Toponogov's theorem and improved Kakeya-Nikodym estimates for eigenfunctions on manifolds of nonpositive curvature
Sogge, Chris
Toponogov's theorem and improved Kakeya-Nikodym estimates for eigenfunctions on manifolds of nonpositive curvature This is joint work with Matthew Blair. Using wave equation techniques and elementary facts from Riemannian geometry, we show that, on negatively curved manifolds, eigenfunctions cannot concentrate near geodesics as measured in $L^2$. From this we obtain improved $L^p$ estimates which complement the sup -norm bounds in this setting obtained by Berard in the 1970s. Time permitting, we shall also discus related joint work with Y. Xi and C. Zhang on period integrals on Riemannian surfaces of negative curvature
 Well, thank you very much for inviting me to this conference. The conference, of course, is called Nonlinear Waves. I have an apology. My talk's about linear things. But it's about estimates that have proved to be useful for proving various types of nonlinear theorems, existence and so on. For instance, the types of estimates that I'll be talking about in the case of domains were used by Burke, LeBeau and Plancheon to prove existence for the critical wave equation and domains in R3 and so on. And also we've been hearing about how good estimates for the torus lead to good existence theorems in that setting. There are also theorems about good existence for NLS for the sphere, for spheres. The frustrating thing is that there's really not good theorems for manifolds of negative curvature. So I'm going to talk about improving LPS estimates and KK adnickadim estimates, whatever that is in that setting. Maybe someday there will be some progress on NLS in that setting. Okay, let's see here. So the setting is that I'll be talking about manifolds without boundary, compact. Of course they have a metric G and the dimension will be two or more. My little Plancheon is negative and lambda j is the frequency of the eigenvalue. Geometers don't put the square there usually. So lambda j is the eigenvalue corresponding to the first order operator, which is the square root of minus little Plancheon. All the eigenfunctions are L2 normalized. dv is, of course, the volume element. The little Plancheon is still a Plancheon coming from the metric. And so a vague question is how can you detect and measure various types of concentration in these guys? And you're going to expand your horizons a little bit and not just concern yourself with eigenfunctions but near eigenfunctions, quasi-modes, whatever that is. And you expect that you might get an extreme concentration at certain types of points, which would lead to bad soup norms or bad LP norms for large p for certain types of points. And you also might expect to get concentration near periodic geodestrics. If you're going to get concentration on a different sort of set, you'd expect the set to be invariant under the flow. So the natural thing is periodic geodestrics. Okay? And I'm going to concentrate on the latter. So geodestic, of course, is bigger than a point, so you might expect to detect concentration along periodic geodestrics in terms of LP norms for relatively small p. And that turns out to be the case. Okay? And you can measure concentration or dispersion in various ways. Well, I'm on a compact manifold, so there's nowhere for the eigenfunctions to disperse. But there is a famous problem, which says, for instance, when you have negative curvature, if you take these probability measures, then they should converge weekly to the uniform measure. So that's called the quantum-unique ergodicity conjecture. And that would be an ideal thing. It's very difficult to prove. It's conjectured in the case of negative curvature by Rudnick and Sardak, and it was proved in some special cases, for instance, by Lyndon Strauss, and that was one of the things that they cited when he won the Fields Medal. Okay, so if this is your goal to try to show that something like this happens, it fails miserably. This should be the volume in the manifold. It fails miserably on the sphere. The sphere is the worst case for everything I'm talking about today, so let's go over that to sort of set up the better things. So of course, the sphere is up there, the n-sphere. It's a subset of our n plus 1 given by that equation. And the eigenvalues of the square root of the Laplacian on the sphere are just given by this formula, basically k, square root of k times k plus n minus 1. And they repeat with a very high multiplicity. That's the highest multiplicity that's allowed because of the sharp vial formula. And what's going on there is there's a lot of periodicity. All the geodesics are periodic with period 2 pi. They're all the great circles. And additionally, if you take p, instead of taking the wave groups that we've been hearing about, but you complete the square by doing that, then these half wave groups are periodic. They're either periodic with period 2 pi in the case of n odd or 4 pi in the case of n even. And so the periodicity of these half wave operators also accounts for the bad behavior of the eigenfunctions. The eigenfunctions are nothing but the spherical harmonics, which are the restriction of homogeneous harmonic polynomials in Rn plus 1 to the n sphere. It is hot here. All right, so let's talk about the worst type of spherical harmonics. The worst type of eigenfunction if you don't like concentration near periodic geodesics. So here's the guy which is highly concentrated on the set where the modulus of x1 plus i x2 is equal to 1. So in other words, the place on the sphere where the other coordinates are 0. It's equator. So it's highly concentrated on the equator. I put that factor k to the n minus 1 over 4 to l2 normalize it. So you can evaluate the modulus of this. It's exactly that, of course, by calculus. You tailor expand the natural log about 1, and you'll find that it's like this. These are called Gaussian beams. See they live essentially in the set where this has the argument here is of size 1. And so that's a k to the minus 1 half tube about the equator. I'll soon be calling my eigenvalues lambda. And so this is living in a tube about the equator where the tube has width lambda to the minus 1 half. For all practical purposes, because of that Gaussian factor, you could just pretend that this guy in terms of his size is the indicator function of that tube times a normalizing factor. You're not really making too many mistakes. And so therefore you can compute its lp norm. So you have this factor out front. And then if you compute the lp norm, you'll get the volume of this tube to the 1 over p, which is k to the n minus 1 over 2 with a times minus 1 over p. So the norms of these guys are on the nose comparable to this. And this works for all p bigger than or equal to 2. So they have bad lp norms. And so here's a picture of some spherical harmonics. So the ones that I was describing, well, this never vanishes. So if you took real parts of this, you get a real eigenfunction, usually study real eigenfunctions. So if you take the real part of this guy, it's called the highest weight spherical harmonics, and it's these guys. As k increases because of the Gaussian beam behavior, like some speakers, almost all the mass is on the equator. Zonal functions, which I won't be talking about, but that's a separate talk. They are functions which are peaking at the north and the south pole through the middle guys. The first guy is pretty boring. It's just the constant function. And then you have spherical harmonics of higher and higher order. And these pictures also denote where it's zero, which is a very interesting problem, the nodal sets. And already, what is this? The spherical harmonics of degree 1, 2, 3, the spherical harmonics of degree 3, you could see this very high concentration of the highest weight spherical harmonics. Okay? All right, so let's talk about saturation of norms. So many years ago, I showed certain LP estimates for eigenfunctions, and there are two ranges. This is critical exponent, which is 2 times n plus 1 of rand minus 1. This should be a familiar exponent to lots of you guys. It's the exponent that pops up in Stryckart's estimates, for instance. So I showed that if you're below that critical exponent, you have exactly the same norms that we were talking about. K is now lambda, and it's k raised to that power, n minus 1 over 2 times the gap between 1 over 2 and 1 over p. And because of the calculations that we did in the previous slide, we see that this estimate up here from the 1980s is sharp. It can't be improved. The universal estimate holds on all compact Riemannian manifolds. It can't be improved because of these bad guys, the highest weight spherical harmonics. I won't be talking about this, but I also proved estimates for larger exponents where the power of lambda turns out to be something different. It turns out to be this thing, n times the gap, 1 half minus 1 over p minus 1 half. Way back when I was doing this, that was the case that I cared about, because that number pops up in harmonic analysis. It's what's called the critical index for bachnery summation. And what I wanted to do back then was to extend some results about harmonic analysis in Euclidean space to harmonic analysis on manifolds. And I was able to do it because of that estimate. And this estimate in red, I put in my paper because you might as well put everything you can do in the paper, but I thought it was a boring estimate. And now this estimate, this red estimate is much more interesting. It's linked to several things, actually including this. So the idea is that it's uniform for many, and then it's equal for any many? Yeah, yeah. The constants involved, of course, depend on the manifold, but the growth rate landed to whatever power always holds. It's what I would call local estimate. You prove it using local techniques. Soon we'll see the wave equation. I forgot to say that one of the reasons that I don't feel so guilty is that I'll show you how you can use wave equation techniques to study harmonic analysis, and especially eigenfunction theory. You prove this estimate, all you need to do is to understand the wave equation for a unit period of time. You don't need to watch a long movie. And that makes sense, right? Because on the sphere, these sorts of wave operators that arrive are periodic. So if you watch the movie for time 2 pi or 4 pi, you know everything. That doesn't happen in the case of manifolds of negative curvature. In that case, in some sense, the target manifold won't turn out to be a hyperbolic space, but the ammunition manifold, as we'll see at the end, will be hyperbolic space, whatever that means. All right. Let's see. You want me asking a simple-minded question? Sure. This definitely points out the periodic orbits. But at least some metrics have invariant tori or more complicated higher-dimensional tori or other things. And those also are recurrent. And those also play a role somehow in spectral mass and products. Yeah. Like, not exactly talking about what you were saying, but a zoolom manifold will also have these bad properties. A zoolom manifold, all the G-dessex are periodic to the same period. And if you're willing to broaden your horizon just a tiny bit and consider what are called quasi-modes, they're disastrous for the same reasons for estimates like this. Okay. Okay. Let's see. All right. So I was going to say this. Knowing when you can improve these estimates for small exponents is something that's only been recently understood. I've worked with Steve Zeldic quite a bit. And moreover, the work of Bayerard from the 1970s says that if you're willing to consider big exponents, in particular p equals infinity, then you get a big improvement. Bayerard's work, which is actually very important for what I'm talking about now, is from the 1970s. The student norm estimate, if you plug in the formula up there, that's the power right here, and if you plug in p equals infinity, it would say the universal bound is this. So in any compact manifold, the supernorms grow at worse like that. That's a theorem that's often attributed to Hornmander. He wrote a famous paper in 1968 called Spectrophunction of Elliptic Operator, which was actually proved in the 1950s by Avokramavich and Levitan. So that's the universal bound. And Bayerard wrote a very beautiful paper in which implicitly you get this improvement if mg has non-positive curvature. And we'll use his techniques quite a bit. It's a global theorem. You have to deal with the wave equation for a large period of time, up to basically log lab done. And then Steve and I have written several papers, and we actually have necessary and sufficient conditions, at least in the real analytic case, of beating this. It's a generic condition. This is not a generic condition. Okay, so let's talk about some related problems. Well I did quite a bit of work. There's actually been a lot of activity on studying the size of nodal onsets. Nodal sets are zeros of real eigenfunctions. There's a conjecture of yaw that the size of the nodal set is n minus one dimensional house-dorff measures should be like lambda. And the lower bounds have been studied quite a bit recently. And before basically 2010, they were pretty bad. They were exponential in lambda. And then we were quite happy we got polynomial lambda, lambda to some power, negative power in high dimensions. And there were two competing techniques. Steve Zeldic and I were using wave equation techniques. And another competitor of ours was Coltigan-Minnikas, use harmonic function theory. And in the work that Steve and I did, this was sort of the central thing, to get lower bounds for L1 norms. And when this first came up, I thought this was weird, because as a harmonic analysis analyst, when you're dealing with L1, it's usually trivial. But when you're dealing with L1 as a harmonic analyst, you're usually proving upper bounds, not lower bounds, big difference. So Steve and I proved this thing right here, that this is a universal bound, that the L1 norm is always bounded from below by this. It's easy to prove using holders inequality. So the L2 norm is one. You holder it. Two is between a one and a P. You take P to be in that range I talked about, between two and that Sturckart's number, two times N plus one, brand minus one, holder it. And then you get the bound that we were talking about for this. Figure out what this powers, do your arithmetic and get that. By this holder argument, you see that if you can beat the estimate that this is controlled by this, you get a better L1 norm, lower bound. And if you feed that better L1 lower bound into the things we were proving, you get better estimates for the size of nodal sets. And I was pretty happy about that because of the work I'll be talking about. We were able to beat the world record for lower bounds of Kolding and Minnikazi by logs. So in particular, in 3D, the Kolding and Minnikazi, Steve and I came up with a different proof later on about the same time. The lower bound that they obtained was not that the size of the nodal set grows like lambda but is bounded for three dimensional manifolds. Not good, but at least it's not exponential. So this proves the superiority of the wave equation. Well, just wait, Sergio. So we held the world record for a little while for a few months. We could show they blow up logarithmically. But there's been an amazing breakthrough by a postdoc, okay, Sodin and Israel Luganov, in 2016, and he completely solved the thing for the lower bound. And darn it, he used very local techniques and harmonic function theory. You get the correct lower bound and upper bound. It's a very famous paper, a series of papers with Donnelly and Pfeffermann, circa 1990. And there was all this just for real analytic manifolds, not for C infinity manifolds, but he proved the correct bound, pretty amazing thing for the lower bound. What is a Corde phone? Lambda. It should be comparable to lambda. Just like if you take cosine of lambda x and you count the number of zeros. Okay, it's going to be like lambda. Okay, so that's pretty amazing. So hopefully wave equation techniques will work. We're still trying. I mean, there's still plenty of open problems about nodal domains and nodal sets, but that's a pretty amazing thing. So these work only in the analytic case? No, so this used to be another slide, and I could brag about this and so on. So I should have still made this another slide. Donnelly and Pfeffermann are only the real analytic case. That's 25 years. There's been really no significant progress, but he, the young mathematician, proved the correct lower bound for the C infinity case. Very nice. Okay. All right. So now we come to the Cacay and Nicodem part of the title. So we've seen that these Q lambdas have almost all of their L2 mass in these shrinking tubes about the equator, this periodic geodesic. Okay, and let's see. I guess I didn't say so, but if you can't beat this lower bound for L1 norms on any manifold, then you could show, if you have a sequence of eigenfunctions that saturate this norm, then you could show that for each