Toponogov's theorem and improved Kakeya-Nikodym estimates for eigenfunctions on manifolds of nonpositive curvature

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Toponogov's theorem and improved Kakeya-Nikodym estimates for eigenfunctions on manifolds of nonpositive curvature
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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
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as if the Over time and and I mean and I will thank you very much for inviting me to this conference the comforts of course called nonlinear waves I have an apology my talks about linear things on-the-spot estimates that had proved to be useful for proving various types of farm non-interference existence a song for instance on the types of estimates that I'll be talking about in the case of domains were you by Burkle Borland plane chanted Truth existence for the critical wave equation into makes an artery and so on and on also we've been hearing about how good estimates for the Taurus lead to good good existence theorem in that setting there also are fears about good existence for analysts for the sphere spheres that the frustrating thing is that there's really not good theorems 4 of manifolds of negative curvature climate talk about improving LP and Kocarnik estimates whatever that is in that setting you know maybe someday there will be some progress on analysts in that setting OK let's see here so the setting is that I'll be talking about on manifold without boundary Compaq cost metric g and the dimension of the 2 or more pledges negative and manages the frequency of the Eigen value jammed stop at the square they're usually sold managing a steely-eyed in value corresponding to the first-order operator which is squaring of Meisel Parchin all the I confront Israel to normalize DVD is cost the volume elemental plus installed plastic coming from the metric and so and they question is How can you detect and married and measure various types of concentration of these guys and Udinese brand arise in school that not just concern yourself with fighting functions but knew right from 2 squads announced whatever that is and on you expect that you might get extreme concentration at certain types of points which would lead to bad supra warrants or bad LP norms for large for certain types .period and you also might expect to get on concentration near Pier attitude 6 intermediate concentration on a different sort of said you'd expect is said to be on invariant under the floor of the natural things attitude OK and I'm going to concentrate on the latter so geodesic accords as a point so you might expect to detect concentration along periodic cheered 6 in terms of LP norms for relatively small piece and that turns out to be the case OK and you know you can measure concentration her dispersion in various ways 1 my Compaq manifold so there's nowhere for the IT functions to disperse but there is a famous problem on which says for instance when you have negative curvature can take these probability managers the nation converge weekly to the uniform measure so that's called the continuity carried on conjecture and that's that there would be an ideal thing is very difficult to prove conjectured in the case and the curvature by RNA could start acting in it was proved in some special cases Prince despite when its trust in government things that they cited when 1 feels no OK so if this is your goal to try to social that something like this happens if it fails miserably this should be on the bottom of the manifold it fails miserably hemisphere this year is the worst case for everything and talking about today's let's go over that to sort of set up the better things so courses gears up there at the end of the year it's a subset of warrantless 1 given by the equation and the idea values of the square double-platinum year are given by this formula basically K squaring UK times Cape was a minus 1 and they repeat with a very high multiplicity that size multiple said allowed because of the vile formula the shop formula and what's going on there is there's a lot of period at all the GDS 6 are periodic with 3 to play the role the great circles In addition only if you take on PC into instead of taking the waiver so we've been hearing about but you complete the square by doing that than these halfway groups are periodic through the periodic with period to pie In the case of an or for pies in then and so the period city of these half wave operators also account for the bad behavior of the IT functions yeah inferences are nothing but the circle harmonics which of the restriction of homogeneous harmonic polynomials in foreign plus 1 to the end sphere it is hard here it's alright to less talk about the worst type of circle harmonics if the worst type of violence I can function if you don't like concentration p-adic so here's a guy which is on a highly concentrated on the set were the modulus of X 1 plus I X 2 is equal to 1 so in other words has set the place on the sphere were the other according to 0 equator so it's highly concentrated on the on the equator I put that factor K to the and whenever foretell to normalizing so you can evaluate the modules of this case is exactly that a course by calculus e-tailer expand the natural