Equivariant wave maps on a class of rotationally symmetric manifolds
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Title 
Equivariant wave maps on a class of rotationally symmetric manifolds

Title of Series  
Part Number 
17

Number of Parts 
21

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CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
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Release Date 
2016

Language 
English

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Abstract 
Joint work with Qidi Zhang (Shanghai). We study a class of rotationally invariant manifolds, which we call admissible, on which it is possible to prove smoothing and Strichartz estimates for the wave equation. This class includes asymptotically flat manifolds but also perturbations of real hyperbolic spaces \mathbb^ for n\ge3, and other manifolds with non vanishing curvature at infinity. Among other results we can prove the global existence of equivariant wave maps from admissible manifolds to general targets, for small initial data of critical regularity H^

00:00
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07:50
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28:54
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36:24
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I had what I'm among the best in the world so I have to thank the organisers from the food politically great places take Bruckner's interesting conference so all of the results that we talk about our government and 1 project that we have achieved John Doe who is in Shanghai and hope that the published by the work in progress and some some tools and also in other people and is so let me recall not for me being less than half of the audience and uh what what what is the problem when Maxim itself as the general definition of this this is not even the most skilled gender possible because of some cases 1 could consider even more generous public but essentially the most generous so we consider maps between 2 manifolds the paddock manifold is Riemannian manifold on the other hand the base manifold is completely Lorentzen manifold the signature decided and the cadences of a weaker notion away map it's a critical point of from for some Lagrangian can do Lagrangian density is our are hands with and there's so it's up to the tracing the H of the World Bank of the Mexican g uh in coordinates it's very easy to see that it this quadratic thing going into the pockets of course and I remember that the signature of H is a hyperbolic softer is hyperbolic looking question as wondered that will mean that the easiest way fool to feel the question is still think of it as a hyperbolic version of the harmonic Napa which they took this very natural denunciation of the house .period map of the question but of course rather different and OK maybe it's something to say to advertise for them because they were known but rejected they pop out in several different theories and including a general relativity and some of the existence of the such accessible some field so that you can get there will not be questions in some dimensions and so on Little said considered more concretely to what kind of aggression we Have you waited so the stand of the situation in which would not substandard is starting from flattening Cosby space of course so in this case and but usually 1 should write the heart time sentences the simplest that but in the case but it is necessary flattening cost based dimension and space that if you expounding coordinates of the you get the system although millenium with corrections the a question you know aggressions which very nice structure so the got because symbols and you get the quadratic formula derivatives which is for this is sometimes called the extrinsic formulation while the intrinsic formulations existing formulation that is nice served from a dramatic point of view from former point of view but there doesn't leave the room Bloomberg and it is more difficult to work on it for use in the formulation of course but it's nice so if you need medicine medical leaders target 94 only 2 of the men and you can just say that I would match Sunday the Mets such that the box you expandable to his electoral proposals 1 of land and you takes values the the target and that so there's no nice way to look at the when the progression but not the most useful so if further even if the target manifold is realized as the 0 locals some functions then you get somebody Nizar expressions global mice expressions for them when the equations of for instance if you a few diabetes as fears that the systematic inclusion of course and you see that clearly the nor form structure all generally patent In and the similar more general expression and you get as the target then is that the uh the onset of some models would function so there you have it all was the same kind of structure came to some functional you time for and but I wanted spring In this case when the base manifold is flat you will have a lot of results now that there is some possibility of well understood that and there are some corner cases which are still under intensive study about the especially twodimensional case but that of course the natural setting is because she problem with the initial date back here if there is stuffed with the mind and body protest which should be commended independent space the 2nd debate about the understand faith and HAS the usual scale of spaces and you look for a functions which are continuously a chance and is 1 in span 1 and that you will need the check and that you have an iceskating care which is given by and so they could because faces agenda was filled with political interests and of you and you have so that you have immediately and the the nature of the expectations on the situation for this problem and it this and completely standard the so the problem is critical when asked his people the promise supercritical Manassas lower than others to use you expect something that occurred and the press is larger than you expect weightlessness in some sense In actually in the critical cases of maybe I should washable too ?question mark here because it is the amendment that in that case of the geometry of target and listen to hold in this case value and should play some more important role trade but basically this is the picture you expect fantasists confirmed from the local theaters of the local during behaves nicely so essentially if as largest weekly object and and tools and you get a look on wood poles announces this was by probably the 1st revision of London the government building that had it in the context of way operations the woman wanted a kind of onedimensional and effectively announce came in on the other hand the litigation for the S smaller than other