eigenfunction there's a geodesic and there's a tube where he has the profile of the high-sway spherical harmonic. Okay, in a large proportion of the tube, he'll have exactly the same size as the high-sway spherical harmonic. I was hoping to be able to use this for nodal lower bounds, but it's obsolete. Okay. So at any rate, if you, so the high-sway spherical harmonic has this mass in these tubes, if you can't beat the L1 lower bound, then you have a guy that looks a lot like the high-sway spherical harmonics. So maybe something to consider is the L2 norms over these types of tubes. So I call the Cacay and Nicodem norm of a function just the supremum of all the L2 norms over these shrinking tubes. Okay? So you consider tubes centered anywhere and their orientation could be arbitrary. Then you're taking the L2 norms over these shrinking tubes. I call it Cacay and Nicodem because these sorts of averages arose in the work of Cordoba, that would be Antonio Cordoba, when he studied Bacna Ries. In his case, he was considering the maximal operator which involved the super averages of your function overall tubes of width delta and length one about a point x. He called that the Cacay and maximal function. And then in 1991, Borgand had a very, very seminal important paper that broke open a whole chapter of harmonic analysis and he switched terminology. He called it the Nicodem maximal function. And then he also considered another problem which is related to the structure of the Cacay set which is the maximal function where you fix the orientation of your tubes but you super overall the centers. So I do both and so I call it the Cacay and Nicodem norm. Okay? So clearly because I L2 normalize my functions, clearly these guys are always less than or equal to one. I'm integrating here over a subset of the manifold. So that's a trivial bound. It cannot be beat by these high weights for harmonics because of what I said before. So the problem is when can you beat it? When can you show that these Cacay and Nicodem norms, you're souping over the averages of overall geodesic tubes, when are these a little low of one as the eigenvalue goes to infinity? Okay? And this came up I guess about five years ago in my work and then shortly before that in work of Borgan which anticipated this. So Borgan was interested in linking improved LP estimates with improved what are called restriction estimates for eigenfunctions. For restriction estimates what you do is you restrict your eigenfunctions to a unit length geodesic and you take the L2 norm. Bertrand and Setkoff in 2007 proved this universal bound. Okay? That the L2 norm squared of the eigenfunctions over these geodesics are a big O of lambda to the one half. That's saturated by the highest weight, that's a 2D. That's saturated by the highest weight spherical harmonic because you had that normalizing factor which was lambda to the one quarter in 2D. So this is always true and people for a variety of reasons including problems involving nodal domains and so on are interested in trying to improve this. Borgan showed that if you have better L4 norms the numerology is that in 2D the L4 norms always are big O of lambda to the one eighth. If you can beat that estimate then you can beat Bertrand and Setkoff. So Borgan showed that this implies this. It's easy to see that if you can beat this estimate you can beat the Cauchy and Nicodem estimates. Nothing's going on. Cauchy and Nicodem says you're trying to have small L2 norms over a tube. If you get good L2 norms over all slices, right, you just use Fubini's theorem. So let's see. Two implies three and then in my paper I showed that three implies one. Borgan came close to showing that either two or three imply one. He was using techniques that really go back to the work of Cordoba. When you study these eigenfunctions as you'll see in a second you study reproducing operators for the eigenfunctions and their oscillatory integrals. The types of oscillatory integrals that arose in the study of Bach-Norris and in 2D these things are completely known. They're sharp estimates. And there are two ways of proving them. One is through the work of Cordoba and Pfeffermann, it's kind of a geometric approach using these Cauchy and Nicodem maximal operators that I was talking about. Borgan only used that method and he was actually just off by a lambda to the epsilon showing that this implies this. There's a competing approach which is due to Carlisthen, Julian and Hornmander which is based on bilinear techniques. And what I realized is that you could just take the two things, you just split things up into two cases, splice them together and that allows you to show that this implies this and therefore they're all equivalent. And I'll show you why these are all equivalent in a second in more detail. So the central thing is, as I said, taking the biggest possible L2 norms over tubes, the tubes are very narrow and you allow their orientation to be anything and the center to be anything. Okay. And let's see. So Blair and I in a paper that appeared in 2015 but we did the work a few years before that. Backblog, you know. We showed that the same thing works in higher dimensions. In higher dimensions, especially for more dimensions, you really are stuck with using these KK and Nikonin norms. This is too singular, restricting your eigenfunctions to geodesics for technical reasons. So the KK and Nikonin things are natural to use especially in higher dimensions. And so Blair and I showed that the result extends in higher dimensions. If you have small KK and Nikonin norms, then that's exactly occurs when you have small LP norms. They're the same. And in a more recent paper, we showed that you can dominate the LP norm in terms of lambda to the correct power times the L2 norm over the entire manifold raised to a certain power and then this KK and Nikonin thing raised to the theta. This is one minus theta. This is a better estimate than just having L2 norms on the right because this quantity right here as we talked about before is dominated by the L2 norm over M. So this is a stronger estimate than my old theorem where I just had the L2 norm over M to the first power. And of course, this says that if you can improve this, then you can improve that. Just by Holder's inequality and the numerology here, if you can improve this, then you can improve these L2 norms over these tubes. This gives you the hard half. And let's see. In 2014, Zeldic and I showed that in 2D, you can get these improved KK and Nikonin norms if your manifold has non-positive curvature. And in the same paper in 2015, Blair and I showed that for higher dimensions. So armed with these facts and this inequality, you can see that if the manifold has non-positive sectional curvature, you get these improved LPS. And that was a problem that I was interested in for many, many years, you know, motivated by this result. So I was happy when we could prove this. But still, in the theorems that I'm describing on this page, we don't have any, it doesn't give any indication how the norms improve with lambda. There's not a rate like in Bay-Rard's theorem involving the log. Okay? Chris, one quick question on the previous slide. So when you define the KK and Nikonin norm, you define it so that it scales somehow in the same way as the L2 norm? It's related to weight packets. So you take all tubes of width, lambda to the minus one-half, about geodestic. And I don't turn it, I guess I think I didn't know what you mean. I don't turn it into an average. I just soup out over the L2 norms. Okay? Okay. All right. So here's the theorem that we proved, okay, using wave equation techniques and actually using dispersive properties of the wave equations and so on. We showed that these KK and Nikonin norms have a certain decay, okay? And the decay is basically one of our lambda. It's worse in 2D because of the worst dispersive properties for the wave equation in 2D. And it's just a little worse in 3D, okay? And we also showed that if you have non-positive curvature, you can beat the results of Berkshire and Steckhoff by the same amount. Okay? This is when you allow the curvature to be zero, but of course it can't be positive, okay? If you're willing to assume that the curvature is strictly negative, then you get the same decay rate in all dimensions. And there's a simple reason for that. That's because if you have negative curvature, if you're in Rn as soon will be with negative curvature, then instead of having dispersion of 1 over t to the n minus 1 over 2, you have exponential dispersion. 1 over sine hyperbolic of t to that power. And so that helps you out. Okay. All right. So as a corollary, just feeding it into this estimate right here, you get the log improvement of this. Of course, that's going to give you log improvements of the LP norms, okay? And so finally, we're getting something like Bayerard. Maybe not the optimal power, but some power. n minus 1 over 4. This is n minus 1 over 2. Remember, in my case, my eigenfunctions have eigenvalue lambda, or lambda squared instead of lambda for the Laplacian. Okay? All right. So you get this as a corollary. This is not as relevant as it seemed to be before, but you do get an improvement for the L1 norm of eigenfunctions, the lower bound. And that would lead you to an improvement of the size of the nuttlesets, but it's way blown away by the recent breakthrough that I told you about. Okay? All right. All right. So also, and this would be very happy too, because this also did not seem to be within reach. As it turns out, if you take these improved LP estimates over here, and you take Bayerard's estimate and you throw it into a recipe cooked up by Bourguin in 1991, which he used to give a nice proof. It's the same paper I was talking about, the breakthrough paper that he had involving the cacaia and niconin-maximal functions. If you take an argument that he had in this paper, which gave a very simple proof of weak type estimates for the Stein-Thomas restriction theorem, if you throw, if you take his recipe, take Bayerard's estimate and take the improved LP estimates that Blair and I had, it turns out that by using an argument of Bourguin, you can prove weak type estimates, improved weak type estimates for the critical exponent, PC, which is 2 times n plus 1 over n minus 1. I don't feel so guilty about talking about this, because Rowan talked about weak type spaces. And then if you use some more harmonic analysis, something that goes back to Bach and Seger, but actually it follows from an interpolation argument of Bourguin again, you can upgrade those weak type estimates to get LP estimates. And so I'm happy about this, because nobody had attained improved LP estimates for the critical power. If you can improve LP estimates for the critical power, then just by interpolation, you get it for all exponents. You get log logs because of this argument that I'm talking about, but still it's an improvement. It'd be interesting to turn these into logs, but don't know how to do that. So I want to tell you, since this is a wave equation, conference, how the wave equation arises. So I'm going to tell you about how you set up these arguments. You want to prove these improved L2 estimates over these types of tubes for Eigenfunctions. OK, a problem is that, except for very special cases like the torus or the sphere, there's no way you can write down a formula for Eigenfunctions. So you have to attack them through operators that reproduce them. And so here's a typical way of doing this. You choose a Schwartz class function. You want it to be 1 at the origin. This will allow it to reproduce Eigenfunctions. And because you're going to be dealing with the wave equation, you want this Fourier transform to be supported in, say, the unit interval. Then if you let P be the square root of minus the Laplacian, then this function of P, rho of T lambda minus P, is defined by the spectral theorem. So if you act on an Eigenfunction with eigenvalue mu, then it's just multiplication by this Fourier multiplier rho of T of lambda minus mu. T will turn out to be, later on, a small constant times log lambda. OK, so you have that formula. They reproduce Eigenfunctions. So you're trying to show that the integral of your Eigenfunction over these small tubes is small. So if you simply take F to be E lambda, this will be 1. And you get the balance that you want. OK, so if you can prove this estimate, then simply by taking F to be E lambda, you get what you want. So you want, of course, the constants to be independent of lambda and so on. So this is what you need to prove. OK, so how do you do that? Well, you just take this guy right here, and you rewrite it using the Fourier transform. So it's what, 1 over 2 pi. There's a 1 over T because of the dilation. There's rho hat of T over capital T. There's E to the i lambda T. And then E to the minus i Tp dt. And because rho hat is supported between minus 1 and 1, this integral just involves fairly big times, but times which are at most log in size. OK, so that's a formula for this operator that we need to estimate in this way. And then you just use Euler's formula. You add to this operator rho of lambda plus p. P is a positive operator. It'll have this large number. So this guy will have a kernel which is rapidly decreasing. So when I add this operator to this, I'm just making a mistake involving an operator which is trivial, which has a kernel which is rapidly decreasing. When I play this game of adding these two guys, I'll be adding E to the i Tp with a plus sign. OK, and then I just use Euler's formula. So after using Euler's formula, I can replace this by cosine of Tp. So I have to show that this guy has small, when acting on a L2 function, F has small L2 norms over these tubes. I like cosine because cosine by Huygens principle, this kernel will vanish. If I take the kernel, it'll, well, I'm going to lift it up to the universal cover. I'm getting ahead of myself. So I want to use Huygens principle. That's why I introduced cosine. OK, and so now I use some geometry. So by the Cartan-Hadamard theorem, I can rewrite this kernel. I can consider the universal cover of my manifold because my manifold is non-positive curvature. And so I take the universal cover, which is Rn. And the covering map is just the exponential map. OK, I identify, so the exponential map is, of course, the map from the tangent space at any point to the manifold. I identified the tangent space with Rn. And if I were sensible, I would be playing this game where I take the exponential map over P, which is the center of the G-desic. So if I play this game, I can get, I can lift my metric on my manifold using this covering map, this exponential map, to get a metric here. If I'm assuming as I am that my manifold has non-positive curvature, then G tilde, the lifted metric, will also have that property. And then you have this formula right here. The wave kernel for the metric downstairs agrees with the sum overall that, OK, there's more here. So it's best to go over the model case. So the model case, of course, is a torus. The torus you identify with minus pi to pi to the end. And then this is a fundamental domain for the torus. And in addition to having this covering map, you have deck transformations. In the case of the torus, the deck transformations would just correspond to translating by elements of Zn. I'll have a picture for this in a second. And then you have this formula here, which relates the wave equation on your torus, if we're dealing with the torus, with the wave equation upstairs, which would be an Rn. And so you're summing up over all the translations. You're taking the solution of the wave equations kernel upstairs. You're evaluating at a point x in your fundamental domain. And then you're translating the other point y in the fundamental domain. This formula right here, where this would be the Laplacian on the torus, and this would be the Rn Laplacian, this formula right here is easy to prove. It simply follows from the fact that C infinity functions on the torus are in one-to-one correspondence with smooth periodic functions in Rn. You couple that with uniqueness for the Cauchy problems on the torus, and on Rn, you get this formula. And the very same argument works in this more general setting. Okay, I couldn't find very good pictures, but this is supposed to be the picture for fundamental domains for manifolds of negative curvature, compact manifolds of negative curvature. And these are the translates by the deck transformations. I'll tell you why it's a bad picture in a second. But it's a good picture because it depicts the way things, if you have a manifold strictly negative curvature, the geometry from our Euclidean eyes is becoming very, very warped in the angular direction. There's no warpage at all in the radial direction. Okay? Why is this a bad picture? It's a bad picture because it's supposed to really depict the fundamental domain for, say, the two-fold torus, the manifold of curvature, the simplest type of manifold of, compact manifold of curvature minus one. Now if you take the standard torus, S1 cross S1, you get its fundamental