Toponogov's theorem and improved Kakeya-Nikodym estimates for eigenfunctions on manifolds of nonpositive curvature
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00:00
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Transcript: English(auto-generated)
00:15
Well, thank you very much for inviting me to this conference. The conference, of course, is called Nonlinear Waves.
00:21
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, Leboe, and Planchon
00:40
to prove existence for the critical wave equation in 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.
01:03
The frustrating thing is that there's really not good theorems for manifolds of negative curvature. So I'm gonna talk about improving LP estimates and Kakeya Nikodim estimates, whatever that is in that setting. And maybe someday there will be some progress on NLS in that setting.
01:23
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 Planchon is negative. And lambda J is the frequency of the eigenvalue. Geometers don't put the square there usually.
01:43
So lambda J is the eigenvalue corresponding to the first order operator, which is the square root of minus the Laplacian. All the eigenfunctions are L2 normalized. DV is of course the volume element. The Laplacian is the Laplacian coming from the metric. And so a vague question is how can you detect
02:03
and measure various types of concentration of these guys? And you'd expand your horizons a little bit and not just concern yourself with eigenfunctions but near eigenfunctions, quasimodes, whatever that is. And you expect that you might get an extreme concentration
02:20
at certain types of points, which would lead to bad sup norms or bad LP norms for large P for certain types of points. And you also might expect to get concentration near periodic geodesics. If you're gonna get concentration on a different sort of set,
02:41
you'd expect the set to be invariant under the flow. So the natural thing is periodic geodesics. Okay, and I'm gonna concentrate on the latter. So a geodesic of course is bigger than a point. So you might expect to detect concentration along periodic geodesics in terms of LP norms for relatively small P.
03:01
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,
03:21
if you take these probability measures, then they should converge weakly 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.
03:42
It's conjectured in the case of negative curvature by Rudnick and Sarnak. And it was proved in some special cases, for instance, by Lindenstrauss. 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,
04:03
it fails miserably, this should be the volume of 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.
04:20
So, of course, the sphere is up there, the M-sphere. It's a subset of our M plus one 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 M minus one. And they repeat with a very high multiplicity.
04:43
That's the highest multiplicity that's allowed because of the sharp Weyl formula. And what's going on there is there's a lot of periodicity. All the geodesics are periodic with period two pi, they're all the great circles. And additionally, if you take P,
05:02
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 two pi in the case of N odd or four pi in the case of N even.
05:25
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 one to the M-sphere.
05:40
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 or periodic geodesics. So here's a guy which is highly concentrated on the set where the modulus of X1 plus I, X2 is equal to one.
06:05
So in other words, the place on the sphere where the other coordinates are zero, it's equator. So it's highly concentrated on the equator. I put that factor K to the N minus one over four to L2 normalize it. So you can evaluate the modulus of this, okay?
06:23
It's exactly that, of course, by calculus. You Taylor expand the natural log about one 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 one.
06:41
And so that's a K to the minus one 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 one half.
07:03
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 the normalizing factor. You're not really making too many mistakes. Okay, and so therefore you can compute its LP norm.
07:21
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 one over P, which is K to the N minus one over two with a times minus one over P. So the norms of these guys are on the nose comparable to this. Okay, and this works for all P bigger or equal to two.
07:41
So they have bad LP norms. Okay, and so here's a picture of some spherical harmonics. So the ones that I was describing, well, this has, this never vanishes, but if you took real parts of this, right, you'd get a real eigenfunction. Usually you study real eigenfunctions. So if you take the real part of this guy,
08:03
it's called the highest weight spherical harmonics, and it's these guys, okay? As K increases, because of the Gaussian being behavior, like some speakers, almost all the mass is on the equator. Zonal functions. I'm in the audience too.
08:21
Zonal functions, which I won't be talking about, but that's a separate talk, are functions which are peaking at the north and the south pole. They're the middle guys. The first guy's pretty boring. It's just the constant function. And then you have spherical harmonics of higher and higher order. Okay, and these pictures also denote where it's zero, which is a very interesting problem, the nodal sets.
