 reporting on the child's work with Yves Bonoie and well at least in my eyes this work is related to Marcel in a very special way and you know when we first started working on this subject we were both staying in Berkeley and it was the last time actually I was I had a contact with Marcel the setting of was to tell him about this work and that this occasion he told me his memories about his own stay in Berkeley in the early 60s yeah that had been one of the turning points of his career as well as a magical adventure for his whole family so so this talk is an outgrowth of the shun conjecture it states that any quasi-asymetric map from a high-pebbled plane into itself is within bounded distance from a unique harmonic map so this conjecture has been proved true quite recently by Markovitch and then further extended in a joint work with Marcus Lehm to quasi-asymetric maps from a high-pebbled space into itself so what is a quasi-asymetric map well a map is quasi-asymetric whenever it is by elliptics at large scale meaning that the distance between the meters of two points is bounded above and below by a fine functions of the distances between these two points you can take the same see that's not a problem so what is a harmonic map a harmonic map is a critical point for the Dirichlet energy integral and as far as smooth map are concerned a map is harmonic if only if it is a solution of a certain elliptic PD so now we know that quasi-asymetric map from let's say nice how public gram of high public spaces to another gram of high public space do have a boundary value at infinity and that to such maps sure is the same boundary value whenever they are within a bounded distance from each other so another way of rephrasing this conjecture I mean now it's a theorem is to say that we have a preferred representative namely harmonic representative for the family of all quasi-asymetric maps from a G to each D that shares the same boundary value at infinity okay so my goal in this talk is to extend this result by weakening both the assumptions on the spaces we work with and on the map so the theorems the following states that any course embedding between pinched Adam are manifolds is within a bounded distance from a unique harmonic map so what is a pinched Adam are manifold well first it is a Adam are manifold only it's a complete simply connected Romanian manifold and instead of only requiring that it says non-positive curvature I will ask that the curvature namely the sectional curvature is bounded between two negative constants now what is the course embedding the data map between two metric spaces is a course embedding when you can control the distance between the measure of two points in terms the distances of these spots both functions phi 1 and phi 2 that appear in these inequalities are required to be non-decreasing and unbounded okay so it's a bit that you for course embedding you can always assume that the function phi 2 here for the upper bound is an affine function but here you have much more flexibility so it makes quite as a metric maps special cases of course embeddings and I may say that our result is already interesting for a quasi-asometric maps but slightly more general okay we'd like to give examples application of a result so first of all now well I insist on the fact that in our result we do not assume the source and the target manifold to be the same and we do not even require them to have the same dimension so for example an instance of our results give the following any quasi-circle is the boundary value of a unique harmonic map so there are lots of equivalent definitions for quasi-circles which have many can have many different flavors but well the one which is easier for me to use today is to say that quasi- circle the map from s1 to s2 the circle to the two spheres that is the boundary value of a quasi-asometric map and then it is the boundary value of a unique harmonic map okay so another example it will still go from h2 to h3 then take h2 I select a geodesic here and I draw all the orthogonal geodesics this one that I just selected and then I will construct a course embedding of h2 in h3 so what I will do so I will take this which is like a vertical plane here and I will bend it so that this geodesic will go to a curve here and I send it with constant speed and then each one of the geodesics is sent to the geodesic vertical geodesic which is orthogonal here and if you do things properly the map from the geodesic to its image here will be a course embedding of r into h2 and then this construction will give you a course embedding from h2 to h3 and what our theorem states is that this course embedding is from a bounded distance from an harmonic map so before going into some proof so I would like to stay two earlier results the first one by Ponsu and it states that any quasi-asometric map either from the hyperbolic space over the field of quaternions into itself or from the Kelly plane over the octonions into itself is within a bounded distance from an isometry which is of course a special instance of a harmonic maps of this situation is more rigid than the other ones I'm usually dealing with and there is also a similar result due to Kleiner and Leib for maps between higher rank symmetric spaces and well of course what I should have said here is that the first example you should have in mind of a pinstead a Marmene fold is a symmetric space of one point one compact type so now what I want to do it to give you a first and tell