Harmonic coarse embeddings
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8
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Arithmetic meanHyperbolaSpacetimeHarmonic functionBoundary value problemScaling (geometry)TheoremState of matterDistanceInfinityKritischer Punkt <Mathematik>ResultantPlanar graphAmsterdam Ordnance DatumLemma (mathematics)Point (geometry)Hyperbolischer RaumMoving averageSymmetric matrixFamilyGroup representationEinbettung <Mathematik>INTEGRALIsometrie <Mathematik>Glattheit <Mathematik>Musical ensembleFunctional (mathematics)Multiplication signPlane (geometry)Evelyn PinchingElliptischer PseudodifferentialoperatorFood energyHadamard manifoldLecture/Conference
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Harmonic functionCurvatureCircleEinbettung <Mathematik>Directed graphLogical constantFunctional (mathematics)Cartesian coordinate systemGeodesicInequality (mathematics)ResultantDependent and independent variablesDistanceSymmetric matrixCurveDimensional analysisConnected spaceTerm (mathematics)ManifoldEquivalence relationPoint (geometry)Maß <Mathematik>Boundary value problemState of matterPlane (geometry)TheoremRiemannian geometryMetrischer RaumHadamard matrixVertex (graph theory)Evelyn PinchingLecture/Conference
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Harmonic functionDistanceManifoldPoint (geometry)Isometrie <Mathematik>Symmetric spaceMassEinbettung <Mathematik>Series (mathematics)Proof theoryResultantGlattheit <Mathematik>Körper <Algebra>Plane (geometry)Directed graphMatching (graph theory)Strategy gameBoundary value problemBound stateHyperbolischer RaumEvelyn PinchingFamily1 (number)Order (biology)Hadamard matrixRadiusQuaternionDerivation (linguistics)Lecture/Conference
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DistanceEinbettung <Mathematik>Volume (thermodynamics)Point (geometry)LinearizationQuantum stateEstimatorTime domainResultantRadiusGeodesicBoundary value problemArithmetic meanBound stateCategory of beingLimit (category theory)Proof theoryInfinityReduction of orderLemma (mathematics)TendonIsometrie <Mathematik>Lecture/Conference
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AngleSphereMathematical singularityAxiom of choiceEinbettung <Mathematik>Point (geometry)GeodesicRadiusQuantum stateLecture/Conference
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AngleSymmetric spaceDifferential (mechanical device)Cone penetration testEstimatorInfinityCategory of beingTriangleQuaternionDistanceProduct (business)Harmonic functionEinbettung <Mathematik>MeasurementIsometrie <Mathematik>Multiplication signNeighbourhood (graph theory)CircleFunctional (mathematics)Alpha (investment)Point (geometry)Parameter (computer programming)Right angleQuantum stateLogical constantCurveLengthLinearizationGraph (mathematics)Symmetric matrixMany-sorted logicHarmonic analysisSign (mathematics)Lemma (mathematics)Table (information)Numerical analysisLecture/Conference
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Identical particlesIsometrie <Mathematik>Musical ensembleLecture/ConferenceMeeting/Interview
Transcript: English(auto-generated)
00:16
reporting our joint work with Yves Benoit. At least in my eyes, this work is related
00:28
to Marcel in a very special way. When we first started working on this subject, we were both staying in Berkeley, and it was the last time, actually, I had a contact
00:44
with Marcel. The setting off was to tell him about this work. And on this occasion, he told me his memories about his own stay in Berkeley in the early 60s, that had been one of the turning points of his career, as well as a magical adventure for his whole family.
01:05
So this talk is an outgrowth of the Schoen conjecture. It states that any quasi-isometric
01:29
map from a hyperbolic plane into itself is within bounded distance from a unique harmonic
02:00
map. So this conjecture has been proved true quite recently by Markovitch, and then further
02:23
extended in a joint work with Marcus Lem to quasi-isometric maps from the hyperbolic space into itself. So what is a quasi-isometric map? Well, a map is quasi-isometric whenever
02:57
it is biolipchitz at large scale, meaning that the distance between the images of two
03:06
points is bounded above and below by affine functions of the distances between these two points. You can take the same sea. That's not a problem. So what is a harmonic
03:27
map? A harmonic map is a critical point for the Dirichlet energy integral, and as
03:58
a solution of a certain elliptic PDE. So now we know that quasi-isometric maps from let's
04:11
say nice grama phi public spaces to another grama phi public space do have a boundary value at infinity, and that to such maps shares the same boundary value whenever they
04:24
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
04:43
all quasi-isometric maps from h e to h d that share the same boundary value at infinity. So my goal in this talk is to extend this result by weakening both the assumptions on
05:02
the spaces we work with and on the map. So the theorem is defined, states that any
05:30
coarse embedding between pinched Hadamard manifolds is within a bounded distance from
06:08
a unique harmonic map. So what is a pinched Hadamard manifold? Well, first it is a Hadamard
06:26
manifold, namely it's a complete simply connected Riemannian manifold, and instead
06:44
of only requiring that it has non-positive curvature, I will ask that the curvature, namely the sixinar curvature, is bounded between two negative constants.