08:42
And already, what is this? The spherical harmonics of degree one, two, three, the spherical harmonics of degree three, 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
09:04
for eigenfunctions, and there are two ranges. There's this critical exponent, which is two times n plus one over n minus one. This should be a familiar exponent to lots of you guys. It's the exponent that pops up in Strickart's estimates, for instance. So I showed that if you're below that critical exponent,
09:22
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 one over two times the gap between one over two and one over p. And because of the calculations that we did in the previous slide, we see that this estimate up here
09:44
from the 1980s is sharp. It can't be improved. This universal estimate holds on all compact Riemannian manifold. 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,
10:01
where the power of lambda turns out to be something different. It turns out to be this thing, n times the gap, one half minus one over p, minus one half. Way back when, 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 Bockner resummation.
10:22
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.
10:40
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 it's true 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.
11:02
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
11:22
the wave equation for a unit period of time. You don't need to watch a long movie, okay? And that makes sense, right? Because on the sphere, these sorts of wave operators that arrives are periodic. So if you watch the movie for time two pi or four pi, you know everything.
11:42
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.
12:01
All right, let's see. Do you mind 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,
12:21
and would those also play a role somehow in spectral asymptotics? Like not exactly talking about what you were saying, but a zone manifold will also have these bad properties. A zone manifold, all the geodesics 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,
12:42
they're disastrous for the same reasons for estimates like this, okay? Okay, let's see. All right, so I was gonna say this. Knowing when you can improve these estimates for small exponents is something that's only been recently understood.
13:01
I've worked with Steve Zelditch quite a bit, and moreover, the work of Berard from the 1970s says that if you're willing to consider big exponents, in particular p equals infinity, then you get a big improvement. Berard's work, which is actually very important for what I'm talking about now, is from the 1970s.
13:26
The sup 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.
13:40
So in any compact manifold, the sup norms grow at worst like that. That's a theorem that's often attributed to Hormander. He wrote a famous paper in 1968 called Spectral Function of Elliptic Operators, which was actually proved in the 1950s by Avikumovich and Levitin. So that's the universal bound, and Berard wrote a very beautiful paper
14:03
in which implicitly you get this improvement if mg has nonpositive curvature. Okay, and we'll use his techniques quite a bit. Okay, it's a global theorem.
14:21
You have to deal with the wave equation for a large period of time, up to basically log lambda. Okay, 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.
14:44
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 sets. Nodal sets are zeros of real eigenfunctions. There's a conjecture of Yao that the size of the nodal set,
15:01
it's n minus one dimensional Hausdorff measures, should be like lambda, okay? And the lower bounds have been studied quite a bit recently and before basically 2010, they were pretty bad. They were exponential and lambda. And then we were quite happy we got polynomial lambda.
15:22
Lambda is to some power, negative power in high dimensions. Okay, and there were two competing techniques. Steve Zelditch and I were using wave equation techniques, and another competitor of ours was Kulting and Minicazzi use harmonic function theory.
15:40
Okay, 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,
16:00
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 Holder's inequality. So the L2 norm is one, you holder it. Two is between a one and a P.
16:21
You take P to be in that range I talked about, between two and that Strychart's number, two times N plus one over N minus one, holder it. And then you get the bound that we were talking about for this. Figure out what this power is, do your arithmetic and get that. By this Holder argument, you see that if you can beat the estimate
16:41
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.
17:01
We were able to beat the world record for lower bounds of Kolig and Minicazzi by logs. So in particular, in 3D, the Kolig and Minicazzi, 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 it's bounded
17:20
for three-dimensional manifolds. Not good, but at least it's not exponential. So this proves the superiority of the wave equation, I guess. 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 of Sodin and Israel Lugonov in 2016.
17:46
And he completely solved the thing for the lower bound, okay? 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
18:02
with Donnelly and Pfefferman, 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 Kolig bound? Lambda. Should be comparable to lambda.
18:20
Just like if you take cosine of lambda x and you count the number of zeros. Okay. It's gonna 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.
18:40
So this works only in the analytic case, you see? 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 Pfefferman are only the real analytic case. That's 25 years. There's been really no significant progress, but he, this young mathematician, proved the correct lower bound for the C-infinity case.
19:02
Very nice. Okay. All right. So now we come to the Cacaean 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,
19:21
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 can 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
19:43
where he has the profile of the highest weight spherical harmonic. Okay, and a large proportion of the tube, he'll have exactly the same size as the highest weight spherical harmonic. I was hoping to be able to use this for nodal lower bounds, but it's obsolete.