log about 1 you find it it's like this this recall Garcia means see they live essentially in the Senate where this has on the argument here is of size 1 and so that's a on K 2 the minus one-half I want to about the equator I'll soon be calling my values and so this is living in a tube about the equator where the 2 has with wedded to the minus 1 for all practical purposes because of their Galston factor just pretend that this guide to resume his size it is on the indicator function of that 2 times of normalizing factor not really making too many mistakes k and so therefore you have computers cell pin on so you have this factor out front and then if you compete LP you'll get the volume of this to to the whenever P which is catered in minus 1 or 2 with the ties minus 1 repeat so the nor these guys are on the nose comparable to this In the source for all people are equal 2 so they battled the norms OK and
so here's a picture of some of the spherical harmonics so the ones that I was describing well
this has on this event never vanishes that if you took real parts of this when you get a religous function usually study related functions so you take the real part of this guy called the highest Whitstable
monarchs in it's these guys the SK increases because of the gas being behavior like some speakers almost all the masses on the equator the normal functions on fears the zonal functions which I won't be talking about that that's that's a suffered heart or functions which are peaking at the North and the South Pole 3rd the middle guys In the 1st as pretty boring says the constant function and then you are spherical harmonics of higher and higher water tenders these pictures over the words 0 which is very interesting problem the novel set in already what is this Will the Circle reminds us to be 1 2 3 producer her comics injury 3 you could see this very high concentration of the highest waits for core markets OK all right so let's talk about saturation of norms so many years ago I showed on sir Nelke estimates frying functions and they're there to ranges as critical exponent which is 2 times and plus 1 of reminisced 1 this should be a familiar exported to let you guys it's the exponent pops up and stripper for instance so I showed that if you're below that critical exponents you have exactly the same norms that we're talking about case now land and is Katie real array erase that power and minus 1 over 2 times the gap between or whatever to whatever repeat and because of the calculations that we did in the previous slide we see that this estimate appears from the 1980's is shopping can be improved this universal estimate holds on all Compaq reminding them fully can't be improved because of the bad guys the highs with 4 months I will be talking about this but I also approved OMB estimates for larger exponents were the power of land that turned out to be something different it turns out to be this thing and times the gap one-half will repeat mice 1 half way back when when I was doing this that was the case that I cared about because that number pops up and harmonic analysis is what's called a critical and extra batteries summation and what I wanted to do back then was to extend results about harmonic analysis the nuclear space to harmonic analysis on manifold and I was able to do it because of that estimate an assessment red I put my paper because you can muzzle put everything we do in the paper but I thought it was a boring us and now this estimate this reticence much more interesting is linked to several things in actually putting them get used to it will remain in the area the constants involved of course depend on the manifold but the 2 of them growth rate landed to the whatever power always hold it's it's what I would call local estimates you prevent using local techniques Soon Will walls will see the wave equation I forgot to say that 1 of the reasons 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 I approached 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 lock watch a lot of movies it and that makes sense right because understeer these sorts of wave operators a rise of periodic so if you watch the movie from time to Paia for pie 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 hyperbolic space but the ammunition and for will CPI and will be hyperbolic space whatever that means all right let's see you use of the word was should the only thing that periodic orbits sun but at least some of America's have invariant for on the properties I admit of using and nearly all of recurrent and rules also play a role somehow instead yes yes but not exactly talk about what you were saying but it was all men of the walls of the bad property it was all manifold all the G . 6 a periodic to the same period and appear willing to broaden your horizons just a tiny bit and consider what are called Squires he notes the the disastrous for for the same reasons for estimates like this OK OK let's see alright so I was going to say they're on knowing when you can improve this these estimates for small exponent is something it's only been recently understood on my work receives all quite a bit and moreover the work of a rock from 1970 says that if you're willing to consider big exponents in particular equals infinity then you get a big improvement their odds work which is actually very important forum talking about now is from the 1970 the sitting on estimate if you plug into the formula at their best the power right here and if you plug on equals infinity it would say the universal bound is this so any complex manifold the soup norms grow at worst like that that's a Turin is often attributed to a horn mentor here at a famous paper 1968 call Specter Fox of elliptic operator was actually proved in the 1950's buyout of mileage to Levittown so that's universal bound and their
