Wave maps on hyperbolic space

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Wave maps on hyperbolic space
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The Cauchy problem for wave maps on hyperbolic space exhibits several features of a different nature than the corresponding problem on flat space. In this talk we'll focus on the question of gauge choice and we'll sketch the proof of small data global well-posedness and scattering in high dimensions using the moving frame approach introduced by Shatah and Struwe. In this setting of a curved domain, the argument will rely crucially on the fact that the main dynamic equations in Tao's caloric gauge are scalar, rather than the tensorial equations that arise in say, the Coulomb gauge. This talk is based on joint work with Sung-Jin Oh and Sohrab Shahshahani.
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notion of a gage 1st we haven't we will suppress scattering unbuttoned
appropriation of I had this is sorry this is a constant absence is this is a measure of Infinity's so so yes it's it's it's it's given by the initial data so so so should reduced to a constitutional point on part of ancestors of Italy so I went and I in and that the target is on no and is adjusted by this based on reminding manifold Savannah geometry so including the frivolity space sphere of yet again and is a nice reminder for fear of what this is and this is a small data is and for for a larger later than than the dynamics for inadequate depends on look a lot on the target of but here is so small that a problem so here's friend of the OK so I think a few remarks so so 1 this is on some of small data critical theory and for the way problem and wider as a critical tool Watson's critical and I have this scaling variance but in the case of dancing legal guardian then found think study was to think that you have to 1 is invariant is unique and so what's form here invariant under the scaling of equipment the scaling think so as a solution and so is Youlanda effects due to London wonder and so is the analog of of the small that's going critical of global result focused on nothing more about this this this problem including minute I saw a cover remarks why wiry doing this so so sometimes I wanted the ball 1 it's just a it's a natural starting point a study on the geometric with equation on a curved background it's so nice conceptual curvature and geometry and all in a head start and that there's Wallace is that the linear theory on each diesel so linear theory has been well studied so on Seoul's solutions this equation was studied in particular from Celesta got so the dismisses persevering and so the global circuits estimates on approval of free equations and hyperbolic space and this is an American right all the on provisions the borders to the kind of home I'll mention a few on this for the shorter equation His work by announcement stuff flying In order to Banneker on and for the wave equation and hyperbolic space I we realize circuits estimates of Metcalfe and Taylor and on Ontario careful each other so well and so the case was still trying to answer the question why stay on 2nd wife of high dimensions well OK again this is starting a high dimensions is an easier problem because we have better dispersion and higher dimensions without prompted becomes quite difficult on outside a cemetery dimensions to 3 it too is the most interesting "quotation mark injured intervention because seemed the cost to the critical sold off Mom the energy spaces age 1 cross-sell to Seoul really interested in studying this problem and so on In the energy space would be forced to on lower starting with this theory on a high dimensions to develop some technique needed to address a low-dimensional kits on OK so tables for is easier to starting point and focus on our maybe auditors which of wide will in some is we've taken hold a look at the energy critical case but under a senator reduction so so with my colleagues we studied the unhcr integral case under coordination symmetry and some mean so vigorously with maps from Oct cross HQ taking values and the last 2 aged from 2 different from two-dimensional suspected pockets and court initial symmetry means that the map respects the action of additional multi-domain and pocket and in this case we saw on qualitatively different behavior than 1 sees in the Euclidean problem comes I will talk to much about this today but not many of you see 1 of us talked about it and some of these works in the past always find interesting is that so even in the case of a major target on a two-dimensional case undergo officials monitoring there exists on many of the final energy come harmonic maps and and he's playing into the role of anemic so they're all on as part of a stable and in particular for Thonotosassa but stable but the global as well as globally hasn't honored stable summoning an unstable solvent resolution army and in that case it should target so you can look at and official data on many geometric propagation of initial data you can identify that tells you which but harmonic nappies scatter to business is in stark contrast to including cases where it's well-known say from the sixties in the work of the workable Sampson that there are no I'm fine energy from harmonic maps from to into a manifold over negative sexual curvature a leader on that case we have to target His brother more interesting there is again on a continuous family from maps and these displays some interesting on stability properties that I won't get into today but it relates to the and talk of shop Miguel on Monday mean that 1 sees in fact anonymously so slow decay so there it's stable harmonic maps 1 sees anonymously slow from 1 maps arbitral sold it's so both these properties are interesting and our goal in studying the way the problem outside of symmetry is to try to and then study non core additional provisions of reform ,comma