domain, say the two torus, by chopping it up. You chop it up once, that gives you a cylinder, then you chop it up like this, you unroll it, and you get a square. So if you do that for the two-fold torus, surface, you have to make twice as many cuts. So there should be eight sides instead of seven sides. And this picture, they're seven sides. It's not my picture. So when you deal with, there's lots and lots of pictures for fundamental domains, okay, of hyperbolic quotients, where the picture depicts the punk-ray disc model or the upper-half-space model. This is the model corresponding to using exponential coordinates, geodesic normal coordinates, which is the right sort of coordinates for the way I'm going to be looking at these problems. Okay, so I couldn't find a good picture. Well you guys know this, I'm sure. So if you have a manifold of negative curvature, of course, the sum of the angles adds up to less than pi, okay. This isn't so relevant. And so now let me tell you about how we're going to try to use the wave equation, how we're going to prove these things. Okay, hopefully I have enough time. So this is the guy that I need to estimate. This is the kernel of the operator that I need to estimate. I'm trying to show that if I act on a function f and I restrict this guy to a tube like this, I get something which is small. Okay, so let's erase something. So what I'll do, as I talked about, is I'll use exponential coordinates. So here's my fundamental domain. My geodesic will be something like this. I can always use geodesic normal coordinates about what I call P right here. And then I can extend, this is a geodesic gamma. Okay, I can think of this as a geodesic on the manifold or a geodesic upstairs. And then since I'm working upstairs, I can extend this. And if I choose geodesic normal coordinates like I've been talking about, this will just be the line. So this is the fundamental domain. And these are translates of it, like in that picture. Okay, so here's one that hits the extension of the geodesic. And then here's, there's a whole bunch of them, like in the picture, actually exponentially many of them. Okay, if the radius of a ball is T, then the number of these fundamental domains is really large. The number of fundamental domains could grow exponentially. If the curvature is negative, it'll look like that. Okay, and so there's sort of two types. So here's alpha of D, and here's another alpha of D. So if you think about Hormander's theorem of propagation of singularities and so on, if you have some experience with this, okay, it's a disaster, as I said, because the number of nonzero terms in the sum is actually growing exponentially. You won't get anything because of Huygens' principle and the fact that this integral is supported between minus T and T. You won't get anything from fundamental domains that live outside of this ball. So that contributes nothing to the sum. So you'll have a whole bunch of terms. It'll be exponentially many terms. It'll turn out that it's fairly easy to show that individually they're all well behaved. They actually have perfectly good norm. You just have to add them up. So it'll turn out that there's two types of fundamental domains. The ones that intersect this geodastic and then everything else. The terms that intersect the geodastics actually, the number of terms just grows linearly. It grows like T. But there's exponential in T terms that aren't there. So you expect the main contributions to come from these sorts of guys. And you'd be kind of happy if you can show that if you consider these guys that intersect this geodastic and are within the ball of radius T, if you could put a small cone through them or if you could put a fairly large cone through them. So if you can control this angle theta, it depends on T, if you could put inside of there by choosing this constant C appropriately. If you could put inside of there an angle which is bigger than or equal to lambda to minus delta where delta is very small, you might be in good shape. And it turns out that you can because of topodinagof. So I wanted to find clipart with a poodle but I couldn't. So if you have your friend who is a girl in this case walking away from you in Euclidean space, then of course the angle that she makes with the horizon is basically 1 over T. It's just decaying like 1 over T. In hyperbolic space, this angle is like 1 over the sine hyperbolic of T. It's decaying exponentially. So topodinagof says that if you assume that your curvature is pinched from below, which it automatically is, I'm dealing with manifolds of non-positive curvature. They're compact. So I have a lower bound on the curvature. I might as well assume the lower bound is minus 1. I could just multiply the metric by a constant to achieve that. When I lift the metric upstairs, on the universal cover up, preserve that lower bound. So I can always assume that the curvature is pinched from below by minus 1. And then topodinagof says that this angle we're talking about is bounded from below by the corresponding angle in hyperbolic space. And because that angle is what I just told you about, that allows me to get this lower bound. And that's terrific because I can add up these guys. I have universal bounds for all of them. I'll have this 1 over log here. They'll come with nice bounds. And they'll also come with jth term. They'll also come with something which is of this size if that's a jth term. And so that allowed me to add things up. And I get different bounds if I'm in dimension 2, 3, or higher. And then finally, because I have one minute, let me tell you about how to handle everything else. So these guys are good. I have to tell you how to handle these guys that are outside of, that do not intersect, that avoid this cone. So all I do is I take this operator and I compose with a pseudo differential operator that localizes to this cone we're talking about and then everything else. So q theta is localizing to this cone in Fourier transform space. So this is my angle theta. So these guys will really only see these guys because of this picture. So this piece is very good. So let's ignore it. So therefore, I only have to consider this operator composed with this. And then I can undo, I can go back to this formula. So I take this operator and compose it with this. And then I have to prove estimates for this. I can use Euler's formula again to really reduce it to this. So this composed with this, I need good estimates. And this involves an average. And this is a unitary operator. So since I've run out of time, what I really need to do is to show that if I take this sort of thing, I'm rushing through this, acting on an f dt, then I get good L2 norms over these tubes if f is supported in a tube. And because this involves an average over an interval of length capital T, and since these are unitary operators, I could just reduce to this. That's a trivial reduction. And here's why it works. Here's the wave equation. So at the end of the day, I'm trying to prove good estimates for L2 norms of this expression inside of this small tube. Oh, I forgot to put in this cutoff. That's important. L2 norm over the tube. I've got to the punch line. So I need to estimate this. So having this pseudo differential cutoff means that I'm just only considering waves that are traveling at an angle theta or more from the direction of the central axis. So if I want to prove this estimate, I just have to know if I have a particle inside of the tube, and it forms this angle, how long will it take till it exits the tube? And the time, the escape time, it's easy to see, is going to be less than or equal to lambda to the minus 1 half, and then you're going to have to lose, of course, if the angle is small. You lose like that. So this will be trivial because f is supported in the tube. I'm taking the L2 norm of the tube if time is bigger than this. So really, I just have to estimate this. I'm rushing through this. And then I just use synergy estimates. I just bring the L2 norm out, and I use the fact that this is bounded to L2 on the manifold with norm 1. And so when I play this game, I'm going to get a gain which is like this. And that's a really terrific gain if my angle is like this. That'll give me a total gain which is like lambda to the minus 1 half plus delta. And because I'm just trying to beat logs, that's way more than I need to handle the contributions of this remaining piece. And it looks like I'm over time. Sorry about that. So you had the same correctly. You used a wave equation just at the end. Yeah, just at the end. Just at the end. Domain of dependence you use, but that could be. Domain of dependence. No, it's actually very... I didn't emphasize this as much as I should have because I forgot to say I really, really use the global param... Hadamard parametrics of Bayard. It's pretty subtle to use the wave equation. You're using... It's... You run into disaster if you didn't have Huygens principle and things like that. So you really need that... Yes, really, really, really. Yes, and I need very, very... So everything goes wrong logarithmically, and that's implicit in his parametric... parametrics. So I use quite a bit... We use quite a bit about the wave equation in hyperbolic type manifolds. Very sensitive to that. So this parametric is only productive logarithm in time. Yeah, exactly. Did you have a question, Walter? In negative sectional curvature case, there's a huge amount known about the geodesic flow. Yes, the geodesic flow, yeah. You're using like one geodesic, or sort of even a part of the geodes, because you're not translating it around by the deck transformation. Right. Just using one geodesic and coming face. Is there, if we say strictly negative sectional curvature, does the situation get better? Yes, it'll get better. For instance, it's just a technical theorem that I showed you gets better. And the most interesting case, if you have... The most interesting, I consider it all geodesics, but the interesting geodesics are the periodic geodesics. So if you have a periodic geodesic, you can put one of these tubes about it, right? And then you can care about how the estimates depend on the length of the tube. And there are other problems that are also very related to the types of things that you're bringing up. A slide which I skipped. There's something called period integrals. And the log improvements that you get, are they expected to be optimal? Or is it possible? No, well, it depends. So there are some far out conjectures. So it's conjectured that by, somewhat by Tarnac and by physicists, that if you have strictly negative curvature in 2D, for instance, then the eigenfunctions, as they are in the torus, should be essentially bounded. But people are way, way far away from proving that. This was from the 1970s. And nobody's improved on this. Just like this conjecture is implicitly, is from the late 80s, but except in very special cases, nobody's proved them. So there's optimism, but. Your result implies the improvement of the trigger to the goal piece for the Schrodinger equation on the Schreiber 24? That's a good question. It should. It should. I haven't looked into that. Actually, I didn't really think about that till I was here feeling guilty about the fact that there was nothing nonlinear in my talk. Spectral problems, not linear problems. You multiply a lambda by a solution. Okay. One more question. Great. Thank you. Yeah, go ahead. Thank you.
 Well, thank you very much for inviting me to this conference.  The conference, of course, is called Nonlinear Waves.  I have an apology.  My talk's about linear things.  But it's about estimates that have proved to be useful for proving various types of  nonlinear theorems, existence and so on.  For instance, the types of estimates that I'll be talking about in the case of domains  were used by Burke, LeBeau and Plancheon to prove existence for the critical wave equation  and domains in R3 and so on.  And also we've been hearing about how good estimates for the torus lead to good existence  theorems in that setting.  There are also theorems about good existence for NLS for the sphere, for spheres.  The frustrating thing is that there's really not good theorems for manifolds of negative  curvature.  So I'm going to talk about improving LPS estimates and KK adnickadim estimates, whatever  that is in that setting.  Maybe someday there will be some progress on NLS in that setting.  Okay, let's see here.  So the setting is that I'll be talking about manifolds without boundary, compact.  Of course they have a metric G and the dimension will be two or more.  My little Plancheon is negative and lambda j is the frequency of the eigenvalue.  Geometers don't put the square there usually.  So lambda j is the eigenvalue corresponding to the first order operator, which is the  square root of minus little Plancheon.  All the eigenfunctions are L2 normalized.  dv is, of course, the volume element.  The little Plancheon is still a Plancheon coming from the metric.  And so a vague question is how can you detect and measure various types of concentration  in these guys?  And you're going to expand your horizons a little bit and not just concern yourself  with eigenfunctions but near eigenfunctions, quasi-modes, whatever that is.  And you expect that you might get an extreme concentration at certain types of points,  which would lead to bad soup norms or bad LP norms for large p for certain types of points.  And you also might expect to get concentration near periodic geodestrics.  If you're going to get concentration on a different sort of set, you'd expect the set  to be invariant under the flow.  So the natural thing is periodic geodestrics.  Okay?  And I'm going to concentrate on the latter.  So geodestic, of course, is bigger than a point, so you might expect to detect concentration  along periodic geodestrics in terms of LP norms for relatively small p.  And that turns out to be the case.  Okay?  And you can measure concentration or dispersion in various ways.  Well, I'm on a compact manifold, so there's nowhere for the eigenfunctions to disperse.  But there is a famous problem, which says, for instance, when you have negative curvature,  if you take these probability measures, then they should converge weekly to the uniform  measure.  So that's called the quantum-unique ergodicity conjecture.  And that would be an ideal thing.  It's very difficult to prove.  It's conjectured in the case of negative curvature by Rudnick and Sardak, and it was proved in  some special cases, for instance, by Lyndon Strauss, and that was one of the things that  they cited when he won the Fields Medal.  Okay, so if this is your goal to try to show that something like this happens, it fails  miserably.  This should be the volume in the manifold.  It fails miserably on the sphere.  The sphere is the worst case for everything I'm talking about today, so let's go over  that to sort of set up the better things.  So of course, the sphere is up there, the n-sphere.  It's a subset of our n plus 1 given by that equation.  And the eigenvalues of the square root of the Laplacian on the sphere are just given  by this formula, basically k, square root of k times k plus n minus 1.  And they repeat with a very high multiplicity.  That's the highest multiplicity that's allowed because of the sharp vial formula.  And what's going on there is there's a lot of periodicity.  All the geodesics are periodic with period 2 pi.  They're all the great circles.  And additionally, if you take p, instead of taking the wave groups that we've been hearing  about, but you complete the square by doing that, then these half wave groups are periodic.  They're either periodic with period 2 pi in the case of n odd or 4 pi in the case of n  even.  And so the periodicity of these half wave operators also accounts for the bad behavior  of the eigenfunctions.  The eigenfunctions are nothing but the spherical harmonics, which are the restriction of homogeneous  harmonic polynomials in Rn plus 1 to the n sphere.  It is hot here.  All right, so let's talk about the worst type of spherical harmonics.  The worst type of eigenfunction if you don't like concentration near periodic geodesics.  So here's the guy which is highly concentrated on the set where the modulus of x1 plus i  x2 is equal to 1.  So in other words, the place on the sphere where the other coordinates are 0.  It's equator.  So it's highly concentrated on the equator.  I put that factor k to the n minus 1 over 4 to l2 normalize it.  So you can evaluate the modulus of this.  It's exactly that, of course, by calculus.  You tailor expand the natural log about 1, and you'll find that it's like this.  These are called Gaussian beams.  See they live essentially in the set where this has the argument here is of size 1.  And so that's a k to the minus 1 half tube about the equator.  I'll soon be calling my eigenvalues lambda.  And so this is living in a tube about the equator where the tube has width lambda to  the minus 1 half.  For all practical purposes, because of that Gaussian factor, you could just pretend that  this guy in terms of his size is the indicator function of that tube times a normalizing  factor.  