proved very wrote a very beautiful paper in which implicitly it doesn't prevent if energy it is not positive curvature it and will use Asustek takes quite a bit it is a global theorem you have to do with the way equation for large periods tied it up to basically walked with OK and Steven I didn't have written several papers and we actually have a necessary and sufficient conditions at least in the real analytic case of beating this it's a generic condition was not a generic condition OK so listed about some related problems will I did quite a bit of work there's accidental lot of activity and studying the size of nodal nodal onset nodal center zeros of realigning functions there's a conjecture of Yao that the size of the Noble said is that money is one-dimensional house top-matched it should be like and we can aid and the lower bounds haven't studied quite a bit recently and before before basically 2010 there were pretty bad the exponential land and they were quite happy we got polynomial and lenders to some power negative powered higher dimensions OK and there are 2 competing competing on techniques Steve zealots and I were using wave equation techniques and another to cook a competitor of ours was call the imminent hazard use Herman function theory can and into the work of Steven I did this was sort of the central thing to get lower bounds for L 1 works and when this 1st came up I thought this was weird because there's a harmonic analysis analysts when you're doing fell 1 is usually trivial but when you're dealing without 1 is a harmonic analyst usually pretty up about that lower values big difference so Steven I proved this thing right here that the this is a universal bound that the L-1 1 warrant is always that banner from below by this it's easy to prove using holders inequality so the Alton is 1 you holder attitudes between a 1 in a to be in that range talked about between 2 minister cast number 2 times and plus 1 remains 1 holder and then you get the at the bound we were talking about for this figure this powers due written taking it that by this holder argument you see that if you can't beat the estimate that this is controlled by this you get a better L 1 1 lower about and if you feel easy if you feed that better lower bound into the things were pretty you'd better estimates for the size and noble sets and I was pretty happy about that because because of the work of the talking about we're all we're allowed to beat the world record for a lower bounds of Corleone in the Kazakh Man locks on so in particular in 3 D the Collier many causes Steven I came up with a different proof later on about the same time that the law about they obtained was not that the size of the Nuttall said grows like land is founded and for three-dimensional levels that good but we smell exponential this is of the way well despite suggest I so we held the world record for a little while for a few months we could show they blow up longer claim but there's an amazing breakthrough by a postdoc can of soda Israel gone off in 2016 and complete all the To the lower back case and started he's very local techniques and harmonic function very he did attract banning an upper bound it's a very famous paper through a series of papers of Dimling far from and circa 1990 and on their result to Cyrillic metals Memphis Symphony manifold but he proved the key proved the crackdown pretty amazing things for the lower back the the court in the land be comparable to land just like the co-signed Land because the number 0 I think is going on land OK so that's pretty amazing so hopefully wave equation techniques will work were were still trying there's still plenty of open problems but no novel domains animals sets but that's a pretty amazing things this was going no so so this used to be another slide and I could brag about this and so and so I should estimate is another slight Donnelly suffering only the real analytic case that 25 years has been really no significant progress but he's mathematician prove the crackle lower bound from the the the case drainage brokers are 8 so now we come to the conclusion that couldn't part of the title so we've seen that these on cue Landis have almost all of the L 2 mosques in this the shrinking choose about the equator this spearheaded Janessa OK on and on let's see I guess I didn't say so but aren't there if you can't beat this floor of L 1 warms on any manifold then you can show if you have a sequence of mining functions and saturate this norm than you could so show that 3 giant function there's a GSA and there's a 2 where he has profile the Switzer ,comma can a large proportion of the 2 he'll have exactly the same size as the highest weight to require monarch I was hoping to be able to use this for nodal law rounds but it's obsolete