so study the the full fury of an orchid there's there's a course that many more financial might have sponsored centuries-old OK so let me I'm leaders decided that this is motivation talk a bit about how to prove this gear so 1st why is it so difficult is your problem and I have small US move complex exported data all of the main difficulty by the end of career quadratic derivative nonlinear OK and on and it's well known that that this is not pejorative meaning that you can't come close integration argument on based solely on estimates for 4 blocks cancel so-and-so in fact 1 went 1 can't disclose using summits simulator techniques this was a difficult for the case Wednesday is already this small data call attracted quite a lot of research over the interest over the 90's in early 2000 but there is fundamental work by Oklahoman and on inclined suburb for the shop the subcritical local theory and then the bridges breakthrough work of depart and then followed by How in that round 2001 who approved the analog of the sphere on and the case was saying as already and when the target manifold as this year OK and this is followed and led by the work of Clendenin ski and who who established the same for marginal targets and so on and then looking Krieger on as well for the for the case of the of hyperbolic part OK so let me let me and honor awarded to a bit more the history Man at Bologna just say what Wiskowski insights so that the key incentives of town was that found the builders say this was that it would look into the 2nd round the board was that they did the gage
structure of the wave equation and can be used to read normalize away the number attributed part of our nominee area so we performed a gage gage transform which made which render this quadratic the release the worst part of this quadratic the interaction of perturbed of amenable to preserve analysis look at and this idea of using a gage structure on a rowboat rose the arms harmonic publisher and work of Phillip To the harmonic maps Europe to her management focus on then so is breakthrough worker tell so this is the case I think the festival and there is a deep civil it's implication of this of the of the of the Tao Theriot profile theory and high dimensions and due to shut us through and none had defined siphonophore will embark on their own 2000 and flurry of ordinarily thousands and in particular to focus on the work of shutdowns to and as if there was a general approach to apply here so it's structure tends to realized was that you could perform a is similar deep implication wanted the former global gage transformed and some innate so they've they've been refunded and a stage transformer which in particular they they formulated the derivatives formulations and chemical engaged folks went 2nd on HBO this was that after if you put yourself in the coal engage the whole number on it is essentially becomes good cubic a higher and hence amenable to procure but of analysis so let me out so this is roughly the approach we used on powerful the cool engages in fact very poorly suited to the case of a curved background demanding laughter resorted to a different technique all roughly follow the approach so when the outline how this goes looking sold Japan's through their work in the derivative formulation which means that sounds so they work with the idea is to work with the EU on instead of you and the is a common is a one-for-one over and with values and you start thinking lies in the public Wendell and it has the most advantage of being winner is the 1 format of this section this bundle but so is the linear object and so on the idea of the idea is to on shoes for DePaul in terms of the year on the 1st episode Tuesday global Frank Field on the election intended vectors very smoothly and you start again which is a vector bundle over an OK so why is such an objective was just 1 of a global often move frame exists because OK so that the basis of contractual manifold cases are across HT so the bases contract mostly always at least 1 such friends OK such as the level of Springfield and then the choice of the gives rise to a connection me some particularly connections on the graduate of money starting and 1 can be expressed in terms of this connection its architectural 1 form a false they often where here this is the location connection on the on the base manifold and was users extends that would attend served attends a bundle on taking that tender signatories and this insupportable OK ,comma and solicitors again appears in the in the formulation of the way problem and now will express dealing this frame sold on the rights of fines ,comma 1 form MCI such that the DU is sir that maybe the the stateside I J and down the component or the components of DU as well as with Florida's time so the offer you is the of OK so now the wave of problems and becomes the following give crosses system so so you wear translates known this sigh formulation to the offer trial physical 0 and then we have the following portion freed property of our connections this is a locker relations the other outside data by debate site a 0 on the way back from him this difficult system and what about this connection form a while this satisfies so a satisfies a girl tape formula so on this kind of greatest as me it was helpful so bereft up F is the the curvature to form on the pullback on and since the pullback rock collector Richard this is nothing other than the remind heritage and served on the target and on that side of the side of those factors the pullback and on the curvature among and so a satisfies this equation but is otherwise undetermined so we have some freedom here freedom announced that John choice of which then determines an and so actually make a particular choice there they require a divergent free the DAB especially divergent 3 equal 0 this called a cold conditions note so this is the shuttle should framework for a for