20:01
Okay, so at any rate, if you, so the highest weight spherical harmonic has its mass in these tubes. If you can't beat the L1 lower bound, then you have a guy that looks a lot like the highest weight spherical harmonics. So maybe something to consider is the L2 norms over these types of tubes.
20:21
So I call the Kakeya-Nikoden norm of a function just the supremum of all the L2 norms over these shrinking tubes. Okay, so you consider tube centered anywhere and their orientation could be arbitrary and you're taking the L2 norms over these shrinking tubes.
20:42
I call it Kakeya-Nikoden because these sorts of averages arose in the work of Cordoba, that would be Antonio Cordoba when he studied Bachner Ries. In his case, he was considering the maximal operator, which involved the sup over averages
21:00
of your function over all tubes of width delta and length one about a point x. He called that the Kakeya maximal function. And then in 1991, Bourguin had a very, very seminal important paper that broke open a whole chapter of harmonic analysis and he switched terminology. He called it the Nikoden maximal function.
21:22
And then he also considered another problem, which is related to the structure of the Kakeya set, which is the maximal function where you fix the orientation of your tubes, but you sup over all the centers. So I do both and so I call it the Kakeya-Nikoden norms.
21:41
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 highest weight spherical harmonics because of what I said before. So the problem is when can you beat it?
22:01
When can you show that these Kakeya-Nikoden norms, you're suping over the averages of all 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
22:22
in work of Bourguin, which anticipated this. So Bourguin 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.
22:45
Burch, Duran and Secoff in 2007 proved this universal bound. Okay, that the L2 norm squared of the eigenfunctions over these geodesics are big O of lambda to the one half. That's saturated by the highest weight, that's a 2D.
23:01
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.
23:20
Bourguin 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 could beat Burch, Duran and Secoff. So Bourguin showed that this implies this. It's easy to see that if you can beat this estimate, you can beat the Kekan-Nikoden estimates.
23:42
Nothing's going on. Kekan-Nikoden says you're trying to have small L2 norms over a tube. If you get good L2 norms over all slices, you just use Fubini's theorem. So let's see, two implies three. And then in my paper, I showed that three implies one.
24:04
Bourguin 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,
24:24
and they're oscillatory integrals. They're the types of oscillatory integrals that arose in the study of Bachner-Ries. 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 Pfefferman.
24:43
This is kind of a geometric approach using these Kakeya-Maximal, or Nikita-Maximal operators that I was talking about. Bourguin 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,
25:01
which is due to Carlos and Julien and Hormander, 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, okay?
25:24
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.
25:43
So Blair and I, in a paper that appeared in 2015, but we did the work a few years before that, backlog, you know, we showed that the same thing works in higher dimensions. In higher dimensions, especially four or more dimensions, you really are stuck with using these K-N-N norms.
26:04
This is too singular, restricting your eigenfunctions to geodesics for technical reasons. So the K-N-N things are natural to use, especially in higher dimensions. And so Blair and I showed that the result extends in higher dimensions.
26:21
If you have small K-N-N norms, then that 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
26:40
raised to a certain power. And then this K-N-N thing raised to the theta. So the 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, okay? So this is a stronger estimate than my old theorem
27:00
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, okay, if you can improve this, then you can improve these L2 norms over these tubes. This gives you the hard half.
27:21
And let's see, in 2014, Zelditch and I showed that in 2D, you can get these improved K-N-N norms, okay? If your manifold has non-positive curvature and in the same paper in 2015, Blair and I showed that for higher dimensions.
27:40
So armed with these facts and this inequality, you can see that if the manifold has non-positive sectional curvature, you get these improved L-P estimates. 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 their theorems
28:01
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 Bayreard's theorem, involving the log. Well, one quick question on the previous slide. So when you define the Kajan coding norm, you define it so that it scales
28:21
somehow in the same way as the L2 norm, or? It's related to wave packets. So it's, you take all tubes of width lambda to the minus one half, about a geodesic. And I don't, I don't turn it, I think, 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.
28:41
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 equation and so on. We showed that these Kajan Nikoden norms have a certain decay, okay? And the decay is basically one over lambda.