You're not really making too many mistakes.  And so therefore you can compute its lp norm.  So you have this factor out front.  And then if you compute the lp norm, you'll get the volume of this tube to the 1 over  p, which is k to the n minus 1 over 2 with a times minus 1 over p.  So the norms of these guys are on the nose comparable to this.  And this works for all p bigger than or equal to 2.  So they have bad lp norms.  And so here's a picture of some spherical harmonics.  So the ones that I was describing, well, this never vanishes.  So if you took real parts of this, you get a real eigenfunction, usually study real eigenfunctions.  So if you take the real part of this guy, it's called the highest weight spherical harmonics,  and it's these guys.  As k increases because of the Gaussian beam behavior, like some speakers, almost all the  mass is on the equator.  Zonal functions, which I won't be talking about, but that's a separate talk.  They are functions which are peaking at the north and the south pole through the middle  guys.  The first guy is pretty boring.  It's just the constant function.  And then you have spherical harmonics of higher and higher order.  And these pictures also denote where it's zero, which is a very interesting problem, the  nodal sets.  And already, what is this?  The spherical harmonics of degree 1, 2, 3, the spherical harmonics of degree 3, you  could see this very high concentration of the highest weight spherical harmonics.  Okay?  All right, so let's talk about saturation of norms.  So many years ago, I showed certain LP estimates for eigenfunctions, and there are two ranges.  This is critical exponent, which is 2 times n plus 1 of rand minus 1.  This should be a familiar exponent to lots of you guys.  It's the exponent that pops up in Stryckart's estimates, for instance.  So I showed that if you're below that critical exponent, you have exactly the same norms  that we were talking about.  K is now lambda, and it's k raised to that power, n minus 1 over 2 times the gap between  1 over 2 and 1 over p.  And because of the calculations that we did in the previous slide, we see that this estimate  up here from the 1980s is sharp.  It can't be improved.  The universal estimate holds on all compact Riemannian manifolds.  It can't be improved because of these bad guys, the highest weight spherical harmonics.  I won't be talking about this, but I also proved estimates for larger exponents where  the power of lambda turns out to be something different.  It turns out to be this thing, n times the gap, 1 half minus 1 over p minus 1 half.  Way back when I was doing this, that was the case that I cared about, because that number  pops up in harmonic analysis.  It's what's called the critical index for bachnery summation.  And what I wanted to do back then was to extend some results about harmonic analysis in Euclidean  space to harmonic analysis on manifolds.  And I was able to do it because of that estimate.  And this estimate in red, I put in my paper because you might as well put everything you  can do in the paper, but I thought it was a boring estimate.  And now this estimate, this red estimate is much more interesting.  It's linked to several things, actually including this.  So the idea is that it's uniform for many, and then it's equal for any many?  Yeah, yeah.  The constants involved, of course, depend on the manifold, but the growth rate landed  to whatever power always holds.  It's what I would call local estimate.  You prove it using local techniques.  Soon we'll see the wave equation.  I forgot to say that one of the reasons that I don't feel so guilty is that I'll show  you how you can use wave equation techniques to study harmonic analysis, and especially  eigenfunction theory.  You prove this estimate, all you need to do is to understand the wave equation for a unit  period of time.  You don't need to watch a long movie.  And that makes sense, right?  Because on the sphere, these sorts of wave operators that arrive are periodic.  So if you watch the movie for time 2 pi or 4 pi, you know everything.  That doesn't happen in the case of manifolds of negative curvature.  In that case, in some sense, the target manifold won't turn out to be a hyperbolic space, but  the ammunition manifold, as we'll see at the end, will be hyperbolic space, whatever that  means.  All right.  Let's see.  You want me asking a simple-minded question?  Sure.  This definitely points out the periodic orbits.  But at least some metrics have invariant tori or more complicated higher-dimensional tori  or other things.  And those also are recurrent.  And those also play a role somehow in spectral mass and products.  Yeah.  Like, not exactly talking about what you were saying, but a zoolom manifold will also have  these bad properties.  A zoolom manifold, all the G-dessex are periodic to the same period.  And if you're willing to broaden your horizon just a tiny bit and consider what are called  quasi-modes, they're disastrous for the same reasons for estimates like this.  Okay.  Okay.  Let's see.  All right.  So I was going to say this.  Knowing when you can improve these estimates for small exponents is something that's only  been recently understood.  I've worked with Steve Zeldic quite a bit.  And moreover, the work of Bayerard from the 1970s says that if you're willing to consider  big exponents, in particular p equals infinity, then you get a big improvement.  Bayerard's work, which is actually very important for what I'm talking about now, is from the  1970s.  The student norm estimate, if you plug in the formula up there, that's the power right  here, and if you plug in p equals infinity, it would say the universal bound is this.  So in any compact manifold, the supernorms grow at worse like that.  That's a theorem that's often attributed to Hornmander.  He wrote a famous paper in 1968 called Spectrophunction of Elliptic Operator, which was actually  proved in the 1950s by Avokramavich and Levitan.  So that's the universal bound.  And Bayerard wrote a very beautiful paper in which implicitly you get this improvement  if mg has non-positive curvature.  And we'll use his techniques quite a bit.  It's a global theorem.  You have to deal with the wave equation for a large period of time, up to basically log  lab done.  And then Steve and I have written several papers, and we actually have necessary and  sufficient conditions, at least in the real analytic case, of beating this.  It's a generic condition.  This is not a generic condition.  Okay, so let's talk about some related problems.  Well I did quite a bit of work.  There's actually been a lot of activity on studying the size of nodal onsets.  Nodal sets are zeros of real eigenfunctions.  There's a conjecture of yaw that the size of the nodal set is n minus one dimensional  house-dorff measures should be like lambda.  And the lower bounds have been studied quite a bit recently.  And before basically 2010, they were pretty bad.  They were exponential in lambda.  And then we were quite happy we got polynomial lambda, lambda to some power, negative power  in high dimensions.  And there were two competing techniques.  Steve Zeldic and I were using wave equation techniques.  And another competitor of ours was Coltigan-Minnikas, use harmonic function theory.  And in the work that Steve and I did, this was sort of the central thing, to get lower  bounds for L1 norms.  And when this first came up, I thought this was weird, because as a harmonic analysis analyst,  when you're dealing with L1, it's usually trivial.  But when you're dealing with L1 as a harmonic analyst, you're usually proving upper bounds,  not lower bounds, big difference.  So Steve and I proved this thing right here, that this is a universal bound, that the L1  norm is always bounded from below by this.  It's easy to prove using holders inequality.  So the L2 norm is one.  You holder it.  Two is between a one and a P. You take P to be in that range I talked about, between  two and that Sturckart's number, two times N plus one, brand minus one, holder it.  And then you get the bound that we were talking about for this.  Figure out what this powers, do your arithmetic and get that.  By this holder argument, you see that if you can beat the estimate that this is controlled  by this, you get a better L1 norm, lower bound.  And if you feed that better L1 lower bound into the things we were proving, you get  better estimates for the size of nodal sets.  And I was pretty happy about that because of the work I'll be talking about.  We were able to beat the world record for lower bounds of Kolding and Minnikazi by logs.  So in particular, in 3D, the Kolding and Minnikazi, Steve and I came up with a different proof  later on about the same time.  The lower bound that they obtained was not that the size of the nodal set grows like  lambda but is bounded for three dimensional manifolds.  Not good, but at least it's not exponential.  So this proves the superiority of the wave equation.  Well, just wait, Sergio.  So we held the world record for a little while for a few months.  We could show they blow up logarithmically.  But there's been an amazing breakthrough by a postdoc, okay, Sodin and Israel Luganov,  in 2016, and he completely solved the thing for the lower bound.  And darn it, he used very local techniques and harmonic function theory.  You get the correct lower bound and upper bound.  It's a very famous paper, a series of papers with Donnelly and Pfeffermann, circa 1990.  And there was all this just for real analytic manifolds, not for C infinity manifolds, but  he proved the correct bound, pretty amazing thing for the lower bound.  What is a Corde phone?  Lambda.  It should be comparable to lambda.  Just like if you take cosine of lambda x and you count the number of zeros.  Okay, it's going to be like lambda.  Okay, so that's pretty amazing.  So hopefully wave equation techniques will work.  We're still trying.  I mean, there's still plenty of open problems about nodal domains and nodal sets, but that's  a pretty amazing thing.  So these work only in the analytic case?  No, so this used to be another slide, and I could brag about this and so on.  So I should have still made this another slide.  Donnelly and Pfeffermann are only the real analytic case.  That's 25 years.  There's been really no significant progress, but he, the young mathematician, proved the  correct lower bound for the C infinity case.  Very nice.  Okay.  All right.  So now we come to the Cacay and Nicodem part of the title.  So we've seen that these Q lambdas have almost all of their L2 mass in these shrinking tubes  about the equator, this periodic geodesic.  Okay, and let's see.  I guess I didn't say so, but if you can't beat this lower bound for L1 norms on any  manifold, then you could show, if you have a sequence of eigenfunctions that saturate  this norm, then you could show that for each eigenfunction there's a geodesic and there's  a tube where he has the profile of the high-sway spherical harmonic.  Okay, in a large proportion of the tube, he'll have exactly the same size as the high-sway  spherical harmonic.  I was hoping to be able to use this for nodal lower bounds, but it's obsolete.  Okay.  So at any rate, if you, so the high-sway spherical harmonic has this mass in these tubes, if  you can't beat the L1 lower bound, then you have a guy that looks a lot like the high-sway  spherical harmonics.  So maybe something to consider is the L2 norms over these types of tubes.  So I call the Cacay and Nicodem norm of a function just the supremum of all the L2 norms  over these shrinking tubes.  Okay?  So you consider tubes centered anywhere and their orientation could be arbitrary.  Then you're taking the L2 norms over these shrinking tubes.  I call it Cacay and Nicodem because these sorts of averages arose in the work of Cordoba,  that would be Antonio Cordoba, when he studied Bacna Ries.  In his case, he was considering the maximal operator which involved the super averages  of your function overall tubes of width delta and length one about a point x.  He called that the Cacay and maximal function.  And then in 1991, Borgand had a very, very seminal important paper that broke open a  whole chapter of harmonic analysis and he switched terminology.  He called it the Nicodem maximal function.  And then he also considered another problem which is related to the structure of the Cacay  set which is the maximal function where you fix the orientation of your tubes but you  super overall the centers.  So I do both and so I call it the Cacay and Nicodem norm.  Okay?  So clearly because I L2 normalize my functions, clearly these guys are always less than or  equal to one.  I'm integrating here over a subset of the manifold.  So that's a trivial bound.  It cannot be beat by these high weights for harmonics because of what I said before.  So the problem is when can you beat it?  When can you show that these Cacay and Nicodem norms, you're souping over the averages of  overall geodesic tubes, when are these a little low of one as the eigenvalue goes to infinity?  Okay?  And this came up I guess about five years ago in my work and then shortly before that  in work of Borgan which anticipated this.  So Borgan was interested in linking improved LP estimates with improved what are called  restriction estimates for eigenfunctions.  For restriction estimates what you do is you restrict your eigenfunctions to a unit length  geodesic and you take the L2 norm.  Bertrand and Setkoff in 2007 proved this universal bound.  Okay?  That the L2 norm squared of the eigenfunctions over these geodesics are a big O of lambda  to the one half.  That's saturated by the highest weight, that's a 2D.  That's saturated by the highest weight spherical harmonic because you had that normalizing factor  which was lambda to the one quarter in 2D.  So this is always true and people for a variety of reasons including problems involving nodal  domains and so on are interested in trying to improve this.  Borgan showed that if you have better L4 norms the numerology is that in 2D the L4 norms  always are big O of lambda to the one eighth.  If you can beat that estimate then you can beat Bertrand and Setkoff.  So Borgan showed that this implies this.  It's easy to see that if you can beat this estimate you can beat the Cauchy and Nicodem  estimates.  Nothing's going on.  Cauchy and Nicodem says you're trying to have small L2 norms over a tube.  If you get good L2 norms over all slices, right, you just use Fubini's theorem.  So let's see.  Two implies three and then in my paper I showed that three implies one.  Borgan came close to showing that either two or three imply one.  He was using techniques that really go back to the work of Cordoba.  When you study these eigenfunctions as you'll see in a second you study reproducing operators  for the eigenfunctions and their oscillatory integrals.  The types of oscillatory integrals that arose in the study of Bach-Norris and in 2D these  things are completely known.  They're sharp estimates.  And there are two ways of proving them.  One is through the work of Cordoba and Pfeffermann, it's kind of a geometric approach using these  Cauchy and Nicodem maximal operators that I was talking about.  Borgan only used that method and he was actually just off by a lambda to the epsilon showing  that this implies this.  There's a competing approach which is due to Carlisthen, Julian and Hornmander which  is based on bilinear techniques.  And what I realized is that you could just take the two things, you just split things  up into two cases, splice them together and that allows you to show that this implies  this and therefore they're all equivalent.  And I'll show you why these are all equivalent in a second in more detail.  So the central thing is, as I said, taking the biggest possible L2 norms over tubes,  the tubes are very narrow and you allow their orientation to be anything and the center  to be anything.  Okay.  And let's see.  So Blair and I in a paper that appeared in 2015 but we did the work a few years before  that.  Backblog, you know.  We showed that the same thing works in higher dimensions.  In higher dimensions, especially for more dimensions, you really are stuck with using  these KK and Nikonin norms.  This is too singular, restricting your eigenfunctions to geodesics for technical reasons.  So the KK and Nikonin things are natural to use especially in higher dimensions.  And so Blair and I showed that the result extends in higher dimensions.  If you have small KK and Nikonin norms, then that's exactly occurs when you have small  LP norms.  They're the same.  And in a more recent paper, we showed that you can dominate the LP norm in terms of lambda  to the correct power times the L2 norm over the entire manifold raised to a certain power  and then this KK and Nikonin thing raised to the theta.  This is one minus theta.  This is a better estimate than just having L2 norms on the right because this quantity  right here as we talked about before is dominated by the L2 norm over M.  