the wave equation and also that the sell-off remark you used by nominal back and stuff this is the 2nd is that this change in China which has since and and said that all the time so answer I would my my my my while I was during at the holder and harmonic map formulation in this this this and this is that was not within the union of all Castaldo who leave they say Sure sure know of course that has also allows a mention on the little paper from the early 80's which you can always in a small data setting you can always the question we have to solve this divergence equation for find such that difference and this is this is always be done in a small there's a upsetting the fundamental paper fullback from 82 I think so he can thank to the all so this is the Colgate set up and what's where we go from here at the OK so here we are we just now orders differentiate these these equations and obtain analytic equation for a and a wave equation for PCI on so we differentiate them in the following the declaration of a mail for the curvature in terms of being known this the following on from the weather cleared from sigh some of so this is equal to or from the city for the dollar was insistence on Milford said that OK I'm sorry
this is a complicated looking equation but in the case signal
is already it simplifies quite nicely because these are curvature terms for the 0 0 in the case of amendment flat and awards all again but the left-hand side simply becomes costs and other components of the missile components ways 1 form but for each component is the scale policy a is equal to of the right-hand side of the of New and the 2 scalar on down version of sigh his home the so we have the following coupled system elliptical wishing for a nominee with reverberation for applying NL shall during analyzes this system on using L P estimates work with the center-left experience estimates for a and Stricker existence was with equations and particularly prove control over on the proper estimates to other controlling normal for for this trip and and they're able to close the small data on Friday OK so this is very nice is very simple very simple argument and to prove the small data theory for 4 with maps released as simple and in today's at this point I'm so where does that so-and-so mentioned that comes in it's a handling this term so it kind of dimensions you can so you have to look where we're working we had we analyzes term so the right-hand side of stadiums in on the right side needs to go in L 1 T L 2 x saying intervention for for example and we have no choice but to put this term but now and In on and in time and an L 2 and an X because we control 1 this is this is intervention for age 2 and so we control 1 1 2 derivatives of you which is 1 derivative of sorts of suspicion should go to that means it's but a and L no 1 and T and on and on Finnerty Annex OK and this is this is this is that at the pump I hope that the point is is that is that we have no choice but always you put this now Infiniti and and lower dimensions we can't control at the 1 end and no 1 in the weekend before the fella and ability and senior more more sophisticated structure the right-hand side neighborly there's no structure parent here which 1 relied on Lord mentioned very delicate analysis and the simple spaces as well on the basis to systems are not quite enough to calls for higher dimensions we barely admission for the threshold for 2 simple argument on contributors work and it's because of this this difficult from here 1 sorry a parade of
this holiday extending this now
to the setting of a curved background will OK this now or in trouble because we have from the curvature terms here ends a left-hand side doesn't produced nice scalar Miletic equation or with quick come and and also what what are these locked inside the objects this you can you can see is the Hodge on what policy of the 1 former 8 and this is the highest down version of the 1 form cited cancer equations and there no 1 good estimates known for her for these objects in particular there is no global should protest was known for the French I don't version ,comma acting and for intact in the case of ball states is known that the huddle pass into his role in 1 forms in fact a failure of dominance of the stressful OK so that is valid exterior and just unknown on undisbursed fear in this case obviously the injured so how the main question becomes or 1 initial question becomes have former this problem correctly caseworker background In particular how do we deal with the issue of tens reality on the on the left-hand side for the main dynamic variables so the anemic variable in this cases our or a often the site of the and the answer on Parliament comes isn't that a nice way to get around this if we use Intel's caloric which was introduced in a different context but but miraculously deals with this issue tense reality and beautiful so has tells Flora gage work on as for the next Informatik questions so the issue this finding framework 1 that deals with no 1 pretends reality of promenade questions and to I'm still the point of this whole thing was to read normalizing on linearity to make this amenable to perturb of analysis here so I just can't say enough about this but from the that in this case on a L but we saw elliptic equation in a meeting with place we write a adds on as creative and resourceful posse members of grading this is a nice of our partners abounded about the geometry services like the passing of sites where for example so a is that they can be expressed as in terms of inverse derivative of squares and so we see here in linearity that we have the money becomes essentially cu and sigh of his all so how we deal with this reality and also on suitably we know more OK and so the town's caloric it'll do both these thanks for so hot work and it's a really beautiful idea and and it's kind of an American so recklessly simple so we choose