OK so at any rate if you survive with her ,comma harmonic is as as mass in these tubes if you can't beat the L-1 1 lower bound then you have a looks a lot like the price was for or monarchs so maybe something to consider is the Elton awesome these types of troops so I could I call they contend they couldn't norm the function just supremo of all the L 2 norms over the shrinking tubes content so you consider To centered anywhere and their orientation can be arbitrary the new taking the Alton odds on the shrinking to I call it naked and because the sorts of averages arose in the work of coral by there would be Antonia accord over when he studied batteries In his case he was considering the maximal operator which which involved the soup over averages your function overall tubes Of with Delta and linked whine about it .period access he called that the 2 Cammack small function and then in 1991 Borghi and had a very worrisome all-important paper that broke open a whole chapter of harmonic analysis and switch terminology he called it the naked natural function and then he also considered another problem which is related to the structure of which is natural function where you fix the orientation of the tubes but she Super Raul the centers so I do both and so I called that Nicanor Maxwell particularly was OK so clearly because I L to normalize my functions clearly these guys are always less written warnings on trip and integrating here over a subset of the manifold so that's a trivial bound they cannot be beat by these highs with spherical harmonics because of what I said before so promise when can you believe it when the new show that these aren't care Nikitin warms Mississippi over the averages overall doesn't tubes whether these are little of 1 as the Agnelli goes to infinity 20 and how this came up his about 5 years ago in on my work and then shortly before that the work of organ which anticipated this so arm for him was interested in linking improved Delpy estimates with improved where called restriction estimates for writing functions for restriction estimates which you do is you restrict your funds to a unit like Jurassic and you take the L North Bergeron said often 2007 proved this universal back they that the deal square of the IT functions cover this year 6 are overland to the one-half that's saturated by the highest weight density that saturated by the highest weight circle monarch as you that normalizing factor which was landed to the one-quarter 2 days so this is always true and people for a variety of reasons including novels problems involving Donald evaders 1 are interested in trying to improve this for him showed that if you had better L for norms-the numerology extended into DVD on arms always a big of land to the 180 if you can't beat that estimate than you can beat Burke trends that coffee sober again showed that this implies that it's easy to see that if you can beat this estimated beat the Taconic in estimates nothing's going on occasion and says you're trying to have small L 2 norms over a 2 if you get Goodell to norms overall slices prejudice used to be there so let's see 2 implies 3 my paper I showed that 3 implies 1 Britain came close to showing that the the 2 or 3 imply 1 he was using techniques that really go back to the work of scored over when you study these are Meighan functions as you'll see in a 2nd new study reproducing fonts reproducing operators for the IT functions and there also printable theory also for intervals the types of osetra intervals that arose in the study batteries and into the these are some the things are completely Nelda sharp estimates and there are 2 ways of proving that 1 is on through the work of Cordell opened Pfeffer menaces ,comma geometric approaches in these 2 Cammack small firms make axle operators of talking about Bergen only use that method he was actually dissolved by a lender to the ups for ensuring that these that this implies that there's a competing approach which is due to Carlos Angelina which is based on by Linda techniques and when I realize is you just take the 2 things you just put things up and 2 cases splice them together and that allows you to show that this implies this is therefore all equivalent and I'll show you why these are all equivalent in the 2nd in more details the case so on the central thing is as I said it's taking the the biggest possible L 2 norms over overdue over to the 2 vivid narrow and you allow the orientation to be anything in the center to be anything OK and
let's see so blaring iron in a paper that appeared in 2015 but we did the work 2 years before that on that for you we showed that the same thing works and higher dimensions In higher dimensions especially for war dimensions you really are
stuck with using his head and warns this is to singular restricting writing functions in June assets for technical reasons so the and things are a natural due to to to use especially hard dimensions