So this is a stronger estimate than my old theorem where I just had the L2 norm over  M to the first power.  And of course, this says that if you can improve this, then you can improve that.  Just by Holder's inequality and the numerology here, if you can improve this, then you can  improve these L2 norms over these tubes.  This gives you the hard half.  And let's see.  In 2014, Zeldic and I showed that in 2D, you can get these improved KK and Nikonin norms  if your manifold has non-positive curvature.  And in the same paper in 2015, Blair and I showed that for higher dimensions.  So armed with these facts and this inequality, you can see that if the manifold has non-positive  sectional curvature, you get these improved LPS.  And that was a problem that I was interested in for many, many years, you know, motivated  by this result.  So I was happy when we could prove this.  But still, in the theorems that I'm describing on this page, we don't have any, it doesn't  give any indication how the norms improve with lambda.  There's not a rate like in Bay-Rard's theorem involving the log.  Okay?  Chris, one quick question on the previous slide.  So when you define the KK and Nikonin norm, you define it so that it scales somehow in  the same way as the L2 norm?  It's related to weight packets.  So you take all tubes of width, lambda to the minus one-half, about geodestic.  And I don't turn it, I guess I think I didn't know what you mean.  I don't turn it into an average.  I just soup out over the L2 norms.  Okay?  Okay.  All right.  So here's the theorem that we proved, okay, using wave equation techniques and actually  using dispersive properties of the wave equations and so on.  We showed that these KK and Nikonin norms have a certain decay, okay?  And the decay is basically one of our lambda.  It's worse in 2D because of the worst dispersive properties for the wave equation in 2D.  And it's just a little worse in 3D, okay?  And we also showed that if you have non-positive curvature, you can beat the results of Berkshire  and Steckhoff by the same amount.  Okay?  This is when you allow the curvature to be zero, but of course it can't be positive,  okay?  If you're willing to assume that the curvature is strictly negative, then you get the same  decay rate in all dimensions.  And there's a simple reason for that.  That's because if you have negative curvature, if you're in Rn as soon will be with negative  curvature, then instead of having dispersion of 1 over t to the n minus 1 over 2, you have  exponential dispersion.  1 over sine hyperbolic of t to that power.  And so that helps you out.  Okay.  All right.  So as a corollary, just feeding it into this estimate right here, you get the log improvement  of this.  Of course, that's going to give you log improvements of the LP norms, okay?  And so finally, we're getting something like Bayerard.  Maybe not the optimal power, but some power.  n minus 1 over 4.  This is n minus 1 over 2.  Remember, in my case, my eigenfunctions have eigenvalue lambda, or lambda squared instead  of lambda for the Laplacian.  Okay?  All right.  So you get this as a corollary.  This is not as relevant as it seemed to be before, but you do get an improvement for the  L1 norm of eigenfunctions, the lower bound.  And that would lead you to an improvement of the size of the nuttlesets, but it's way  blown away by the recent breakthrough that I told you about.  Okay?  All right.  All right.  So also, and this would be very happy too, because this also did not seem to be within  reach.  As it turns out, if you take these improved LP estimates over here, and you take Bayerard's  estimate and you throw it into a recipe cooked up by Bourguin in 1991, which he used to give  a nice proof.  It's the same paper I was talking about, the breakthrough paper that he had involving  the cacaia and niconin-maximal functions.  If you take an argument that he had in this paper, which gave a very simple proof of weak  type estimates for the Stein-Thomas restriction theorem, if you throw, if you take his recipe,  take Bayerard's estimate and take the improved LP estimates that Blair and I had, it turns  out that by using an argument of Bourguin, you can prove weak type estimates, improved  weak type estimates for the critical exponent, PC, which is 2 times n plus 1 over n minus  1.  I don't feel so guilty about talking about this, because Rowan talked about weak type  spaces.  And then if you use some more harmonic analysis, something that goes back to Bach and Seger,  but actually it follows from an interpolation argument of Bourguin again, you can upgrade  those weak type estimates to get LP estimates.  And so I'm happy about this, because nobody had attained improved LP estimates for the  critical power.  If you can improve LP estimates for the critical power, then just by interpolation, you get  it for all exponents.  You get log logs because of this argument that I'm talking about, but still it's an  improvement.  It'd be interesting to turn these into logs, but don't know how to do that.  So I want to tell you, since this is a wave equation, conference, how the wave equation  arises.  So I'm going to tell you about how you set up these arguments.  You want to prove these improved L2 estimates over these types of tubes for Eigenfunctions.  OK, a problem is that, except for very special cases like the torus or the sphere, there's  no way you can write down a formula for Eigenfunctions.  So you have to attack them through operators that reproduce them.  And so here's a typical way of doing this.  You choose a Schwartz class function.  You want it to be 1 at the origin.  This will allow it to reproduce Eigenfunctions.  And because you're going to be dealing with the wave equation, you want this Fourier transform  to be supported in, say, the unit interval.  Then if you let P be the square root of minus the Laplacian, then this function of P, rho  of T lambda minus P, is defined by the spectral theorem.  So if you act on an Eigenfunction with eigenvalue mu, then it's just multiplication by this  Fourier multiplier rho of T of lambda minus mu.  T will turn out to be, later on, a small constant times log lambda.  OK, so you have that formula.  They reproduce Eigenfunctions.  So you're trying to show that the integral of your Eigenfunction over these small tubes  is small.  So if you simply take F to be E lambda, this will be 1.  And you get the balance that you want.  OK, so if you can prove this estimate, then simply by taking F to be E lambda, you get  what you want.  So you want, of course, the constants to be independent of lambda and so on.  So this is what you need to prove.  OK, so how do you do that?  Well, you just take this guy right here, and you rewrite it using the Fourier transform.  So it's what, 1 over 2 pi.  There's a 1 over T because of the dilation.  There's rho hat of T over capital T. There's E to the i lambda T. And then E to the minus  i Tp dt.  And because rho hat is supported between minus 1 and 1, this integral just involves fairly  big times, but times which are at most log in size.  OK, so that's a formula for this operator that we need to estimate in this way.  And then you just use Euler's formula.  You add to this operator rho of lambda plus p.  P is a positive operator.  It'll have this large number.  So this guy will have a kernel which is rapidly decreasing.  So when I add this operator to this, I'm just making a mistake involving an operator which  is trivial, which has a kernel which is rapidly decreasing.  When I play this game of adding these two guys, I'll be adding E to the i Tp with a  plus sign.  OK, and then I just use Euler's formula.  So after using Euler's formula, I can replace this by cosine of Tp.  So I have to show that this guy has small, when acting on a L2 function, F has small  L2 norms over these tubes.  I like cosine because cosine by Huygens principle, this kernel will vanish.  If I take the kernel, it'll, well, I'm going to lift it up to the universal cover.  I'm getting ahead of myself.  So I want to use Huygens principle.  That's why I introduced cosine.  OK, and so now I use some geometry.  So by the Cartan-Hadamard theorem, I can rewrite this kernel.  I can consider the universal cover of my manifold because my manifold is non-positive curvature.  And so I take the universal cover, which is Rn.  And the covering map is just the exponential map.  OK, I identify, so the exponential map is, of course, the map from the tangent space  at any point to the manifold.  I identified the tangent space with Rn.  And if I were sensible, I would be playing this game where I take the exponential map  over P, which is the center of the G-desic.  So if I play this game, I can get, I can lift my metric on my manifold using this covering  map, this exponential map, to get a metric here.  If I'm assuming as I am that my manifold has non-positive curvature, then G tilde, the  lifted metric, will also have that property.  And then you have this formula right here.  The wave kernel for the metric downstairs agrees with the sum overall that, OK, there's  more here.  So it's best to go over the model case.  So the model case, of course, is a torus.  The torus you identify with minus pi to pi to the end.  And then this is a fundamental domain for the torus.  And in addition to having this covering map, you have deck transformations.  In the case of the torus, the deck transformations would just correspond to translating by elements  of Zn.  I'll have a picture for this in a second.  And then you have this formula here, which relates the wave equation on your torus, if  we're dealing with the torus, with the wave equation upstairs, which would be an Rn.  And so you're summing up over all the translations.  You're taking the solution of the wave equations kernel upstairs.  You're evaluating at a point x in your fundamental domain.  And then you're translating the other point y in the fundamental domain.  This formula right here, where this would be the Laplacian on the torus, and this would  be the Rn Laplacian, this formula right here is easy to prove.  It simply follows from the fact that C infinity functions on the torus are in one-to-one correspondence  with smooth periodic functions in Rn.  You couple that with uniqueness for the Cauchy problems on the torus, and on Rn, you get  this formula.  And the very same argument works in this more general setting.  Okay, I couldn't find very good pictures, but this is supposed to be the picture for  fundamental domains for manifolds of negative curvature, compact manifolds of negative curvature.  And these are the translates by the deck transformations.  I'll tell you why it's a bad picture in a second.  But it's a good picture because it depicts the way things, if you have a manifold strictly  negative curvature, the geometry from our Euclidean eyes is becoming very, very warped  in the angular direction.  There's no warpage at all in the radial direction.  Okay?  Why is this a bad picture?  It's a bad picture because it's supposed to really depict the fundamental domain for,  say, the two-fold torus, the manifold of curvature, the simplest type of manifold of, compact manifold  of curvature minus one.  Now if you take the standard torus, S1 cross S1, you get its fundamental domain, say the  two torus, by chopping it up.  You chop it up once, that gives you a cylinder, then you chop it up like this, you unroll  it, and you get a square.  So if you do that for the two-fold torus, surface, you have to make twice as many cuts.  So there should be eight sides instead of seven sides.  And this picture, they're seven sides.  It's not my picture.  So when you deal with, there's lots and lots of pictures for fundamental domains, okay,  of hyperbolic quotients, where the picture depicts the punk-ray disc model or the upper-half-space  model.  This is the model corresponding to using exponential coordinates, geodesic normal coordinates,  which is the right sort of coordinates for the way I'm going to be looking at these problems.  Okay, so I couldn't find a good picture.  Well you guys know this, I'm sure.  So if you have a manifold of negative curvature, of course, the sum of the angles adds up to  less than pi, okay.  This isn't so relevant.  And so now let me tell you about how we're going to try to use the wave equation, how  we're going to prove these things.  Okay, hopefully I have enough time.  So this is the guy that I need to estimate.  This is the kernel of the operator that I need to estimate.  I'm trying to show that if I act on a function f and I restrict this guy to a tube like this,  I get something which is small.  Okay, so let's erase something.  So what I'll do, as I talked about, is I'll use exponential coordinates.  So here's my fundamental domain.  My geodesic will be something like this.  I can always use geodesic normal coordinates about what I call P right here.  And then I can extend, this is a geodesic gamma.  Okay, I can think of this as a geodesic on the manifold or a geodesic upstairs.  And then since I'm working upstairs, I can extend this.  And if I choose geodesic normal coordinates like I've been talking about, this will just  be the line.  So this is the fundamental domain.  And these are translates of it, like in that picture.  Okay, so here's one that hits the extension of the geodesic.  And then here's, there's a whole bunch of them, like in the picture, actually exponentially  many of them.  Okay, if the radius of a ball is T, then the number of these fundamental domains is really  large.  The number of fundamental domains could grow exponentially.  If the curvature is negative, it'll look like that.  Okay, and so there's sort of two types.  So here's alpha of D, and here's another alpha of D.  So if you think about Hormander's theorem of propagation of singularities and so on,  if you have some experience with this, okay, it's a disaster, as I said, because the number  of nonzero terms in the sum is actually growing exponentially.  You won't get anything because of Huygens' principle and the fact that this integral  is supported between minus T and T. You won't get anything from fundamental domains that  live outside of this ball.  So that contributes nothing to the sum.  So you'll have a whole bunch of terms.  It'll be exponentially many terms.  It'll turn out that it's fairly easy to show that individually they're all well behaved.  They actually have perfectly good norm.  You just have to add them up.  So it'll turn out that there's two types of fundamental domains.  The ones that intersect this geodastic and then everything else.  The terms that intersect the geodastics actually, the number of terms just grows linearly.  It grows like T. But there's exponential in T terms that aren't there.  So you expect the main contributions to come from these sorts of guys.  And you'd be kind of happy if you can show that if you consider these guys that intersect  this geodastic and are within the ball of radius T, if you could put a small cone through  them or if you could put a fairly large cone through them.  So if you can control this angle theta, it depends on T, if you could put inside of there  by choosing this constant C appropriately.  If you could put inside of there an angle which is bigger than or equal to lambda to  minus delta where delta is very small, you might be in good shape.  And it turns out that you can because of topodinagof.  So I wanted to find clipart with a poodle but I couldn't.  So if you have your friend who is a girl in this case walking away from you in Euclidean  space, then of course the angle that she makes with the horizon is basically 1 over T. It's  just decaying like 1 over T. In hyperbolic space, this angle is like  1 over the sine hyperbolic of T. It's decaying exponentially.  So topodinagof says that if you assume that your curvature is pinched from below, which  it automatically is, I'm dealing with manifolds of non-positive curvature.  They're compact.  So I have a lower bound on the curvature.  I might as well assume the lower bound is minus 1.  I could just multiply the metric by a constant to achieve that.  When I lift the metric upstairs, on the universal cover up, preserve that lower bound.  So I can always assume that the curvature is pinched from below by minus 1.  And then topodinagof says that this angle we're talking about is bounded from below  by the corresponding angle in hyperbolic space.  And because that angle is what I just told you about, that allows me to get this lower  bound.  And that's terrific because I can add up these guys.  I have universal bounds for all of them.  I'll have this 1 over log here.  They'll come with nice bounds.  And they'll also come with jth term.  They'll also come with something which is of this size if that's a jth term.  And so that allowed me to add things up.  And I get different bounds if I'm in dimension 2, 3, or higher.  And then finally, because I have one minute, let me tell you about how to handle everything  else.  So these guys are good.  I have to tell you how to handle these guys that are outside of, that do not intersect,  that avoid this cone.  