you about the overall strategy of the proof and then I'm stuck so I asked before my talk how to get to black boss town and somebody told me there was something to ah you have to work for it yourself okay yeah thank you I was expecting something to hang from the okay so how do we prove this result so we start with this course embedding and we want to find a harmonic map which is within a bounded distance from F so what we do is to solve the family of bounded directly problems with boundary conditions F on a family of balls that exhaust the source manifold x and what we hope for actually what we prove is that the series of harmonic maps will converge to a harmonic map which is within a bounded distance from F so in order to make this strategy work the right move is to begin by smoothing F out that a target manifold is had a Marmene fold so you can use a center of mass procedure basically and then this built a smooth map which is within a bounded distance from F and which is smooth nope it's life you take the image of a small ball and you take a center of mass of the image and you do not I mean it's not a problem may assume that have you smooth with bounded derivatives those step two is just what I told you about you fix the point and origin X and for any radius you consider the harmonic map which is defined on this big ball with center O and reduce R to Y and which is solution of the Dirichlet problem with value F on the boundary so the existence uniqueness and regularity of the solution of this Dirichlet problem are granted due to results by Hamilton, Schoen and Ohlalmek. Now steps three, these steps it's actually the core of the proof and it's consistent in providing a uniform bound for the distance between these two maps the initial map the initial course embedding F and this harmonic map HR solution of the Dirichlet problem what I mean by uniform bound that is it does not depend on the radius R so once this step is completed we are good because you just have to use the standard compactness procedure to get the theorem so the main ingredient here in this result is the Chang Lema so what does it tell it tells us that in this setting when you have a bound on a harmonic map you have a bound on its differential so you know how F behaves it's a course embedding the HRs are within a bounded distance from F so you have a bound on the HR locally and on their differential a little elliptic regularity gives us bounds on higher derivatives and then you just use a school and this tells us that the the family of these maps HR converges or at least sub converges to a harmonic map which is within a bounded distance from F okay so this is a proof for existence and I don't want to go into the proof of uniqueness because I don't have time to so what I would like to do in the time that remains it to get into to give you a few details on how this proof of step 3 goes not so many details and so as I said we fix this point O and we consider this harmonic maps HR that coincide with F on the and we want a uniform bound for the distance between these two maps that I will denote by DR so what I will do is I will proceed by contradiction until in a sense we get an explicit bound and proceed by contradiction meaning that for some are we assume that this distance between these two maps is very large so this is a ball I'm starting with now this distance is rich at some point it's not and with some well of course it is not reached the boundary because on the boundary the two maps coincide okay but with some work it's not abuse but with some work we can see that when the distance between the two maps is large this distance is reached far away from the boundary so I don't say it's obvious you need to work the distance is reached far away from the boundary so that I can drew a large ball with a large reduced L that sits comfortably inside the domain of definition of HR and then I will completely forget about this ball and focus my attention on this one this is supposed to be the center and what I will do is study the images under basically the images under map F of just easy craze starting at this point okay so the key point that allows us to state our result not only for quasi isometric maps but also for course embeddings is the following the point is that we have information concerning the images of just easy craze and of course embedding and namely what we know that for almost all what we improve is that for almost all jdc craze with origin x0 its image goes linearly to infinity and we have a similar result for couple of jdc craze for almost all pair of jdc craze with origin x0 the images of course under F the course embedding pull away from each other linear well obviously I'm being really vague in my statements but okay and just to give you a flavor of the proofs it just use volume estimates and the Borel Cantelli lemma so that in this sense course embeddings share property that we know are true for quasi isometric maps okay so now using this results I will be able to choose wisely to jdc craze with origin x0 psi and ether and I will study the images under the map F so actually I will be only interested on this point itl and psi L where the jdc craze intersect the sphere here and what I will prove basically using this but not this we also the fact that if it's a course in billing once again now what I will prove that the angle seen from the image of this point under F between the images of these two points I will prove that this angle well perhaps is not large but