and so on Blair and I showed that the result extends and higher dimensions if you have small contain nicotine warns then think that exactly occurs when you have small LP norms for the same and on and more recent paper we showed that you can dominate the LP nor in terms of lambasted the crack power times the to normal over the entire manifold raced to a certain power and then there's the Kennecott negative things race to them they said 1 might say that this is a better estimate and is having L 2 norms on the right because this this quantity right here as we talked about the force dominated by the L to normal round it's so this is a stronger estimate than my old here were just had the to number of around to the 1st power and of course the says that if you can improve this then you can prove that despite holders and equality in the numerology here OK if you can improve this then you can improve these Elton norms over these 2 this disk is regarded and let's see in 2014 Zelda Janise showed that and to date on you can get is improved and norms if the manifold as not positive curvature and in the same paper 2015 Blair and I should therefore higher dimensions so armed with the facts in this inequality you can see that if the manifold is not positive section of curvature you get is improved LPs and that was a problem that that I was interested for many many years he know motivated by this result from so I was happy when we could prove this but still In the theorems that I'm describing this page we don't have any any any arms it does give me any indication how the norms improve land and there's not a rate likened their odds chairman following the loss the categories will be questioned this so when you find that you wouldn't want to find something you still on the same way as the more than it's related to wage packets so it's on the take all troops of which landed to the minus 1 have about on Ojeda said and I don't I don't turn it I think I guess I think you know what you mean I don't turn into marriage and Desue bowed over the Yale to norms OK alright so here's that we proved using wave equation tactics and actually using this person properties of a wave equations on which showed that these Nikitin norms have asserted decay Kate and the case basically 1 Orlando it's worsened to duty because of the worst 1st of properties for the wave equation into Anastase lowers the 380 and we also show that if you have not positive curvature you can beat the results of Berkshire inspect topped by the same amount OK this is when you allow the curvature to be 0 but Corsican positive that if you're willing to assume that the curvature strictly negative then you get the same decay rate in all dimensions and there's a simple reason for that test because if you have negative curvature of urine all-rounders with soon will be with negative curvature that instead of having the dispersion of whatever T the Emas 1 virtue you have exponential dispersion 1 over the assigned hyperbolic teacher that power and so that helps you out OK all right so
as a corollary to disputing it into this estimate right
here you get lot improvements of theirs Of course that's going
ideologue improvements of the LP norms of him and so finally were getting something like there are maybe not the optimal power but somehow investment for this is the most whatever to two-member I in my case may I functions have I land a land when the square instead of land for the little plastic intent Bob alright so you get this is a corollary to this is not as relevant as it seemed to be before but use you do get an improvement for the L 1 of iron functions law about and that would lead you to an improvement of the size of the Nuttall says the different way blown away by the recent breakthrough told you about him
alright so on also and this was and this may be very happy to because this this on also did not seem to be within reach as it turns out if you take these improved
LP estimates over here and you take their odds estimate and you throw it into a recipe cooked up by Borg and in 1991 which he used to give a nice princess same coveralls talking about the breakthrough paper that he had involving the care and the Nicanor Maxwell functions if you take an argument that he and this paper which gave a very simple proof of weak type estimates for the Stein Tomas restrictions there if you throw if you take his recipe take their estimate and take the improved Delpy estimates that Blair and I had it turns out that by using an argument were again you can prove we type estimates improved we type estimates for the critical expert PC which is 2 times and was 1 of romance 1 I don't feel so guilty about talk about this could row and talked about we tight spaces and then if you use more harmonic analysis something that goes back to Bock and steer but actually is followed 2 files from interpolation argument were again again you can upgrade those we type estimates to get help he estimates that and so I'm happy about this because nobody had attained improved LP estimates for the critical powers if you can if you can prove improved LP estimates for the critical power than just by interpolation is there for all experiments include you get lot logs because of this argument that I'm talking about the still improvement and be back in the interesting in to turn this into logs that deny to do that
