensor bundle,  taking tensor-staking 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  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 a wave map translates now in this psi formulation  to d alpha 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 like this.  Let me write this as AB, AB, F, 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-Struve 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-Struve framework  for the wave map equation.  And also the similar framework used by Namhad Ullimbeck  and Stefanoff.  I should say that this actually goes back to the Chant Chatan Ullimbeck.  Chine.  The Chant Chatan Ullimbeck on the Schrodinger map.  Oh, OK.  So yeah, so I was attributing it to Heller  and the harmonic map formulation.  In the first time that they wrote it like that.  Was in the Schrodinger map.  OK, sorry.  What 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 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 set up.  And where do we go from here?  Sorry for the.  OK, so here we just now will 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 beta equals,  and the following nonlinear wave equation for psi.  Psi alpha psi 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 components,  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 b 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, Chattano-Struve analyzed this system using LP  estimates for, so standard elliptic theory estimates for A,  and then strict-arts estimates for this wave equation.  And in particular, they prove control over,  they prove operator estimates, L2D,  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.  So the dimension comes in handling this term.  So in high enough dimensions, you  have to look where can we analyze this term.  So the right-hand side needs to go in L1, T, L2,  x in dimension 4, 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 4.  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 in T and L infinity in x.  OK, and this is the point is that we have no choice but to always  put this in L infinity in T. And then in lower dimensions,  we can't control A in L1 in T. We have a failure of intergrability.  And so you need to rely on more sophisticated structure  on the right-hand side.  And, ably, 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,  based on strict rate testaments, aren't quite enough to close.  So in high dimensions, though, this we barely.  At dimension 4 is a threshold for which a simple argument  on strict rate testaments work.  And it's because of this difficult term here.  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 test  that'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  is 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 a 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 bound to 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.  So how do we deal with tensor reality and also suitably  renormalize?  OK.  And so tau's caloric age 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  u.  So 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 the Gaussian 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, where should I go here?  Oh, I can put all this up, I guess.  OK, so it converges the heat flow  under this nice bootstrap assumptions.  It converges to a fixed point as s goes 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 over, 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  caloricate, these are Tau's caloricate 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 caloricate 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 write as before, except we also have ds of u.  This 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 that this is manifestly scalar,  and 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 can suitably renormalize the non-linearity.  And also we deal with tensorality of the main equations.  Yes?  So we just assume it's the same u infinity?  Yes, it's all the same u infinity.  Yeah, so this is fixed by the data  and by our bootstrap assumptions.  So just we bootstrap, 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 little with 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 the Gaussian, right?  In the case of this is sigma is rd, for a second.  And the Fourier side is e to the minus s xc squared.  F naught, 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 little of 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 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 little 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 little Paley resolution  just by integrating, by integration.  So this is ds over s, and this is our little Paley projection.  OK, so here, little 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 little  Paley of the little Paley via heat flow.  So this is our candidate for main dynamic variable,  and it's a nonlinear little 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 d s, so a s is equal to 0, is the same thing as d s  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 is the only one that remains is d s, 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, d s psi alpha,  d s prime, which is now using this caloric gauge condition,  and we're going to s to infinity, capital D,  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 d s, a minus this integral of f s,  sorry, alpha, 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 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 d s psi alpha.  And now we can pull the s derivatives out.  And remember, this is just d s, but we  introduce a curvature term.  So this is d s 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 we  would actually act on like this.  So this becomes the scalar del inversion of psi s  is equal to d s 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  a alpha psi s minus lambda alpha a alpha psi s.  OK, so this becomes the main dynamic equation  that we'll then estimate using strict.  This is now a scalar equation.  We'll then estimate this using strict.  It's 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 heat flow  and wave map equation don't commute.  So w restricted to s equals 0 since we've  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 is at heat time.  Oh, 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 s, which are the low frequency part  of our solution, 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 improve 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 Navman-Steffanov-Ulimbeck  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-wastemates allow us  to 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 gauge  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 perturbed of in this small data  setting, but they become very problematic  in the large data setting.  So the cloricage 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 cloricage setting.  So in fact, this appears in our analysis in a subtle way.  But it becomes very important in the low data setting,  and it was very important in particular to work  on Schrodinger maps by Bejanaro, UNESCO, Kenig, and Tatar.  I hope I get the names.  We're all right.  OK.  Oh.  I'll stop there.  Thanks.  Thank you.  Thank you.  Thank you.  Thank you.  Thank you.  Any questions?  So I didn't understand what depends  on the source being the hyperbolic space, in what 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 Stryker's estimates for the Dalen-Version  hyperbolic space.  And so if you have a different geometry,  so none of this relied in particular on hyperbolic geometry  what I talked about today, but then if you want to start  to improving estimates, you need something like this.  So is it true that if you have good Stryker's estimates?  Then this works, yeah.  So it means that if, for instance, take  a sigma to be the Euclidean space with asymptotically  flat metric, you should get the simulator?  Yes.  Yeah, in particular.  This is part of my thesis.  I did something like this.  Yeah.  No, it's all right.  It's a long time.  OK.  Yeah.  So some other questions for you, Max?  OK, so let's thank you.  Thank you.
Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_0_599.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_600_1415.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_1416_5315.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_5316_5963.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_5964_8219.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_8220_9851.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_9852_10199.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_10200_12611.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_12612_13391.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_13392_15023.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_15024_21839.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_21840_24647.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_24648_26627.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_26628_29615.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_29616_29903.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_29904_33587.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_33588_34091.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_34092_36407.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_36408_36815.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_36816_37799.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_37800_38927.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_38928_56315.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_56316_56399.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_56400_58319.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_58320_65183.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_65184_65627.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_65628_72191.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_72192_73439.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_73440_74351.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_74352_75431.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_75432_76091.png Andrew_LAWRIE_-_Wave_maps_on_hyperbolic_space_76092_76941.png