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please note I did not receive a lot of attention anyway it is not too difficult to construct a very stronger examples of important as this this is written like this but actually our at our own country examples of government initiatives target is sphere but in my 2nd in the much after that that paper but I think it shouldn't be difficult to check but also in higher mentioned you guessed that essentially the same maybe not that strong because here you can write to 2 different weak solutions but uh it's something pretty stronger In Boston as the results of 1 the critical cases you see that not everything is nice suits following written by the powerful that actually you but it's a good space so you do have you can construct solutions which only meets all the sequences of smokeless and approximating solutions but said at the same time you can prove that the social flow is not well behaved the case so you have some troublesome because you can do this that explicit and that's what has gone deleterious nice problem the interesting problem for that with my preparation and and so you have a good result force holding back and this is also expected so if you have small Dayton attended to and and targets are reasonable so you don't really need of course it any special assumption the target the apart from Clinton once completeness of because a small date back as they did when only a notice so small region of the target so essentially this is this is what is expected in on the other hand for last they got an this problem becomes already fragile difficult in you can blocking for time assessing senior brought up when the target is sphere National Lottery 1 and if the targeted as a whole .period geometry then it's more difficult but still cannot produce blocked a motorway notice that lowdimensional blockbuster in and if the exceeds about lowdimensional I don't think anybody has yet proved that role for this problem so this is interesting OK anyway so this is the general picture there are many results but for this other than the generous oftstated resolve that sees something for large classes of goods to the case is especially interesting because the political space is also energy space which one's soul you have the question of what maybe the local so you can expect little dissident to be continued on not too so this is really the geometry of the target that comes into play here and so can Nova also several people from here in this city started this agenda going in a complete study it anyway for instance if the target is the twodimensional CAD I brought if the target is the name of a monarch space you have noble assistance and there is someone going to to block specification and got through the nice solid resolution because of was skin so L but system related result but maybe around here now let's focus on what could release them over corresponding to Roger solutions said the agreement would knapsack the corresponding to the final solutions for the way the creation and aunt that of course is that some of the general situation actually ,comma inside the point of this was the 1st case of started and but said this isn't the subject of this book so let us consider a special case now end not the fact it is additionally symmetric manifold so that you have an idea to coordinate and then the kite 2 is just the fundamental economists fear that as and when Swann and you have this coefficient essentially domestic is completely determined by dysfunction here of course so if you speak different fashion you get different geometries and right at you you do some last stand of the computations as for the rest of us emotionally thinking this is it for this kind of target that I remember that the flooded in the basement the missile flattened costly for this kind of product that the question of a undergoes a dramatic simplification of course and let me just show you how to think about this this coefficient this this function G so for instance if you have a sinus function with another 0 you have a loss of manifold anarchists fear if you have an function growing and you get an open geometry like they have report expects so wanted to sign him so that it provided when that's our maps which are completed the couple Soedradjad component of you and I do love born satisfied 2 different sets of regulations so he assumed that 5 component of your depends only on of course diamond honor that added that it will of course the space and kind depends on the on the underlying value of the space component and because his base and and you get that kind has satisfied just the money from progression from the fears and so with which give some rigidity to the the dimensions of course and and you have to to the Adelaide the incubators for formula Of course and the 5 satisfies nice equation which is just that around the island Cimmino wave equation but we've done some singularity here which makes this is the greatest thing and it can dispose of course is still H and of book so this is the provided and there will not be operational and this was the 1st model and consider the cost of the communities and of course the previous results is often mentioned above previously applied to this question state had Bloomberg assistance for small data and actually brought results are known for precisely this this model that is the source of much of the 4th in you also have the global existence of large solutions provided the target is sufficiently open so this is a good lighting conditions on domestic of foreign money for them and can't commit acts access to this 1 can so not Mississippi
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chair of very spotted picture in the sense that they didn't mention that many many resources there is a lot of the ongoing work on an understanding of what mechanism and classifying the possible rates of all that and there are some more precise results leave even enough for the school data a and the existence of the this was just a very general introduction so now that the the program that