it is bounded below by a positive constant that does not depend neither on the choice of L nor of the radius R and on the other hand I will prove that this angle has to be small and it will be my contradiction okay so how do we prove that this angle is smaller actually so we prove the same angle is small when L and R are too large how do we do this now actually I don't I mean whatever I don't care about the angle of these guys what I know why they don't want to enter common what I know is that I choose wisely these two geodesics and that the images of these two geodesics will pull away from each other so if L is large enough I mean it's not only a consequence of that effect that they pull away but how did you pull away step-by-step F being quite a course in bearing and what I what I am able to prove that if I choose these two geodesic rays wisely then the angle between their images will not be small you have to have my word on this I cannot give the proof okay but on the other hand I will prove that practically however I choose style and ethyl the angle will be small and this will be a contradiction but it should be small okay this makes sense and what really makes the thing work is this it's property that course embeddings share with crisis and isometric maps the point is that you don't have the information for all geodesic rays but only for almost all as it would be I mean for crisis so to prove this angle is small so I will do what I will be interested in I will take as an origin the geodesic ray that goes from the image of X not from by under F or under the harmonic map and I will introduce the image under the harmonic map of psi L as well as the image under the quasi-asometric map F of psi L I will do the same drawing with eta and then what I will prove is that this angle is small it would be the same with eta and so this would say that it's not like I have drawn on the on the blackboard of course but it would say that both F of psi L and F of eta L are within a small angle from this geodesic ray so the angle between them cannot be large okay so we have two different arguments one for this angle and one for this angle here and so how does it go well for the first one I will be interested in this geodesic segment and its image under the harmonic map it goes from the image of X not under HR to the image of psi L under HR and so we have a bound for HR because we know F is course embedding and we know the distance between HR and F so we have information we have a bound for this map and the Chang Le-Mat tells us that we have also a bound for its differential so we have a bound for the length of this curve on the other hand I state that this curve cannot go too close to this point why that it's because the map the function which is a distant function to a fixed point from an harmonic map is a subharmonic function so say this point had an image which was too close from this point you have a bound on the differential of the harmonic map so you know the map will would stay too close from this point on a neighborhood whose size you control and then you would control we'll have a contradiction using the subharmonicity of this function because the the value here has to be less or equal than the mean value on this circle here it is really too small here it's not too large so it will not compensate okay so this proves basically that if the distance between your two maps is too large then this angle is really small and for the other one I will do the last triangle I'm interested in on this other blackboard then you want to have an estimation for this angle though you will estimate the gram of product of this triangle so that it is large so this distance here is large because as I have said the image under F of a geodesic ray it goes away to infinity linearly this is the first statement on the table on the blackboard right there so this distance is greater than a constant time L now this distance here well it's certainly smaller than the distance between F and H this is HR sorry okay and well I'm chatting a little bit here I can choose psi and an eta so that this is larger than dr minus alpha L over 2 so why that it's basically the same argument as it was here using this homonicity of this function because well it this is well it's this okay same argument there and so this is larger than whatever L over 2 which is large if L is chosen large this means that the angle is small and then I have the contradiction that I predicted okay so I think it's time to question concerning the harmonic map that you find which is unique can you say something about the fact it's an embedding or no I don't think it is even in easy instances but one thing I should say that I've been cheating shamelessly here what I told you there is makes perfect sense when we're working from symmetric space to another one because when I say almost all everybody knows what I'm speaking of when you want to extend this to Pinstead and Marmin if all you have to know which respect to which measure you are working with and this will be harmonic measures and to do this you have to have estimates on the harmonic measures for cones any other questions the pond service are which you quoted at the beginning is that that's very special because of the sort of quaternions and octonions yeah so it's very cheap and so the harmonic map which is within bounded distance of the quasi asymmetry actually is the identity I mean that there are not so many quasi symmetric maps okay let's thank Dominic again