07:01
Now what is a coarse embedding? Say that a map between two metric spaces is a coarse embedding when you can control the distance between the image of two points in terms
07:33
of distances of these points. Both functions, phi 1 and phi 2, that appear in these inequalities
07:45
are required to be non-decreasing and unbounded. So it's obvious that for coarse embedding you can always assume that the function phi 2 here for the upper bound is an affine function,
08:05
but here you have much more flexibility. So it makes quasi-isometric maps, special cases of coarse embeddings, and I may say that our result is already interesting for quasi-isometric
08:22
maps, but slightly more general. Okay, so I would like to give examples, application
08:53
of our result. So first of all, while I insist on the fact that in our result we do not
09:02
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 gives the following. An equisicircle is the boundary value of any harmonic map. So there are lots
09:57
of equivalent definitions for quasi-circles, which can have many different flavors, but
10:05
well, the one which is easier for me to use today is to say that a quasi-circle, the map from S1 to S2, the circle to the 2-sphere, that is the boundary value of a quasi-isometric map, and then it is the boundary value of a unique harmonic map. Okay, so another
10:23
example, it will still go from H2 to H3, then take H2, I select the geodesic here, and I draw all the orthogonal geodesics, this one that I just selected, and then I will
10:53
construct a coarse embedding of H2 in H3. So what I will do, so I will take this,
11:03
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 a constant speed, and then each one of the geodesics is sent
11:20
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 coarse embedding of R into H2,
11:41
and then this construction will give you a coarse embedding from H2 to H3, and what our theorem states is that this coarse embedding is from a bounded distance from a harmonic map. So before going into some proofs, I would like to stay to earlier
12:05
results. The first one by Penseu, and it states that any quasi-isometric map,
12:24
either from the hyperbolic space over the field of quaternions into itself, or from the Cayley plane over the octonions into itself, is within a bounded distance from
12:49
an isometry, which is of course a special instance of a harmonic map, so this situation is more rigid than the other ones I'm usually dealing with, and there is also a similar
13:04
result due to Kleiner and Leib for maps between higher-rank symmetric spaces.
13:23
Well of course, what I should have said here is that the first example you should have in mind of a Pinstad, a Marmene default, is a symmetric space of one quan, one compact type.
13:42
Okay, so now what I want to do is to give you 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 Blackboard's town,
14:11
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 Blackboard.
14:52
So how do we prove this result? So we start with this coarse embedding,
15:08
and we want to find a harmonic map which is within a bounded distance from f. So what we do is to solve a family of bounded Dirichlet problems with boundary conditions f, on a family
15:26
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
15:49
by smoothing f out. So the target manifold is a Hadamard manifold, so you can use a
16:08
center of mass procedure, basically. And then this built smooth map, which is within a bounded distance from f, and which is smooth, it's life. You take the image of
16:32
a small ball, and you take a center of mass of the image, and you do not, I mean, it's not a problem. You may assume that f is smooth with bounded derivatives.
16:59
So step two is just what I told you about. You fix the point and origin in x,
17:09
and for any radius, you consider the harmonic map, which is defined on this big ball with
17:26
center o and radius r, and which is solution of the Dirichlet problem with value f on the
17:41
boundary. So the existence, uniqueness, and regularity of the solution of this Dirichlet problem are granted due to results by Hamilton, Chen, and Olavnek.
18:12
Now, step three. This step is actually the core of the proof, and it consists in providing
18:27
a uniform bound for the distance between these two maps, the initial map, the initial
18:47
solution of the Dirichlet problem. So what I mean by uniform bound, that is, it does
19:03
not depend on the radius r, of course. So once this step is completed, we are good,
19:39
because you just have to use the standard compactness procedure to get the theorem.
19:54
So the main ingredient here in this result is the Chang lemma. So what does it tell?
20:06
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 coarse embedding.
20:44
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 sculley. And this tells us that the family of these maps HR
21:10
converges, or at least subconverges, to a harmonic map which is within a bounded distance from f. So this is a proof for existence, and I don't want to go into the
21:32
So what I would like to do in the time that remains is to give you a few details on how
21:41
this proof of step 3 goes, not so many details.
22:56
So as I said, we fix this point O, and we consider these harmonic maps HR that coincide
23:19
with f on the bound available. And we want a uniform bound for the distance between these
23:36
two maps that I will denote by dr.
23:43
So what I will do is I will proceed by contradiction. Actually, in a sense, we get an explicit bound. And proceed by contradiction, meaning that for some r, we assume that
24:12
this distance between these two maps is very large. So this is a ball I'm starting with.