so that 1 side so at its base in the harmonic Mackey flow but for for a wave Matthews we start with so start with where our goal is to provide free estimates of these so start with this move waiver of you yaks and will use its image as initial data so on time and we use it as initial data for the harmonic people solved on wages a new time variable s solve DS minus policy new expensive you think you think you and financial data time as equals 0 0 TX each time the result of a Vermont people so under mild pushed up assumptions on falls well behaved as there was in this small data setting and in particular converges on same constant not on teen acts as a spokesman for August and we have worshiped of him all I can think of something OK so converges that people adjustments which of assumptions mergers to a fixed point as as close affinity for each gene acts on and now all of whom were not choose so now the next step is to choose and then France and the on new start to land which is now sits over what's interesting here too since all revenue with is a new time variable asks what's it's over here and there's a canonical choice for this frame at equals infinity since we commercial constant that we simply choose the same set of words normal vectors I'm overreach .period acts in demeanor as equals infinity so yes attitude and crane P interconnection for so it is a function of and act in the next and we require that being said the acts converters to sustain fixed frame as a and on and on our 2nd requirements and this is tells caloric at tells corkage conditions is that the frame be unparalleled to require that the the physical 0 which is the same as requiring that a yes it was so this article or a gage conditions and also Trident motivate on both these choices 2nd ,comma four-story Dunham are candidate for a new dynamic variable and so express arm up you in this friend so right as before except when I is the site as a whole .period Egypt's IS and again and still the offer new planes in flight of from an our candidate for conveying dynamic variable beats IS OK which is now this it's over Seoul's you this is manifestly and scalar services field questions soul show now that if we just work with clients we on satisfy both those 2 questions always been suitably Norbury normalize among linearity and also we deal with tense relevant questions you yes it's all the same you infinity yet so was fixed by the data and by a push-up assumptions and system reboot strapped with also assume that the map converters to infinity from property OK Our so why is this so why is this a natural choice for the dynamic variable on let me try to motivate this with really simple example namely that of the linear and heat flow on Euclidean space and uses little telling theory sold so analogy fitted with allegedly floor so given function of not we can solve the heat equation with his data and the solution to the equation so effervescent nothing other than on convolution with accounted for in the case of business there is already a 2nd In the side is new to the minors has 6 square if not the Gallician adapted to on the ball on a frequency CSE lesson on customers who have gussied up this ball and frequency space and so this dance the high frequencies of the of the function at located down below frequencies so low frequencies the dental frequencies we can
multiply by a scandal plus will find little pellet production of frequencies comparable estimates would have that being the low frequencies with this this operator and into the the posting the knots which is a kind of policy for glass and nothing other than since F solves the equation is just ask the yesterday but so this is our this band frequencies stands high frequencies and was roughly like production on frequencies like Estimize one-half and what in fact prove all the nice things a little Pelley theory including this where function estimate on in this framework Euclidean space so particular on and what's nice is that we can recover From this little pillar resolution just by integrating the operations progress and this is our little co-productions OK so that the 1 sold here little Kelly production under frequency customized half is given by the estimate of oral proper initial functions were function and this analogy here so will we use the little bit the harmonica peaceful resolution of our not you as from geometric an analog of Littlewood pillar of the little television so this is this is our candidate for main dynamic variable annuities subject geometric from or nonlinear little appealing the composition of our you OK so let's let's just briefly on the show resolves the remaining issues and we have cost the this 1 analog of its reconstruction formula which is where precisely where tells caloric gage conditions and so on so the rate TX convergence to affixed to infinity and yesterday that converts to fix infinity as discussed the and so 1 can see that all the other derivative components soaps I offer and along with the connection formed these they're forced to converge to 0 at the planning and moreover we have is caloric conditions so DS SO AS is equal to 0 is the same thing as DOS and his partial and again this palace means for any that the curvature if we had s any 1 of the components here as of this remember on involves derivatives of a a commentator of any with itself so this system is the only 1 remains in the mail right side colorful as is the integral my signal from To infinity by the fundamental theorem of countless biggest sale is prime which is now using this Court gage condition of most affinity on capital D so this is the time at which is then and now we distortion free properties such nothing other than the Alpha so we can we can recover the all these derivatives that PCI flows from sigh as derivatives of players and same with the so after is on TSA and cynical of efforts the last of which is now the pullback curvature