So all I do is I take this operator and I compose with a pseudo differential operator  that localizes to this cone we're talking about and then everything else.  So q theta is localizing to this cone in Fourier transform space.  So this is my angle theta.  So these guys will really only see these guys because of this picture.  So this piece is very good.  So let's ignore it.  So therefore, I only have to consider this operator composed with this.  And then I can undo, I can go back to this formula.  So I take this operator and compose it with this.  And then I have to prove estimates for this.  I can use Euler's formula again to really reduce it to this.  So this composed with this, I need good estimates.  And this involves an average.  And this is a unitary operator.  So since I've run out of time, what I really need to do is to show that if I take this  sort of thing, I'm rushing through this, acting on an f dt, then I get good L2 norms over  these tubes if f is supported in a tube.  And because this involves an average over an interval of length capital T, and since  these are unitary operators, I could just reduce to this.  That's a trivial reduction.  And here's why it works.  Here's the wave equation.  So at the end of the day, I'm trying to prove good estimates for L2 norms of this expression  inside of this small tube.  Oh, I forgot to put in this cutoff.  That's important.  L2 norm over the tube.  I've got to the punch line.  So I need to estimate this.  So having this pseudo differential cutoff means that I'm just only considering waves  that are traveling at an angle theta or more from the direction of the central axis.  So if I want to prove this estimate, I just have to know if I have a particle inside of  the tube, and it forms this angle, how long will it take till it exits the tube?  And the time, the escape time, it's easy to see, is going to be less than or equal to  lambda to the minus 1 half, and then you're going to have to lose, of course, if the angle  is small.  You lose like that.  So this will be trivial because f is supported in the tube.  I'm taking the L2 norm of the tube if time is bigger than this.  So really, I just have to estimate this.  I'm rushing through this.  And then I just use synergy estimates.  I just bring the L2 norm out, and I use the fact that this is bounded to L2 on the manifold  with norm 1.  And so when I play this game, I'm going to get a gain which is like this.  And that's a really terrific gain if my angle is like this.  That'll give me a total gain which is like lambda to the minus 1 half plus delta.  And because I'm just trying to beat logs, that's way more than I need to handle the  contributions of this remaining piece.  And it looks like I'm over time.  Sorry about that.  So you had the same correctly.  You used a wave equation just at the end.  Yeah, just at the end.  Just at the end.  Domain of dependence you use, but that could be.  Domain of dependence.  No, it's actually very...  I didn't emphasize this as much as I should have because I forgot to say I really, really  use the global param...  Hadamard parametrics of Bayard.  It's pretty subtle to use the wave equation.  You're using...  It's...  You run into disaster if you didn't have Huygens principle and things like that.  So you really need that...  Yes, really, really, really.  Yes, and I need very, very...  So everything goes wrong logarithmically, and that's implicit in his parametric...  parametrics.  So I use quite a bit...  We use quite a bit about the wave equation in hyperbolic type manifolds.  Very sensitive to that.  So this parametric is only productive logarithm in time.  Yeah, exactly.  Did you have a question, Walter?  In negative sectional curvature case, there's a huge amount known about the geodesic flow.  Yes, the geodesic flow, yeah.  You're using like one geodesic, or sort of even a part of the geodes, because you're  not translating it around by the deck transformation.  Right.  Just using one geodesic and coming face.  Is there, if we say strictly negative sectional curvature, does the situation get better?  Yes, it'll get better.  For instance, it's just a technical theorem that I showed you gets better.  And the most interesting case, if you have...  The most interesting, I consider it all geodesics, but the interesting geodesics are the periodic  geodesics.  So if you have a periodic geodesic, you can put one of these tubes about it, right?  And then you can care about how the estimates depend on the length of the tube.  And there are other problems that are also very related to the types of things that you're  bringing up.  A slide which I skipped.  There's something called period integrals.  And the log improvements that you get, are they expected to be optimal?  Or is it possible?  No, well, it depends.  So there are some far out conjectures.  So it's conjectured that by, somewhat by Tarnac and by physicists, that if you have strictly  negative curvature in 2D, for instance, then the eigenfunctions, as they are in the torus,  should be essentially bounded.  But people are way, way far away from proving that.  This was from the 1970s.  And nobody's improved on this.  Just like this conjecture is implicitly, is from the late 80s, but except in very special  cases, nobody's proved them.  So there's optimism, but.  Your result implies the improvement of the trigger to the goal piece for the Schrodinger  equation on the Schreiber 24?  That's a good question.  It should.  It should.  I haven't looked into that.  Actually, I didn't really think about that till I was here feeling guilty about the fact  that there was nothing nonlinear in my talk.  Spectral problems, not linear problems.  You multiply a lambda by a solution.  Okay.  One more question.  Great.  Thank you.  Yeah, go ahead.  Thank you.
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Well, thank you very much for inviting me to this conference. The conference, of course, is called Nonlinear Waves. I have an apology. My talk's about linear things. But it's about estimates that have proved to be useful for proving various types of nonlinear theorems, existence and so on. For instance, the types of estimates that I'll be talking about in the case of domains were used by Burke, LeBeau and Plancheon to prove existence for the critical wave equation and domains in R3 and so on. And also we've been hearing about how good estimates for the torus lead to good existence theorems in that setting. There are also theorems about good existence for NLS for the sphere, for spheres. The frustrating thing is that there's really not good theorems for manifolds of negative curvature. So I'm going to talk about improving LPS estimates and KK adnickadim estimates, whatever that is in that setting. Maybe someday there will be some progress on NLS in that setting. Okay, let's see here. So the setting is that I'll be talking about manifolds without boundary, compact. Of course they have a metric G and the dimension will be two or more. My little Plancheon is negative and lambda j is the frequency of the eigenvalue. Geometers don't put the square there usually. So lambda j is the eigenvalue corresponding to the first order operator, which is the square root of minus little Plancheon. All the eigenfunctions are L2 normalized. dv is, of course, the volume element. The little Plancheon is still a Plancheon coming from the metric. And so a vague question is how can you detect and measure various types of concentration in these guys? And you're going to expand your horizons a little bit and not just concern yourself with eigenfunctions but near eigenfunctions, quasi-modes, whatever that is. And you expect that you might get an extreme concentration at certain types of points, which would lead to bad soup norms or bad LP norms for large p for certain types of points. And you also might expect to get concentration near periodic geodestrics. If you're going to get concentration on a different sort of set, you'd expect the set to be invariant under the flow. So the natural thing is periodic geodestrics. Okay? And I'm going to concentrate on the latter. So geodestic, of course, is bigger than a point, so you might expect to detect concentration along periodic geodestrics in terms of LP norms for relatively small p. And that turns out to be the case. Okay? And you can measure concentration or dispersion in various ways. Well, I'm on a compact manifold, so there's nowhere for the eigenfunctions to disperse. But there is a famous problem, which says, for instance, when you have negative curvature, if you take these probability measures, then they should converge weekly to the uniform measure. So that's called the quantum-unique ergodicity conjecture. And that would be an ideal thing. It's very difficult to prove. It's conjectured in the case of negative curvature by Rudnick and Sardak, and it was proved in some special cases, for instance, by Lyndon Strauss, and that was one of the things that they cited when he won the Fields Medal. Okay, so if this is your goal to try to show that something like this happens, it fails miserably. This should be the volume in the manifold. It fails miserably on the sphere. The sphere is the worst case for everything I'm talking about today, so let's go over that to sort of set up the better things. So of course, the sphere is up there, the n-sphere. It's a subset of our n plus 1 given by that equation. And the eigenvalues of the square root of the Laplacian on the sphere are just given by this formula, basically k, square root of k times k plus n minus 1. And they repeat with a very high multiplicity. That's the highest multiplicity that's allowed because of the sharp vial formula. And what's going on there is there's a lot of periodicity. All the geodesics are periodic with period 2 pi. They're all the great circles. And additionally, if you take p, instead of taking the wave groups that we've been hearing about, but you complete the square by doing that, then these half wave groups are periodic. They're either periodic with period 2 pi in the case of n odd or 4 pi in the case of n even. And so the periodicity of these half wave operators also accounts for the bad behavior of the eigenfunctions. The eigenfunctions are nothing but the spherical harmonics, which are the restriction of homogeneous harmonic polynomials in Rn plus 1 to the n sphere. It is hot here. All right, so let's talk about the worst type of spherical harmonics. The worst type of eigenfunction if you don't like concentration near periodic geodesics. So here's the guy which is highly concentrated on the set where the modulus of x1 plus i x2 is equal to 1. So in other words, the place on the sphere where the other coordinates are 0. It's equator. So it's highly concentrated on the equator. I put that factor k to the n minus 1 over 4 to l2 normalize it. So you can evaluate the modulus of this. It's exactly that, of course, by calculus. You tailor expand the natural log about 1, and you'll find that it's like this. These are called Gaussian beams. See they live essentially in the set where this has the argument here is of size 1. And so that's a k to the minus 1 half tube about the equator. I'll soon be calling my eigenvalues lambda. And so this is living in a tube about the equator where the tube has width lambda to the minus 1 half. For all practical purposes, because of that Gaussian factor, you could just pretend that this guy in terms of his size is the indicator function of that tube times a normalizing factor. You're not really making too many mistakes. And so therefore you can compute its lp norm. So you have this factor out front. And then if you compute the lp norm, you'll get the volume of this tube to the 1 over p, which is k to the n minus 1 over 2 with a times minus 1 over p. So the norms of these guys are on the nose comparable to this. And this works for all p bigger than or equal to 2. So they have bad lp norms. And so here's a picture of some spherical harmonics. So the ones that I was describing, well, this never vanishes. So if you took real parts of this, you get a real eigenfunction, usually study real eigenfunctions. So if you take the real part of this guy, it's called the highest weight spherical harmonics, and it's these guys. As k increases because of the Gaussian beam behavior, like some speakers, almost all the mass is on the equator. Zonal functions, which I won't be talking about, but that's a separate talk. They are functions which are peaking at the north and the south pole through the middle guys. The first guy is pretty boring. It's just the constant function. And then you have spherical harmonics of higher and higher order. And these pictures also denote where it's zero, which is a very interesting problem, the nodal sets. And already, what is this? The spherical harmonics of degree 1, 2, 3, the spherical harmonics of degree 3, you could see this very high concentration of the highest weight spherical harmonics. Okay? All right, so let's talk about saturation of norms. So many years ago, I showed certain LP estimates for eigenfunctions, and there are two ranges. This is critical exponent, which is 2 times n plus 1 of rand minus 1. This should be a familiar exponent to lots of you guys. It's the exponent that pops up in Stryckart's estimates, for instance. So I showed that if you're below that critical exponent, you have exactly the same norms that we were talking about. K is now lambda, and it's k raised to that power, n minus 1 over 2 times the gap between 1 over 2 and 1 over p. And because of the calculations that we did in the previous slide, we see that this estimate up here from the 1980s is sharp. It can't be improved. The universal estimate holds on all compact Riemannian manifolds. It can't be improved because of these bad guys, the highest weight spherical harmonics. I won't be talking about this, but I also proved estimates for larger exponents where the power of lambda turns out to be something different. It turns out to be this thing, n times the gap, 1 half minus 1 over p minus 1 half. Way back when I was doing this, that was the case that I cared about, because that number pops up in harmonic analysis. It's what's called the critical index for bachnery summation. And what I wanted to do back then was to extend some results about harmonic analysis in Euclidean space to harmonic analysis on manifolds. And I was able to do it because of that estimate. And this estimate in red, I put in my paper because you might as well put everything you can do in the paper, but I thought it was a boring estimate. And now this estimate, this red estimate is much more interesting. It's linked to several things, actually including this. So the idea is that it's uniform for many, and then it's equal for any many? Yeah, yeah. The constants involved, of course, depend on the manifold, but the growth rate landed to whatever power always holds. It's what I would call local estimate. You prove it using local techniques. Soon we'll see the wave equation. I forgot to say that one of the reasons that I don't feel so guilty is that I'll show you how you can use wave equation techniques to study harmonic analysis, and especially eigenfunction theory. You prove this estimate, all you need to do is to understand the wave equation for a unit period of time. You don't need to watch a long movie. And that makes sense, right? Because on the sphere, these sorts of wave operators that arrive are periodic. So if you watch the movie for time 2 pi or 4 pi, you know everything. That doesn't happen in the case of manifolds of negative curvature. In that case, in some sense, the target manifold won't turn out to be a hyperbolic space, but the ammunition manifold, as we'll see at the end, will be hyperbolic space, whatever that means. All right. Let's see. You want me asking a simple-minded question? Sure. This definitely points out the periodic orbits. But at least some metrics have invariant tori or more complicated higher-dimensional tori or other things. And those also are recurrent. And those also play a role somehow in spectral mass and products. Yeah. Like, not exactly talking about what you were saying, but a zoolom manifold will also have these bad properties. A zoolom manifold, all the G-dessex are periodic to the same period. And if you're willing to broaden your horizon just a tiny bit and consider what are called quasi-modes, they're disastrous for the same reasons for estimates like this. Okay. Okay. Let's see. All right. So I was going to say this. Knowing when you can improve these estimates for small exponents is something that's only been recently understood. I've worked with Steve Zeldic quite a bit. And moreover, the work of Bayerard from the 1970s says that if you're willing to consider big exponents, in particular p equals infinity, then you get a big improvement. Bayerard's work, which is actually very important for what I'm talking about now, is from the 1970s. The student norm estimate, if you plug in the formula up there, that's the power right here, and if you plug in p equals infinity, it would say the