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addresses and what can 1 say at the same level of generality and so the basic results and when the the best manifold is currently you have you have geometry also in the basement of a notional flat geometry dismantled this is a partial list of results and maybe the others but I mean there are not many results in this case fucking get injured AlQaeda problem officials I believe the yield special solutions no result for the look at this probably result minority and the others are the only ones which vital to prove someone more generalities a so the 1st ball here would be to the approval of the global system smaller collided WiMax intersecting famed they provide in case Of course this is just the 1st step but that I'm ready and
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there are some nice there is something nice about Connecticut book OK so 1st of all we would like to see precisely what kind of a question we have to deal with it so now we have to manifolds which had rotational symmetry the first one I called them no the function of the Italian component think age and the 2nd 1 is before the G end the question becomes this 1 can salt of course and are is replaced by them the function H consisted of 5 yellow box speech on on the manifold them continues in this direction is very similar course but you had this difficulty because of the presence of this opportunity indeed the presence of this operator with window valuable coefficients here so what we can we prove our for this aggression and this is a list of what the approval I surely there some explicit statements about the essentially that this horrible nature of results and all the surprises involved here so we what we told is a global assistance of the small Freeland maps and In the critic of space on the added money these arbitrary and the interesting part is that we Line a glass of manifolds this manifolds on this but the competition is not especially difficult to and we can do this so we can reproduce the procedures on flat space you have a view of the usual a version of the theorem with local assistance with elastic bandages the phenomenon difficult modifications you have to work a bit harder to prove Ohio United results because editing here there are some parts of the question which are singular so you don't get actually arbitrary hiring United up to some degree and it is useful because it gives you an unconditional uniqueness of result the department just improving the the bitterly divided the nation data to show you what we can do completely let me also mentioned that we have some work in progress about the life of I'm not often enough to to 1 announce so that the 2 the case that can be done endorses just a technical problem I think I think was removed it and also we have sleekest estimates also history then the Americans should be deficiency metaphor about the data that could be allowed to be Norway that thinks so you have to work harder much harder for this but that it's possible to do essentially just with effective Will this message checked so I hope to win the next 2 months OK so this is the definition of missing manifold so it should trying try to get the because possible generality it it's a list of conditions about you will see quickly that there is essentially a couple conditions which are important so 1st of all the located concluded that age is just a function which drives and medical bills and the base manifold and so I formed this this quantity here and this is secondary but there in the audience got its 2nd round of the of a book industry should probably decreased to 10 but anyway a salt 1st requirement that is asymptotic certify that dysfunction pencil constant for Avalanche witnesses the rate a nonnegative constant this says that there is some complexity at Infinity for this function for use as a basis for their wanted manifolds insists and minus 1 of the summer this is really not a this it difficult to check just by looking at the function that it just says that the are you have a certain number of derivatives and like the 1 the whereabouts of the consisting of 2 satisfy the 2nd condition this 1 is of course a very restrictive and here is that the another restrictive conditions so 1st of all the domestic must ordinarily so you don't have a lot closer the manifold suddenly and this function must be Boston dealer and decreasing so this is a global conditions which is part of satisfy but you should not is that of you should see with the examples that actually this is really effective only on some say someone that so this is nothing but a condition and this is a condition some complex sector because of its it becomes very successfully stifled laughter as he was 10 OK I don't like this is very convincing this list of the likes of conditions but the let me show you some nice examples became so you can appreciate that there are a lot of money falls in this category something which mounted the 1st of all that the monarch space is included it's easy to track and so you see the function age is a constant itching finicky blast something which decays extremely fast and sold all those conditions are easy to satisfy the only 1 which is so restrictive is the concern is this you have performed this function and check that it is positive and the decreasing almost universal criticism at but as I will mention later once we have won many folder which satisfy this condition we can we have an opposite the money falls within condition is open and we can quantify this thing so that we show you what we get immediately you can perturb a channel so that instead of just the parabolic signed you consider any perturbation which satisfies the set of conditions sold to somebody about this but must make a like epsilon action which should be sufficient to so what is the following factors this this of course is increasing the expression so you see that this condition is OK it it is effective for our small and say that for our largest you have a lot of money falls on geometry is included OK this one's for you not to From condition can so we have a whole hope that's but emissions but I don't believe that the mission is the correct term because of a definitive