don't far-right so I wanna tell use and this is a wave equation conference held the wave equation arises alright so you tell you about how you set up these arguments you want proof these improved L 2 estimates over these types of tubes crying in front OK a problem is that you know on except for a very special cases like the poorest of the sphere there's no way you can write down a formula functions so you have to attack them through operators that reproduce them and so here's a here's a typical way of doing this case which is Schwartz class function you wanted to be 1 of the origins of the soul out to reproduce citing functions and because going be dealing with the wave equation want is for a transformed to be supported and say the unit interval that if you let PB square Meisel posture and can then this function of PT role of team might lend the money paid is defined by the spectral the rights if you act on an island function without value new banisters multiplication by this for a multiplier role of tea of land new the wall turned out to be later on a small constant times so you have that formula they reproduce Eigen functions so you're trying to show that the inner role of your item function over these small to this small so if you simply take after the land this will be 1 and you get the bouncer to what OK so they can prove this estimate than simply by taking after the land that you get what you want this so you wanna cost the costs to be independent of land and so on OK so this is what you need to prove OK so how do you do that well you just don't take that this guy right here and you rewrite it using 40 transformed so is what whatever too high there is no 1 ever to because of the dilation there's role had plenty over capital T there's into the island and then into the minds I T P D 2 and because Roe had supported between minus 1 wonders in the world is involves a fairly big time but times which most longer inside OK so that's a formula for this operator that we need to estimate in this way and then you just use a winless formula you add to this operator a role of land plus the pews a positive operator relentless large number so this guy will have a crime which is rapidly decreasing so and operator To this end is making a mistake involving an operator which is trivial which has ,comma which is rapidly decreasing when it play this game of adding these 2 guys I'll be adding body to the I T P with the plus sign of him managers will formula so after using Weathers formula I can replace this co-sign NTP so I have to show this disguise that has small when acting on a L 2 fun to have had a small Elton was over these 2 I like co-signed because co-signed by do males are sorted by on ways principle but this kernel will vanish if on if I take the colonel it all along well under looked it up to universal covering Indiana myself so I want is for use principle that's why introduce coastline OK and so now use some geometries so by the cotton had a modest 0 I can on rewrite this Colonel I can consider the universal cover of my manifold ,comma metaphor was not positive curvature and so I take the universal cover which is Oran and the covering that is just the exponential map can identify the exponential map is a course that the map from the tenders space at any point in the mantle identified them at the tender space with R and anaphora sensible I would be playing this game where take the exponential maps Over P. which is the center of the G-8 so if I played this game I can get I can live my metric ,comma my manifold using this covering that gives this exponential maps to get a metric here if I'm assuming as a that my manifold has not positive curvature GHG chilled that the lifted metric will also their property and then you have this formula right here the way the colonel for the metric downstairs K agrees with you that there's some overall that OK there's more here so you it's best to go over the model case so the model case of course is forests that were she identify with minus pie supplies To the and then on this is a fundamental domain for the honor for a for the Taurus and in addition to having these on discovering that you have that transformations in the case of the Taurus the deck transformations Rogers correspond to translating by elements of Zia other picture for this Sensex and then you have this finally here which relates the armed with equation under Torres if the tossed with the wave equation upstairs which to be orange the case and on so summing up overall the translations are taking the solution of the wave equations coral upstairs here Valley area .period acts your final demand and the new translating the other .period y In the fundamental domain get this formula right here were those Sevierville plastered the Taurus this would be the Oriental plastered this formula right here is easy to prove it's simply follows from the fact that see Infiniti functions on the Taurus are in one-to-one correspondence with smooth periodic functions in our and you couple that with need is for the cause you problems on the Taurus and an iron and you get this formula and the very same argument works in this more general said OK I can find very good pictures but this is supposed to be the picture for fundamental demand for manifolds and negative curvature Compaq notables and negative curvature and these are the translates by the Dec transformations I'll tell you why a bad picture in 2nd but it's a good picture because it depicts the way things if you have a manifold