universal bound is this. So in any compact manifold, the supernorms grow at worse like that. That's a theorem that's often attributed to Hornmander. He wrote a famous paper in 1968 called Spectrophunction of Elliptic Operator, which was actually proved in the 1950s by Avokramavich and Levitan. So that's the universal bound. And Bayerard wrote a very beautiful paper in which implicitly you get this improvement if mg has non-positive curvature. And we'll use his techniques quite a bit. It's a global theorem. You have to deal with the wave equation for a large period of time, up to basically log lab done. And then Steve and I have written several papers, and we actually have necessary and sufficient conditions, at least in the real analytic case, of beating this. It's a generic condition. This is not a generic condition. Okay, so let's talk about some related problems. Well I did quite a bit of work. There's actually been a lot of activity on studying the size of nodal onsets. Nodal sets are zeros of real eigenfunctions. There's a conjecture of yaw that the size of the nodal set is n minus one dimensional house-dorff measures should be like lambda. And the lower bounds have been studied quite a bit recently. And before basically 2010, they were pretty bad. They were exponential in lambda. And then we were quite happy we got polynomial lambda, lambda to some power, negative power in high dimensions. And there were two competing techniques. Steve Zeldic and I were using wave equation techniques. And another competitor of ours was Coltigan-Minnikas, use harmonic function theory. And in the work that Steve and I did, this was sort of the central thing, to get lower bounds for L1 norms. And when this first came up, I thought this was weird, because as a harmonic analysis analyst, when you're dealing with L1, it's usually trivial. But when you're dealing with L1 as a harmonic analyst, you're usually proving upper bounds, not lower bounds, big difference. So Steve and I proved this thing right here, that this is a universal bound, that the L1 norm is always bounded from below by this. It's easy to prove using holders inequality. So the L2 norm is one. You holder it. Two is between a one and a P. You take P to be in that range I talked about, between two and that Sturckart's number, two times N plus one, brand minus one, holder it. And then you get the bound that we were talking about for this. Figure out what this powers, do your arithmetic and get that. By this holder argument, you see that if you can beat the estimate that this is controlled by this, you get a better L1 norm, lower bound. And if you feed that better L1 lower bound into the things we were proving, you get better estimates for the size of nodal sets. And I was pretty happy about that because of the work I'll be talking about. We were able to beat the world record for lower bounds of Kolding and Minnikazi by logs. So in particular, in 3D, the Kolding and Minnikazi, Steve and I came up with a different proof later on about the same time. The lower bound that they obtained was not that the size of the nodal set grows like lambda but is bounded for three dimensional manifolds. Not good, but at least it's not exponential. So this proves the superiority of the wave equation. Well, just wait, Sergio. So we held the world record for a little while for a few months. We could show they blow up logarithmically. But there's been an amazing breakthrough by a postdoc, okay, Sodin and Israel Luganov, in 2016, and he completely solved the thing for the lower bound. And darn it, he used very local techniques and harmonic function theory. You get the correct lower bound and upper bound. It's a very famous paper, a series of papers with Donnelly and Pfeffermann, circa 1990. And there was all this just for real analytic manifolds, not for C infinity manifolds, but he proved the correct bound, pretty amazing thing for the lower bound. What is a Corde phone? Lambda. It should be comparable to lambda. Just like if you take cosine of lambda x and you count the number of zeros. Okay, it's going to be like lambda. Okay, so that's pretty amazing. So hopefully wave equation techniques will work. We're still trying. I mean, there's still plenty of open problems about nodal domains and nodal sets, but that's a pretty amazing thing. So these work only in the analytic case? No, so this used to be another slide, and I could brag about this and so on. So I should have still made this another slide. Donnelly and Pfeffermann are only the real analytic case. That's 25 years. There's been really no significant progress, but he, the young mathematician, proved the correct lower bound for the C infinity case. Very nice. Okay. All right. So now we come to the Cacay and Nicodem part of the title. So we've seen that these Q lambdas have almost all of their L2 mass in these shrinking tubes about the equator, this periodic geodesic. Okay, and let's see. I guess I didn't say so, but if you can't beat this lower bound for L1 norms on any manifold, then you could show, if you have a sequence of eigenfunctions that saturate this norm, then you could show that for each eigenfunction there's a geodesic and there's a tube where he has the profile of the high-sway spherical harmonic. Okay, in a large proportion of the tube, he'll have exactly the same size as the high-sway spherical harmonic. I was hoping to be able to use this for nodal lower bounds, but it's obsolete. Okay. So at any rate, if you, so the high-sway spherical harmonic has this mass in these tubes, if you can't beat the L1 lower bound, then you have a guy that looks a lot like the high-sway spherical harmonics. So maybe something to consider is the L2 norms over these types of tubes. So I call the Cacay and Nicodem norm of a function just the supremum of all the L2 norms over these shrinking tubes. Okay? So you consider tubes centered anywhere and their orientation could be arbitrary. Then you're taking the L2 norms over these shrinking tubes. I call it Cacay and Nicodem because these sorts of averages arose in the work of Cordoba, that would be Antonio Cordoba, when he studied Bacna Ries. In his case, he was considering the maximal operator which involved the super averages of your function overall tubes of width delta and length one about a point x. He called that the Cacay and maximal function. And then in 1991, Borgand had a very, very seminal important paper that broke open a whole chapter of harmonic analysis and he switched terminology. He called it the Nicodem maximal function. And then he also considered another problem which is related to the structure of the Cacay set which is the maximal function where you fix the orientation of your tubes but you super overall the centers. So I do both and so I call it the Cacay and Nicodem norm. Okay? So clearly because I L2 normalize my functions, clearly these guys are always less than or equal to one. I'm integrating here over a subset of the manifold. So that's a trivial bound. It cannot be beat by these high weights for harmonics because of what I said before. So the problem is when can you beat it? When can you show that these Cacay and Nicodem norms, you're souping over the averages of overall geodesic tubes, when are these a little low of one as the eigenvalue goes to infinity? Okay? And this came up I guess about five years ago in my work and then shortly before that in work of Borgan which anticipated this. So Borgan was interested in linking improved LP estimates with improved what are called restriction estimates for eigenfunctions. For restriction estimates what you do is you restrict your eigenfunctions to a unit length geodesic and you take the L2 norm. Bertrand and Setkoff in 2007 proved this universal bound. Okay? That the L2 norm squared of the eigenfunctions over these geodesics are a big O of lambda to the one half. That's saturated by the highest weight, that's a 2D. That's saturated by the highest weight spherical harmonic because you had that normalizing factor which was lambda to the one quarter in 2D. So this is always true and people for a variety of reasons including problems involving nodal domains and so on are interested in trying to improve this. Borgan showed that if you have better L4 norms the numerology is that in 2D the L4 norms always are big O of lambda to the one eighth. If you can beat that estimate then you can beat Bertrand and Setkoff. So Borgan showed that this implies this. It's easy to see that if you can beat this estimate you can beat the Cauchy and Nicodem estimates. Nothing's going on. Cauchy and Nicodem says you're trying to have small L2 norms over a tube. If you get good L2 norms over all slices, right, you just use Fubini's theorem. So let's see. Two implies three and then in my paper I showed that three implies one. Borgan came close to showing that either two or three imply one. He was using techniques that really go back to the work of Cordoba. When you study these eigenfunctions as you'll see in a second you study reproducing operators for the eigenfunctions and their oscillatory integrals. The types of oscillatory integrals that arose in the study of Bach-Norris and in 2D these things are completely known. They're sharp estimates. And there are two ways of proving them. One is through the work of Cordoba and Pfeffermann, it's kind of a geometric approach using these Cauchy and Nicodem maximal operators that I was talking about. Borgan only used that method and he was actually just off by a lambda to the epsilon showing that this implies this. There's a competing approach which is due to Carlisthen, Julian and Hornmander which is based on bilinear techniques. And what I realized is that you could just take the two things, you just split things up into two cases, splice them together and that allows you to show that this implies this and therefore they're all equivalent. And I'll show you why these are all equivalent in a second in more detail. So the central thing is, as I said, taking the biggest possible L2 norms over tubes, the tubes are very narrow and you allow their orientation to be anything and the center to be anything. Okay. And let's see. So Blair and I in a paper that appeared in 2015 but we did the work a few years before that. Backblog, you know. We showed that the same thing works in higher dimensions. In higher dimensions, especially for more dimensions, you really are stuck with using these KK and Nikonin norms. This is too singular, restricting your eigenfunctions to geodesics for technical reasons. So the KK and Nikonin things are natural to use especially in higher dimensions. And so Blair and I showed that the result extends in higher dimensions. If you have small KK and Nikonin norms, then that's exactly occurs when you have small LP norms. They're the same. And in a more recent paper, we showed that you can dominate the LP norm in terms of lambda to the correct power times the L2 norm over the entire manifold raised to a certain power and then this KK and Nikonin thing raised to the theta. This is one minus theta. This is a better estimate than just having L2 norms on the right because this quantity right here as we talked about before is dominated by the L2 norm over M. So this is a stronger estimate than my old theorem where I just had the L2 norm over M to the first power. And of course, this says that if you can improve this, then you can improve that. Just by Holder's inequality and the numerology here, if you can improve this, then you can improve these L2 norms over these tubes. This gives you the hard half. And let's see. In 2014, Zeldic and I showed that in 2D, you can get these improved KK and Nikonin norms if your manifold has non-positive curvature. And in the same paper in 2015, Blair and I showed that for higher dimensions. So armed with these facts and this inequality, you can see that if the manifold has non-positive sectional curvature, you get these improved LPS. And that was a problem that I was interested in for many, many years, you know, motivated by this result. So I was happy when we could prove this. But still, in the theorems that I'm describing on this page, we don't have any, it doesn't give any indication how the norms improve with lambda. There's not a rate like in Bay-Rard's theorem involving the log. Okay? Chris, one quick question on the previous slide. So when you define the KK and Nikonin norm, you define it so that it scales somehow in the same way as the L2 norm? It's related to weight packets. So you take all tubes of width, lambda to the minus one-half, about geodestic. And I don't turn it, I guess I think I didn't know what you mean. I don't turn it into an average. I just soup out over the L2 norms. Okay? Okay. All right. So here's the theorem that we proved, okay, using wave equation techniques and actually using dispersive properties of the wave equations and so on. We showed that these KK and Nikonin norms have a certain decay, okay? And the decay is basically one of our lambda. It's worse in 2D because of the worst dispersive properties for the wave equation in 2D. And it's just a little worse in 3D, okay? And we also showed that if you have non-positive curvature, you can beat the results of Berkshire and Steckhoff by the same amount. Okay? This is when you allow the curvature to be zero, but of course it can't be positive, okay? If you're willing to assume that the curvature is strictly negative, then you get the same decay rate in all dimensions. And there's a simple reason for that. That's because if you have negative curvature, if you're in Rn as soon will be with negative curvature, then instead of having dispersion of 1 over t to the n minus 1 over 2, you have exponential dispersion. 1 over sine hyperbolic of t to that power. And so that helps you out. Okay. All right. So as a corollary, just feeding it into this estimate right here, you get the log improvement of this. Of course, that's going to give you log improvements of the LP norms, okay? And so finally, we're getting something like Bayerard. Maybe not the optimal power, but some power. n minus 1 over 4. This is n minus 1 over 2. Remember, in my case, my eigenfunctions have eigenvalue lambda, or lambda squared instead of lambda for the Laplacian. Okay? All right. So you get this as a corollary. This is not as relevant as it seemed to be before, but you do get an improvement for the L1 norm of eigenfunctions, the lower bound. And that would lead you to an improvement of the size of the nuttlesets, but it's way blown away by the recent breakthrough that I told you about. Okay? All right. All right. So also, and this would be very happy too, because this also did not seem to be within reach. As it turns out, if you take these improved LP estimates over here, and you take Bayerard's estimate and you throw it into a recipe cooked up by Bourguin in 1991, which he used to give a nice proof. It's the same paper I was talking about, the breakthrough paper that he had involving the cacaia and niconin-maximal functions. If you take an argument that he had in this paper, which gave a very simple proof of weak type estimates for the Stein-Thomas restriction theorem, if you throw, if you take his recipe, take Bayerard's estimate and take the improved LP estimates that Blair and I had, it turns out that by using an argument of Bourguin, you can prove weak type estimates, improved weak type estimates for the critical exponent, PC, which is 2 times n plus 1 over n minus 1. I don't feel so guilty about talking about this, because Rowan talked about weak type spaces. And then if you use some more harmonic analysis, something that goes back to Bach and Seger, but actually it follows from an interpolation argument of Bourguin again, you can upgrade those weak type estimates to get LP estimates. And so I'm happy about this, because nobody had attained improved LP estimates for the critical power. If you can improve LP estimates for the critical power, then just by interpolation, you get it for all exponents. You get log logs because of this argument that I'm talking about, but still it's an improvement. It'd be interesting to turn these into logs, but don't know how to do that. So I want to tell you, since this is a wave equation, conference, how the wave equation arises. So I'm going to tell you about how you set up these arguments. You want to prove these improved L2 estimates over these types of tubes for Eigenfunctions. OK, a problem is that, except for very special cases like the torus or the sphere, there's no way you can write down a formula for Eigenfunctions. So you have to attack them through operators that reproduce them. And so here's a typical way of doing this. You choose a Schwartz class function. You want it to be 1 at the origin. This will allow it to reproduce Eigenfunctions. And because you're going to be dealing with the wave equation, you want this Fourier transform to be supported in, say, the unit interval. Then if you let P be the square root of minus the Laplacian, then this function of P, rho of T lambda minus P, is defined by the spectral theorem. So if you act on an Eigenfunction with eigenvalue mu, then it's just multiplication