29:01
It's worse in 2D because of the worst dispersive properties for the wave equation in 2D. And it's just a little worse than 3D, okay? And we also showed that if you have non-positive curvature, you can beat the results of Berkshire and Speckhoff by the same amount. Okay, this is when you allow the curvature to be zero,
29:21
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 we soon will be with negative curvature, then instead of having dispersion
29:41
of one over T to the N minus one over two, you have exponential dispersion. One 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,
30:02
you get the log improvements of this. Of course, that's gonna give you log improvements of the LP norms, okay? And so finally, we're getting something like Berard. Maybe not the optimal power, but some power. N minus one over four? This is N minus one over two. Remember, in my case, my eigenfunctions have eigenvalue lambda,
30:22
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
30:41
of the size of the nodal sets, but it's way blown away by the recent breakthrough that I told you about, okay? All right, all right. So also, and this made me very happy too, because this also did not seem to be within reach. As it turns out,
31:01
if you take these improved LP estimates over here, and you take Berard'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 Kikeya and Nikita Maxwell functions.
31:21
If you take an argument that he had in this paper which gave a very simple proof of weak-type estimates for the Stein-Tomas restriction theorem, if you throw, if you take his recipe, take Berard's estimate, and take the improved LP estimates that Blair and I had, it turns out that by using an argument of Bourguin,
31:42
you can prove weak-type estimates, improved weak-type estimates for the critical exponent, PC, which is two times N plus one over N minus one. 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 Seeger,
32:00
but actually it follows from an interpolation argument of Bourguin again, you can upgrade those weak-type estimates to get LP estimates, okay? And so I'm happy about this because nobody had attained improved LP estimates for the critical power. If you can prove improved LP estimates
32:20
for the critical power, then just by interpolation, you get it for all exponents, okay? 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. Okay. All right, so I wanna tell you, since this is a wave equation conference,
32:41
how the wave equation arises. All right, so I'm gonna tell you about how you set up these arguments. You wanna prove these improved L2 estimates over these types of tubes for eigenfunctions. Okay, a problem is that except for very special cases like the torus or the sphere,
33:02
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, okay? You choose a Schwarz class function. You want it to be one at the origin. This will allow it to reproduce eigenfunctions.
33:21
And because you're gonna 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, okay, then this function of p, rho of t lambda minus p is defined by the spectral theorem, right? So if you act on an eigenfunction with eigenvalue mu,
33:44
then it's just multiplication by this Fourier multiplier, rho of t of lambda minus mu. T will turn out to be later on the small constant times log lambda, okay?
34:01
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 one and you get the bounds that you want. Okay, so if you can prove this estimate, then simply by taking f to be e lambda,
34:22
you get what you want. So you want, of course, the constants to be independent of lambda and so on, okay? So this is what you need to prove. Okay, so how do you do that? Well, you just take this guy right here and you rewrite it using the Fourier transform.
34:44
So it's what, one over two pi. There's a one 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 t p dt.
35:02
And because rho hat is supported between minus one and one, this integral just involves fairly big times, but times which are at most log size. Okay, so that's a formula for this operator that we need to estimate in this way. And then you just use Euler's formula.
35:22
You add to this operator rho of lambda plus p, p is a positive operator, lambda is a large number. So this guy will have a kernel which is rapidly decreasing. So when I add this operator to this, okay, I'm just making a mistake involving an operator
35:40
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 t p with a plus sign. Okay, and then I just use Euler's formula. So after using Euler's formula, I can replace this by cosine of t p.
36:11
So I have to show that this guy, okay, has small, when acting on a L2 function f, has small L2 norms over these tubes. I like cosine because cosine by Duhamel's,
36:23
sorry, by Huygens' principle, this kernel will vanish if I take the kernel, it'll, well, I'm gonna lift it up to the universal cover and getting it out of myself. So I wanna use Huygens' principle, that's why I introduced cosine. Okay, and so now I use some geometry.
36:40
So by the Cartan-Hadamard theorem, I can rewrite this kernel. I can consider the universal cover of my manifold because my manifold has non-positive curvature. And so I take the universal cover, which is Rn, and the covering map is just the exponential map, okay. I identify, so the exponential map is of course
37:02
the map from the tangent space at any point to the manifold. I identify 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 geodesic. So if I play this game, I can get,
37:22
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.
37:41
The wave kernel for the metric downstairs, okay, agrees with the sum over all the, okay, there's more here. So you, 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 n,
38:02
okay, 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.
38:22
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 Rn, okay? And so you're summing up over all the translations. You're taking the solution
38:40
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, okay? 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
39:02
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.