businesses that can be quite far from a chance of course the other advanced laser if we can include all possible to flat spaces and this works using at the the flat spaces HE is precisely art and incompetent but with something which the cases like this simple and that this is such a small small perturbation and and that is in my this is so dramatically different from the other case is that what is important is that Rachel between them this new and age so if it shows a lot into this automatically small in but 7 but of course this out just the 1st 2 cases which comes to mind but that you can it's quite easy to construct in October examples which entered this deceptively dissatisfied central sanctions for instance if you want to construct an arbitrary growing get a hyperbolic exponentially growing medication can do if you want the company nominally going genetics and you can do and the ones who will you find that it's nice because otherwise you just find 1 which satisfies the constraints that we needed to have a lot of good examples but put the pressure all cases there so this is missus the type of conditions which we get when I was a separate division so if age is inadmissible dramatic than any checks on those satisfies a set of conditions like this actually you also need another set of contingent conditions but without acting on up to and some of but especially this is restrictive condition because see stuttering this a necessary that this is small about the system must be small compared with the page which can be put on and so this system the general picture we have of the different parts of the show results of which from freedom to produce examples looking out an official the protest statements soldiers I mean what you completely expect at this point time for this is that we consider the crucial problem for them at the right hand that question between 2 rotationally symmetric manifolds and that this coefficient coming from the 1 that the aggression H infinity but all in the state and is the limit which is assumed to exist this quantities OK so
27:03
in when so far this is not the patient is a standard definition of possible especially true in the innovation is a standard definition of a solar space on the money from them and they also use this notation which is awakened us over the space of type a you the wait is this quantity this in some Martin some was set of coordinates the that you should probably use it right the oceanographic research expression but the fact that the point of the I would get 3 dubbed this quantity as much as possible so that our statements of slightly different statements but essentially they say exactly the same thing that depending on the age affinities so strictly positive at 0 and then so you have to look assistance with sufficiently small data indicative of space and that's it end Of course you have the usual appendix sources said we use a sticker stressing that so you get a additional property of the solutions and also an additional constraint for uniqueness became so you can communicate strongly in this restrictive space OK in that case agent
28:29
fitting 1 0 just have to slightly modified the definition of the HS sold meaning that there is a small part of the the body which is homogeneous nurses knowledge in this technicality could probably completely changed but I mean this is not only for essentially the same results and this is our choice because there is not much freedom in their choice of strikeforce .period cooperate which appear clear the 3rd and
28:57
probably the good over other sponsors could also work OK so this is the main results on Of course as was just mentioning her you have this annoying problem said that you were we need an additional condition for uniqueness soul this solution conditions in some stickers space and you have the nature of standard question about unconditional uniqueness so if it is sufficient that can do that exist other solution would think they initialed at any chance a new industries and infinity and turnover to note that I think this problem was sold out not too long ago by those with depression In the flat case and in the basement of the flat and Atticus is just go below with the originality so you work in some negative space of negativity delighted I mean it's just technical and this and I said but the deadliest simple the problem is that I mean you don't have to estimate some nonunion quantities in summer not in some notes which are not easy to handle and this is more complicated than our situation so I guess that it should be possible to do just under the conditions uniqueness resulted also in this setting but it's 2nd slightly more difficult so we did it with a what you can do is coming and this is just my implemented using techniques that carefully if you just 1st of all you need a higher regularity South but this not difficult as solely for the initial out slightly more regular after Sept an order which is determined by the singularity and many of the land of the United and if Is there should be a slightly more dangerous than the critical of space then you haven't unconditional uniqueness in this space and this is not very large but do not regard as small in the case of not larger analysts so this is all sold more as a croissant and the local just described how it works no and since the situation is very simple again because the question is just and the idea was to question and with some funding efficiently and so it is it's not too difficult to use to no techniques you don't use of his Honda techniques such as the best thing you have to work a little uh I saw that the 1st step is to make his or approach the 1st step is proving to directly on the manifold and a sporting yesterday and so this is the 1 on the way to approval of status testimony came and so forcefully process but against his by some of the players techniques and that which which coordinates sold the aggression is used to attrition would go singular potential going so this is the only a minor part of the it's proof were sold then it you you have to prove to prove plastic for this aggression but since you have already splitting estimates of this can be done I sneak in and despite you have the necessary to which are more than sufficient that you don't even need the day before the strength of the estimate just pointed to prove to implement the