strictly negative curvature the geometry firmer Euclidean eyes is becoming very very warped in the angular direction there's no work Ajit all in the radial direction John wise is a bad picture it's a bad picture because it's supposed to really depict the on the at the fundamental domain for say the army to hold for the manifold of richer the the symbols of manifold Compaq manifold of curvature minus 1 not to take the standard as 1 process once you get its fundamental domain to the 2 Taurus by chopping it up you chop it up once they could use longer than you chop it up like this you enroll you needed square so if you do that for the the but that too old Torres service you have to make twice as many cut search there should be a science instead of 7 sites in this picture the 7 sites from my picture so when you do we need you with some there's lots and lots of pictures for fundamental domain cave hyperbolic quotients where the picture depicts the plant disk model the upper half base model this is the model corresponding to using exponential corner students of normal Coronets which is the right sort a cornerstone for the way I'm going to be looking at these problems OK so I can find a good picture of Will you there's no this century so you have a manifold negative curvature of course the sum of the angles adds up to less than pie I came on this isn't so relevant and and on the nominee telling about how we're going to try to use the wave equation Hourigan approve these things k of women of time so this is the guy that I need to estimate is the colonel the operator that I need to testimony I'm trying to show that if I act on a function that now restrict this to a tube-like isn't get something which is small OK let's erase something show up what I'll do as I talked about is all use exponential courted so on is my fundamental domain my desert will be something like this I can always use 2 dozen normal Hornets about what I called P right here 2 and on then I can extend this is as gamma Kerrigan figured this is indeed as a common manifold origin as they come upstairs and to working upstairs I can extend this and if I choose Jeter signal quarters like him in talking about this will just be the lines so this is the fundamental domain and these are translated alike in that picture of the here's 1 that hits the on extension of G and then years there's a whole bunch of them like in the picture actually exponentially many of them caved radius of a ball against the number the number of these fundamental domains is really large the number from the fundamental domains could grow exponentially the coverages negative it'll look like that and came and so the sort of 2 types of years Alford date and here's another another Alfred so if you think about when Mendes stammer propagation singularities and so on if you have some some experience with this OK it's a disaster as I said because a number of non-zero terms and the summer's exit growing exponentially you won't get anything because of weight principle and the fact that this interval is supported between the until you won't get anything from fundamental domain the live outside of ball so that contribute nothing to the so you have all mentioned terms will be exponentially many charms it'll turn out that it's fairly easy to show them individually they're all well-behaved actually have perfectly good you just have to add them up so it will turn out that there's 2 types of fundamental demands the ones that intersect is on this dude sick and everything else the terms that intersected Judas exactly the number of times just grows linearly grows like tea but there's exponential in TV terms that aren't there so you expect to make contributions to come from the sorts of guys and you'd be kind of happy if you can show that if you consider these guys that intersect the on the students said within the ball reduced if you could put a small combed through them where it didn't put a fairly large come to them so that you can control this angle theta depends on TV if he could put inside of there by choosing this constancy appropriately if could put inside and there an angle which is nearly equal to lend surmise delta where Delta is very small you might be in good shape and it turns out that you can't because it upon it off so I wondered if I could borrow the poodle but couldn't so if you have your your friend was a role in this case walking away from you Euclidean space then it cost the angle that she makes with the rise in case is basically 1 of its case to sticking with 1 team in hyperbolic space this angle is like the 1 over the sign hyperbolic of TV's decaying exponentially some time says that if you assume that you're curvature is pence from below which would automatically as doing manifolds and I'm positive curvature of the compact so I have a law banning the curvature muzzles soon about his mother's 1 caricatures multiplied the metric by constant to achieve that when I left the metric upstairs came on the universal Covelo cover-up reserve that lower back so I can always assumed that the curvature spent from below by minus-1 and upon us says that the single were talking about is bounded from below by the corresponding anyone hyperbolic space because that angles wages told you about that allows me to get this