39:22
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 their 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 of strictly negative curvature,
39:43
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
40:01
the fundamental domain for, say, the two-hole torus, the manifold of curvature, the simplest type 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,
40:20
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-hole torus surface, you have to make twice as many cuts. So there should be eight sides instead of seven sides.
40:42
In this picture, there are 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 Poincare disk model or the upper half space model.
41:00
This is the model corresponding to using exponential coordinates, geodesic normal coordinates, which is the right sort of coordinates for the way I'm gonna 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 and negative curvature, of course, the sum of the angles adds up to less than pi, okay?
41:21
This isn't so relevant. So now let me tell you about how we're gonna try to use the wave equation, how we're gonna prove these things, okay? Hopefully we 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
41:44
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,
42:01
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. It has seven sides. Okay.
42:22
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'll just be the line.
42:41
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,
43:01
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 D, and here's another alpha D.
43:23
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 non-zero terms in the sum is actually growing exponentially. You won't get anything because of Huygens principle
43:40
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 they'll contribute 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.
44:02
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 geodesic and then everything else. The terms that intersect the geodesics, actually the number of terms just grows linearly.
44:22
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 geodesic
44:42
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,
45:03
if you could put inside of there an angle which is bigger and equal to lambda to the minus delta where delta is very small, you might be in good shape. And it turns out that you can because of Topanogoff. So I wanted to find clipart with the poodle,
45:22
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 one over T, it's just decaying like one over T. In hyperbolic space,
45:41
this angle is like one over the sine hyperbolic of T, it's decaying exponentially. So Topanogoff 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,
46:02
I might as well assume the lower bound is minus one, okay, I could just multiply the metric by a constant to achieve that. When I lift the metric upstairs, okay, on the universal cover, I'll preserve that lower bound. So I can always assume that the curvature is pinched from below by minus one, and then Topanogoff says
46:20
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, okay? And that's terrific, because I can add up these guys, I have universal bounds for all of them,
46:41
I'll have this one over log here, they'll come with nice bounds, and they'll also come with the j term, they'll also come with something which is of this size, okay, if that's a j term, and so that allowed me to add things up, and I get different bounds if I'm in dimension two, three, or higher, okay?
47:03
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
47:22
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, okay?
47:41
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, okay?
48:02
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 to get estimates. And this involves an average, okay?
48:22
And this is 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
48:42
if f is supported in a tube. And because this involves an average over an interval 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.
49:01
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.
49:22
That's important. L2 norm over the tube. I've got to the punchline. So I need to estimate this. So having this pseudo differential cutoff means that I'm just only considering waves
49:40
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,
50:05
is gonna be less than or equal to lambda to the minus one half. And then you're gonna 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.
50:21
So really I just have to estimate this. I'm rushing through this and then I just use energy estimates, right? I just bring the L2 norm out and I use the fact that this is bounded in L2 on the manifold with norm one. And so when I play this game, I'm gonna get a gain, which is like this.
50:48
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 one half plus delta. And because I'm just trying to beat logs,
51:01
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. You said correctly you use a wave equation just at the end.
51:24
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 Hadamard parametrics of Bayard. It's pretty subtle to use the wave equation. You're using, you run into disaster
51:44
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 parametricy, parametrics. So I use quite a bit,
52:01
we use quite a bit about the wave equation in hyperbolic type manifolds. Very sensitive to that. So this parametrics is only good after logarithmic time? Yeah, exactly. Did you have a question? In negative sectional curvature case,
52:21
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 geodesic because you're not translating it around by the deck transformations, you're just using one geodesic uncovering case. Is there, if we say strictly negative sectional curvature,
52:42
does the situation get better? Yes, it'll get better. For instance, this is the 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.
53:00
So if you have a periodic geodesics, you could put one of these tubes about it, right, and then you could 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.
53:24
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 conjecture that by, somewhat by Tarnak and by physicists, that if you have strictly negative curvature
53:42
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 priming that. This was from the 1970s, and nobody's improved on this. Just like, this conjecture is implicitly
54:00
is from the late 80s, but except in very special cases, nobody's improved them. So there's optimism, but. Your result implies the improvement of the Stinger technical piece or the Schrodinger equation on such type of info? That's a good question. It should, it should.
54:23
I haven't looked into that. Actually, I didn't really think about that until I was here feeling guilty about the fact that there was nothing non-linear in my talk. Spectral problems, non-linear problem. You multiply lambda by a solution. Any more questions?
54:41
Great. Thank you. Thank you.