witness 6 appointed a metal door for the critical civilian everywhere creation you have some troublesome because there are some similarities in the coefficient of inside and undecided and in effect but just a minute technicalities to prove some project estimates of the decision efficiency not the big problem you have with the work of harder for the higher the United States and and then you have to to read the result in the original court that's OK so you have to go back to them also want knowledge and then I mean that the address of the work is essentially to construct manifolds and we see how the hell out of the class it's just a better show you so this is a change according so that users to reduce the question To this form so I suggest you get stand up there with racial you imprison mentioned by 2 K which is nice because it gives system but the students instances but the properties the problem is that this potential his singular and a critic of the K and you know that if you will always the negative part of this city is too bad because to be give you lose everything so there's some part of risk here and also here essentially have just a here but they're not the intractable a began is so there is a yet bartender association which has been has been taken here and then you have a twoweek but OK so this is a cubic Samina regression with Cingular coefficient so we chose which has the same critical spaces as usual and was and look at it such but this is what I just said that because of the skippers in OK this is just the kind of thing to see that the estimate with improve the full set of estimates with the exception again .period also in this case the South misunderstand us because the that just as the method the usual condition on the admissibility of the acute and this is the indigeneous version this is precisely the sort of estimates we've improved In in some coordinates as saying that you get someone somewhere OK mn let me briefly recall you that there is now an extensive literature on a status estimates and dispersal estimates and sloping estimates of the various perturbations on the stand questions so far and away the question now we needed something which should have been borderline because he is a critic of the case I mean that there are many results also in this formerly the first one of these spousal then the 1st group of backing in the case of the 50 candy cane was a hoot rational started permit those there and then we did some work on small and large magnetic potential something so that there are many of its results and maybe I should also mention that the danger for genetic secondorder sport division of constant deficient cases but small but the battle is probably the best results OK and so the
36:26
1st step is to prove a sporting estimate debt is solvent operator so this is a process ,comma operate on a that then this should must be very very precise canceled I mean it's easy to prove that splitting estimates would go worse coefficients but this is actually shot you have to work little and once you have this in place and put a suitable Ginger valuables to get also that the estimate for the disarmament of Europe "quotation mark disturbed that the possibility and now despite the this gives you immediately is splitting estimated on the following 4 fantasists stand the couple but not as this is of course a shedding the operator and the sharing of rope With this for free and it was from this you get this so difficult you should work a bit harder to transform this sick corresponding splitting estimate for the wave equation so after media 1506 time I have to do this I decided to the general it is so you can do you can patch couple Yuri and so consider any John terminated operated on by spaceage any closer than that said defining if you have a splitting estimate for shredding and you get this thing as a metaphor for the way it did so and attend I'm just going to find coupling proved that can and of course in the case of the relation you get some money worse estimates for this 1 have but it could become a complete 1 half of the working people he said wait that could be cut hitherto because as the materials the weight and the 1 after that he about what we're in OK once you have this estimator for the wave equation then the potential can equal to the present site is just perturbations you hear about the fact you have to work a little because of the it's annoying coefficients which did you wait so that a single such system In but once you have stickers festivities is justly Kaplan the space in which you get contractions and is Morris standard that you have to be carefully there be being too and that can the war after some guys and then you have to do work different indifference basis according to the value of each intriguing again time I think collect just mentioned that yelling on my part in this competition is that you need them To be careful when you need a new Anania estimates the for for this quantity infractions of a basis and this is the only on my part where you have to work a bit harder to to produce this suitable estimate Whitley lead with it we don't have to look out we had some you never some uniform into the answer to which handles this case is kept in a vote at Inc I can stop here that from the thank you you thank you this kind of thing you do is you do you will lose your latest Jr directions the 2 going to the police station Ohio voters whose cost function In view of where it is you have all a because of because of the it the and some input on and interesting examples of course because otherwise they can just go in any direction public the steps to get from radio Hughes channels but a few if you mean the comments saying a setting about the knowledge of data this should be I think contained essentially in the middle of the shipments to should should should work with the striking a deal with the problem strikeforce in this case from the random data so that should be the only essential tool but that was that was to have to check nonIndian geometry and this is of course another story with more than you need a computer different techniques I think this is just a simple approach works but I mean it's a surprisingly effective because you get the relentless basis interests with the most too much of I thought this is normal question them but it's time for you and you think you probe