by this Fourier multiplier rho of T of lambda minus mu. T will turn out to be, later on, a small constant times log lambda. OK, so you have that formula. They reproduce Eigenfunctions. So you're trying to show that the integral of your Eigenfunction over these small tubes is small. So if you simply take F to be E lambda, this will be 1. And you get the balance that you want. OK, so if you can prove this estimate, then simply by taking F to be E lambda, you get what you want. So you want, of course, the constants to be independent of lambda and so on. So this is what you need to prove. OK, so how do you do that? Well, you just take this guy right here, and you rewrite it using the Fourier transform. So it's what, 1 over 2 pi. There's a 1 over T because of the dilation. There's rho hat of T over capital T. There's E to the i lambda T. And then E to the minus i Tp dt. And because rho hat is supported between minus 1 and 1, this integral just involves fairly big times, but times which are at most log in size. OK, so that's a formula for this operator that we need to estimate in this way. And then you just use Euler's formula. You add to this operator rho of lambda plus p. P is a positive operator. It'll have this large number. So this guy will have a kernel which is rapidly decreasing. So when I add this operator to this, I'm just making a mistake involving an operator which is trivial, which has a kernel which is rapidly decreasing. When I play this game of adding these two guys, I'll be adding E to the i Tp with a plus sign. OK, and then I just use Euler's formula. So after using Euler's formula, I can replace this by cosine of Tp. So I have to show that this guy has small, when acting on a L2 function, F has small L2 norms over these tubes. I like cosine because cosine by Huygens principle, this kernel will vanish. If I take the kernel, it'll, well, I'm going to lift it up to the universal cover. I'm getting ahead of myself. So I want to use Huygens principle. That's why I introduced cosine. OK, and so now I use some geometry. So by the Cartan-Hadamard theorem, I can rewrite this kernel. I can consider the universal cover of my manifold because my manifold is non-positive curvature. And so I take the universal cover, which is Rn. And the covering map is just the exponential map. OK, I identify, so the exponential map is, of course, the map from the tangent space at any point to the manifold. I identified the tangent space with Rn. And if I were sensible, I would be playing this game where I take the exponential map over P, which is the center of the G-desic. So if I play this game, I can get, I can lift my metric on my manifold using this covering map, this exponential map, to get a metric here. If I'm assuming as I am that my manifold has non-positive curvature, then G tilde, the lifted metric, will also have that property. And then you have this formula right here. The wave kernel for the metric downstairs agrees with the sum overall that, OK, there's more here. So it's best to go over the model case. So the model case, of course, is a torus. The torus you identify with minus pi to pi to the end. And then this is a fundamental domain for the torus. And in addition to having this covering map, you have deck transformations. In the case of the torus, the deck transformations would just correspond to translating by elements of Zn. I'll have a picture for this in a second. And then you have this formula here, which relates the wave equation on your torus, if we're dealing with the torus, with the wave equation upstairs, which would be an Rn. And so you're summing up over all the translations. You're taking the solution of the wave equations kernel upstairs. You're evaluating at a point x in your fundamental domain. And then you're translating the other point y in the fundamental domain. This formula right here, where this would be the Laplacian on the torus, and this would be the Rn Laplacian, this formula right here is easy to prove. It simply follows from the fact that C infinity functions on the torus are in one-to-one correspondence with smooth periodic functions in Rn. You couple that with uniqueness for the Cauchy problems on the torus, and on Rn, you get this formula. And the very same argument works in this more general setting. Okay, I couldn't find very good pictures, but this is supposed to be the picture for fundamental domains for manifolds of negative curvature, compact manifolds of negative curvature. And these are the translates by the deck transformations. I'll tell you why it's a bad picture in a second. But it's a good picture because it depicts the way things, if you have a manifold strictly negative curvature, the geometry from our Euclidean eyes is becoming very, very warped in the angular direction. There's no warpage at all in the radial direction. Okay? Why is this a bad picture? It's a bad picture because it's supposed to really depict the fundamental domain for, say, the two-fold torus, the manifold of curvature, the simplest type of manifold of, compact manifold of curvature minus one. Now if you take the standard torus, S1 cross S1, you get its fundamental domain, say the two torus, by chopping it up. You chop it up once, that gives you a cylinder, then you chop it up like this, you unroll it, and you get a square. So if you do that for the two-fold torus, surface, you have to make twice as many cuts. So there should be eight sides instead of seven sides. And this picture, they're seven sides. It's not my picture. So when you deal with, there's lots and lots of pictures for fundamental domains, okay, of hyperbolic quotients, where the picture depicts the punk-ray disc model or the upper-half-space model. This is the model corresponding to using exponential coordinates, geodesic normal coordinates, which is the right sort of coordinates for the way I'm going to be looking at these problems. Okay, so I couldn't find a good picture. Well you guys know this, I'm sure. So if you have a manifold of negative curvature, of course, the sum of the angles adds up to less than pi, okay. This isn't so relevant. And so now let me tell you about how we're going to try to use the wave equation, how we're going to prove these things. Okay, hopefully I have enough time. So this is the guy that I need to estimate. This is the kernel of the operator that I need to estimate. I'm trying to show that if I act on a function f and I restrict this guy to a tube like this, I get something which is small. Okay, so let's erase something. So what I'll do, as I talked about, is I'll use exponential coordinates. So here's my fundamental domain. My geodesic will be something like this. I can always use geodesic normal coordinates about what I call P right here. And then I can extend, this is a geodesic gamma. Okay, I can think of this as a geodesic on the manifold or a geodesic upstairs. And then since I'm working upstairs, I can extend this. And if I choose geodesic normal coordinates like I've been talking about, this will just be the line. So this is the fundamental domain. And these are translates of it, like in that picture. Okay, so here's one that hits the extension of the geodesic. And then here's, there's a whole bunch of them, like in the picture, actually exponentially many of them. Okay, if the radius of a ball is T, then the number of these fundamental domains is really large. The number of fundamental domains could grow exponentially. If the curvature is negative, it'll look like that. Okay, and so there's sort of two types. So here's alpha of D, and here's another alpha of D. So if you think about Hormander's theorem of propagation of singularities and so on, if you have some experience with this, okay, it's a disaster, as I said, because the number of nonzero terms in the sum is actually growing exponentially. You won't get anything because of Huygens' principle and the fact that this integral is supported between minus T and T. You won't get anything from fundamental domains that live outside of this ball. So that contributes nothing to the sum. So you'll have a whole bunch of terms. It'll be exponentially many terms. It'll turn out that it's fairly easy to show that individually they're all well behaved. They actually have perfectly good norm. You just have to add them up. So it'll turn out that there's two types of fundamental domains. The ones that intersect this geodastic and then everything else. The terms that intersect the geodastics actually, the number of terms just grows linearly. It grows like T. But there's exponential in T terms that aren't there. So you expect the main contributions to come from these sorts of guys. And you'd be kind of happy if you can show that if you consider these guys that intersect this geodastic and are within the ball of radius T, if you could put a small cone through them or if you could put a fairly large cone through them. So if you can control this angle theta, it depends on T, if you could put inside of there by choosing this constant C appropriately. If you could put inside of there an angle which is bigger than or equal to lambda to minus delta where delta is very small, you might be in good shape. And it turns out that you can because of topodinagof. So I wanted to find clipart with a poodle but I couldn't. So if you have your friend who is a girl in this case walking away from you in Euclidean space, then of course the angle that she makes with the horizon is basically 1 over T. It's just decaying like 1 over T. In hyperbolic space, this angle is like 1 over the sine hyperbolic of T. It's decaying exponentially. So topodinagof says that if you assume that your curvature is pinched from below, which it automatically is, I'm dealing with manifolds of non-positive curvature. They're compact. So I have a lower bound on the curvature. I might as well assume the lower bound is minus 1. I could just multiply the metric by a constant to achieve that. When I lift the metric upstairs, on the universal cover up, preserve that lower bound. So I can always assume that the curvature is pinched from below by minus 1. And then topodinagof says that this angle we're talking about is bounded from below by the corresponding angle in hyperbolic space. And because that angle is what I just told you about, that allows me to get this lower bound. And that's terrific because I can add up these guys. I have universal bounds for all of them. I'll have this 1 over log here. They'll come with nice bounds. And they'll also come with jth term. They'll also come with something which is of this size if that's a jth term. And so that allowed me to add things up. And I get different bounds if I'm in dimension 2, 3, or higher. And then finally, because I have one minute, let me tell you about how to handle everything else. So these guys are good. I have to tell you how to handle these guys that are outside of, that do not intersect, that avoid this cone. So all I do is I take this operator and I compose with a pseudo differential operator that localizes to this cone we're talking about and then everything else. So q theta is localizing to this cone in Fourier transform space. So this is my angle theta. So these guys will really only see these guys because of this picture. So this piece is very good. So let's ignore it. So therefore, I only have to consider this operator composed with this. And then I can undo, I can go back to this formula. So I take this operator and compose it with this. And then I have to prove estimates for this. I can use Euler's formula again to really reduce it to this. So this composed with this, I need good estimates. And this involves an average. And this is a unitary operator. So since I've run out of time, what I really need to do is to show that if I take this sort of thing, I'm rushing through this, acting on an f dt, then I get good L2 norms over these tubes if f is supported in a tube. And because this involves an average over an interval of length capital T, and since these are unitary operators, I could just reduce to this. That's a trivial reduction. And here's why it works. Here's the wave equation. So at the end of the day, I'm trying to prove good estimates for L2 norms of this expression inside of this small tube. Oh, I forgot to put in this cutoff. That's important. L2 norm over the tube. I've got to the punch line. So I need to estimate this. So having this pseudo differential cutoff means that I'm just only considering waves that are traveling at an angle theta or more from the direction of the central axis. So if I want to prove this estimate, I just have to know if I have a particle inside of the tube, and it forms this angle, how long will it take till it exits the tube? And the time, the escape time, it's easy to see, is going to be less than or equal to lambda to the minus 1 half, and then you're going to have to lose, of course, if the angle is small. You lose like that. So this will be trivial because f is supported in the tube. I'm taking the L2 norm of the tube if time is bigger than this. So really, I just have to estimate this. I'm rushing through this. And then I just use synergy estimates. I just bring the L2 norm out, and I use the fact that this is bounded to L2 on the manifold with norm 1. And so when I play this game, I'm going to get a gain which is like this. And that's a really terrific gain if my angle is like this. That'll give me a total gain which is like lambda to the minus 1 half plus delta. And because I'm just trying to beat logs, that's way more than I need to handle the contributions of this remaining piece. And it looks like I'm over time. Sorry about that. So you had the same correctly. You used a wave equation just at the end. Yeah, just at the end. Just at the end. Domain of dependence you use, but that could be. Domain of dependence. No, it's actually very... I didn't emphasize this as much as I should have because I forgot to say I really, really use the global param... Hadamard parametrics of Bayard. It's pretty subtle to use the wave equation. You're using... It's... You run into disaster if you didn't have Huygens principle and things like that. So you really need that... Yes, really, really, really. Yes, and I need very, very... So everything goes wrong logarithmically, and that's implicit in his parametric... parametrics. So I use quite a bit... We use quite a bit about the wave equation in hyperbolic type manifolds. Very sensitive to that. So this parametric is only productive logarithm in time. Yeah, exactly. Did you have a question, Walter? In negative sectional curvature case, there's a huge amount known about the geodesic flow. Yes, the geodesic flow, yeah. You're using like one geodesic, or sort of even a part of the geodes, because you're not translating it around by the deck transformation. Right. Just using one geodesic and coming face. Is there, if we say strictly negative sectional curvature, does the situation get better? Yes, it'll get better. For instance, it's just a technical theorem that I showed you gets better. And the most interesting case, if you have... The most interesting, I consider it all geodesics, but the interesting geodesics are the periodic geodesics. So if you have a periodic geodesic, you can put one of these tubes about it, right? And then you can care about how the estimates depend on the length of the tube. And there are other problems that are also very related to the types of things that you're bringing up. A slide which I skipped. There's something called period integrals. And the log improvements that you get, are they expected to be optimal? Or is it possible? No, well, it depends. So there are some far out conjectures. So it's conjectured that by, somewhat by Tarnac and by physicists, that if you have strictly negative curvature in 2D, for instance, then the eigenfunctions, as they are in the torus, should be essentially bounded. But people are way, way far away from proving that. This was from the 1970s. And nobody's improved on this. Just like this conjecture is implicitly, is from the late 80s, but except in very special cases, nobody's proved them. So there's optimism, but. Your result implies the improvement of the trigger to the goal piece for the Schrodinger equation on the Schreiber 24? That's a good question. It should. It should. I haven't looked into that. Actually, I didn't really think about that till I was here feeling guilty about the fact that there was nothing nonlinear in my talk. Spectral problems, not linear problems. You multiply a lambda by a solution. Okay. One more question. Great. Thank you. Yeah, go ahead. Thank you.
eigenfunctions but near eigenfunctions, geodesics are periodic with period 2 pi. They're the calculations that we did in the previous improvement. Bayerard's work, which is actually allows you to show that this implies this and way as the L2 norm? It's related to weight okay, using wave equation techniques and actually of negative curvature, compact manifolds of inside of there by choosing this constant C manifolds of non-positive curvature. They're estimates for L2 norms of this expression inside This was from the 1970s. And nobody's improved on