24:35
Now this distance is reached at some point, x0. And with some, well, of course, it does
24:45
not reach the boundary, because on the boundary, the two maps coincide. OK, but with some work, it's not obvious, 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
25:01
boundary. So I don't say it's obvious. You need to work. The distance is reached
25:20
far away from the boundary, so that I can draw a large ball with a large radius L
25:45
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
26:03
one. This is supposed to be the center. And what I will do is study the images under map f of geodesic rays starting at this point.
26:22
OK, so the key point that allows us to state our result not only for quasi-isometric maps, but also for cross embeddings is the following. The point is that we have information concerning
26:59
the images of geodesic rays under Koff's embedding, and namely what we know, that
27:06
for almost all, what we improve is that for almost all geodesic rays with origin x0,
27:25
its image goes linearly to infinity. And we have a similar result for a couple of geodesic rays, for almost all pairs of geodesic rays with origin x0.
27:54
The images, of course, under f, the coarse embedding, pull away from each other linearly.
28:25
Well, obviously, I'm being really vague in my statements, but OK. And just to give you a flavor of the proofs, it just use volume estimates and the Borel-Kantelli
28:41
lemma, so that in this sense, coarse embeddings share property that we know are true for quasi-isometric maps. OK, so now using these results, I will be able to choose wisely
29:12
two geodesic rays with origin x0, psi and eta, and I will study their images
29:45
under the map f. So actually, I will be only interested on this point, eta l and
30:16
psi l, where these geodesic rays intersect the sphere here.
30:24
And what I will prove, basically using this, but not only this, also the fact that f is a coarse embedding, once again, what I will prove is that the angle seen from the image of this point, under f, between the images of these two points, what I will
31:01
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.
31:21
And on the other hand, I will prove that this angle has to be small, and it will be my contradiction. So how do we prove that this angle is small? Actually, so we prove
32:45
the same angle is small when l and r are too large. How do we do this? Now actually,
33:04
I mean, whatever, I don't care about the angle of these guys. What I know is that
33:24
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 the fact that they pull away, but how did you pull away step by step, f being a coarse
33:43
embedding. And what I am able to prove is that if I choose these two geodesics 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
34:04
practically, however I choose sial and ethial, the angle will be small, and this will be a contradiction. But it should be small. Okay? This makes sense? And what really makes this thing work is this. This property that coarse embeddings
34:29
share with quiescent 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 coarse
34:41
asymmetric maps. So to prove this angle is small, I will draw what I will be interested in. I will take as an origin the geodesic ray that
35:02
goes from the image of x0 from under f or under the harmonic map, and I will introduce the image under the harmonic map of sial, as well as the image under the quiescent asymmetric map f of xial. I will do the same drawing with eta. And what I will prove
35:26
is that this angle is small. It would be the same with eta. And so this would say that, it's not like I've drawn on the blackboard, of course, but it would say
35:41
that both f of sial and eta 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
36:00
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
36:30
harmonic map. It goes from the image of x0 under HR to the image of sial under HR. And so we have a bound for HR because we know f is coarse embedding and we know the
36:46
distance between HR and f. So we have information, we have a bound for this map. And the Chang'e Lemma 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
37:06
cannot go too close to this point. Why that? It's because the map, the function,
37:22
which is a distant function to a fixed point from an harmonic map, is a sub-harmonic function. So if, say, this point had an image which was too close from this point, you have
37:45
a bound on the differential of the harmonic map. So you know the map would stay too close from this point on a neighborhood whose size you control. And then you would have a contradiction
38:02
using the sub-harmonicity of this function because 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. So this proves basically that if the distance
38:22
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
38:51
blackboard. Then you want to have an estimation for this angle. So you will estimate the
39:09
gram of product of this triangle and show that it is large.
39:28
So this distance here is large because, as I have said, the image under f of a geodesic
39:40
ray, it goes away to infinity linearly. This is the first statement 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.
40:02
This is hr, sorry. And I'm cheating a little bit here. I can choose psi and eta so that this is larger than dr minus alpha l over 2. So why that? It's basically the same
40:20
argument as it was here using the sub-harmonicity of this function because, well, it's the same argument there. And so this is larger than whatever alpha l over 2, which is large if
40:41
l is too large. This means that the angle is small. And then I have the contradiction that I predicted. So I think it's time to stop.
41:18
I have a question concerning the harmonic map that you find, which is unique. Can you
41:22
say something about the fact it's an embedding? No. I don't think it is. Even in easy instances.
41:40
First, one thing I should say that I've been cheating shamelessly here. What I told you there makes perfect sense when we're working from a symmetric space to another one. Because when I say almost all, everybody knows what I'm speaking of.
42:04
When you want to extend this to Pinsted and Marmenifold, 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.
42:26
Any other questions? The pons sous-vissat which you quoted at the beginning, that's very special because of the quaternions and octonions. Yes, it's very rigid. So the harmonic map, which is within bounded distance of the quasi-isometry
42:44
actually is the identity. I mean that there are not so many quasi-isometric maps. OK, let's thank Dominique again for a clear talk.