scientists so often and so we see in a is now quadratic and on science and Scialfa and weakened since we can recover apply also from science we can now recover all the elephants from just say long as well and finally the finally the the equation for PCI yes is mostly a scalar phenomena way equation the conclusion so if we think the outflow in the office of SIS within commuting to the office of the U.S. of knowledge we pull the estimate is out on remember this is just the Espo injuries occur Richard terms is the S the deal flow tile floors and implores but the curvature year writing needs to have his at Alfa SA and I'll call this W. This is a scalar now deal "quotation mark office is a scalar this only of again we can expand this saw the left-hand side so this is just the components so this 1st guy acts like this is like the offer and sale of times notices of intensive and with sentiment this so this becomes the scalar down version of science is equal to DSW so this term plus half of us of we get by expanding its product here minus 2 in Belfast what the rest of minus an awful awful wireless and mobile phone and OK so this becomes the main dynamic equation that will then estimate using strict is now scalar equation adjustment is using Stryker its estimates on from skeletal policy and unskilled over hyperbolic space and let me about about this nonlinear structure here so what's W W is is the alpha Scialfa and this is at 0 also on so this is number this is the way inadequate insulation should be 0 on but only 0 s equals 0 1 because radical otherwise we've for commuting the wave not the Hermite people with which the new so W. restricted as he 0 since we were aware that this is equal to 0 in it satisfies on WB show satisfies a & variant heat equation which result from as he goes from as equals 0 so this is this is at this is that from the time of this equation occurs at the time of the equation for each 1 so solved on Net 0 2 s and a member of the analogy to the army 2 of the UN linear heat for this is that the time between 0 is like a high-frequency part of all of of the function OK so W W knows the high frequencies of the initial map arises from the high frequencies on and in the rest of the linearity on his other 4 terms involve just the frequencies at S & M lower so and so little in particular so this involves frequencies as prime bigger requests which are the low-frequency part of our our solutions are with our Matthew you and we see this in these equations here so a receipts I'll find a also is involved in rules of from S to infinity so this is this is like but this is this is the nonlinear structure rises from the low-frequency component of map itself OK so this is a this arm so 1 can of it so this is kind of where the analysis now starts so study this equation and improve for estimates an let me just remark that the In the initial shut off to reproach I said Where is the crucial they did not say is that we have we use elliptic theory to estimated and this is the same and the number of Safonov will approach you looked at varied estimate they on what's the
replacement for that wants nothing other than the regularity theory for the harmonic people skis so we recover a and Scialfa just by integration by phone out fear here but we use regularity theory for the for the parabolic equation to skillet obscure pro-war equations to prove estimates on the was made the loss of state aid in terms of players on and sigh often terms of science a couple other last remarks so 1 system advertising for the caloric ages of this is we can see this is like nicely on sold all this will work in the 2 setting as well on and invective of the treaty setting that I laid out earlier on because it is relies on understanding the harmonic people which is a well-studied equation somebody works nicely lined large settings as long as you have a handle on the harmonica people and in the cases were you want to study the wave of equations Tom you have such such shipments understanding source for larger setting work work at Colgate is a little bit of a disaster when she will want you want you leave small at of very blood difficulty with this looked I'm here I'm in particular it is quadratic interactions of a which then become relevant art which I didn't even write down here on which the perturbed over the small setting but they become probably the problematical arches so accorded as nice margin settings and the last of the the remark was last-minute is that the worst part of the linearity so it also is is better than that the worst interactions among the so here I said that a was roughly in the hole and it said in a is roughly like Converse derivative of sites where is small setting and here
the the worst direction given dangerous high high low interactions with high high on In inputs for the low output in the hit with members derivative becomes dangerous the cultivation this interaction just doesn't appear on the court courtyard setting so it's 1 factor on this appears in our analysis and an some way it's very portending the important on the loaded setting it was very important particularly work ensuring about but million-euro to Canada and the talk with the work of the of the the question is he will use the source of the good activities but so the sold not much on meant what I said here but on the crucial factors that at some point when Aoun start analyzing this equation and use Stryker its estimates on 4 of the numbers hyperbolic space and so if you have a different geometries anonymous relied in particular hyperbolic geometry was part of the day but then if you when you start to provide estimates on I think you need something like this for the future of the menace than this works here so it is that it for instance they think about the future of the possible release of the U.S. yes it's Emmy and particularly From my the that something this is not always wanted yeah few young as