lower abound kick and that's terrific because I can add up these guys have universal bounce roll them all have this 1 of a lot here don't come with with a nice balance and the law also coming to the Jake turned the also crime was something which is of this size that's a date time and so that allow me to add things up in a different bounced predicted the timing dimension to 3 were higher state and then finally because I am 1 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 they're outside of on that does not intersect that avoid so all I do is I take this operator and a composer with this year differential operator that localizes to this call we're talking about and everything else so cute failure is localizing to this common 40 transformed space so this is my angle OK so these guys will really only see these guys because in this picture so this piece is very good solar Signoret so therefore I only have to consider this operator composer with theirs and then I can do the acting go back to this formula can so take this operated a composer with this and then I have to prove estimates for this I can use Willis final again To really reduced it to this so this proposal is indeed a good estimates and above an average look at and this is unitary operator systems that 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 another 2 then again good lt norms of these tubes if efforts supported an to and because it involves an average over an interval length capital T and since these are unitary operators like it is reduced to this that interview reductions and here's why works with the wave equation so at the end of the day I'm trying to prove good estimates for L 2 norms of this expression cited the small too no I forgot to put in this this cut off that's important Delta in honor of the 2 I've got to the punch line so just make this so having considered a fragile cut off means that at this only considering ways the traveling and enable fade or more from the direction of the central axis so if I want a previous estimate I just have to know if I have a particle incited the 2 and forms the single how long will it take told acts tube and the time this state times it's easy to see is going to be less their equal to landed to the minus one-half and then you'd have to lose of course the angle small you lose like that so this'll be trivial because EPA's supported the the 2 men taking the ultimate 2 it's time is bigger than theirs so really I just have to estimated this I'm rushing through this and then I just use energy as demonstrators bring the Altoona out and I use the fact that this is valid and to whom the manifold with Norm 1 and so when I play this game I need to get a gain which is like this and that's a really terrific gain if my angle is like this that we made total game which is like Lambert the minus 1 half plus delta and because I'm just trying to be logs that way more than I need to handle the contributions of this remaining piece in books like an overtime sorry about that fact and that's what we it was just yeah is the UN's just at the end of their useful to me and I know essentially there and if I did emphasize as much as they should because I've got to say I really really use the global Parham had my parametric said there are it's pretty subtleties the way the question using it you you run into disaster if you didn't have on Williams principal things like that yes really really really yes and I need very very so everything goes wrong 1 rhythmically mess implicit in his parametric the parametric so I use quite a bit with quite a bit about the way the question in hyperbolic type manacles very sensitive to that the chances the law yeah exactly 2 acquittals maybe this is a huge amount of unit yes the geodesic exploded yeah you're using like what you you know what not translating right and you have to do is to the name the of religion and business leaders yes it'll get better for instance is there in this the technical film that I showed you get better and on that the most interesting case if you have the money that the most interesting I consider all the assets but the interesting Jena 6 of the periodic 2 6 so if you have periodic Genesys put 1 of these tubes about it right now you can care about how the estimates depend on the length of the 2 and on the other problems that are also very related to the types of things had to ring up to the slide which I skipped for something called .period intervals In the case of the people no 1 will independence so there were some far-out conjectures SOS conjectured that arm 5 somewhat by foreign physicists that if you have strictly negative curvature into differences and the iron functions as it as they are in the Taurus should be essentially bounded but you know people are way way far away from proving that this was from the 1970 and nobody's interest on just like the simplest this conjecture is implicitly is from the late eighties but accepted very special cases nobody's perfect so there's optimism but your resulting pool the the question is well that's a good question it it should it should I looked into that accident really think about that until I was here feeling guilty about the fact that there nothing monument talk through long broke the mold .period where the virus what was the few