Compactness of conformal metric with a critical integrability assumption
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Carron, Gilles
This is a joint work with Clara Aldana (Luxembourg) and Samuel Tapie (Nantes). In 1993, M. Gursky has shown a compactness result in
 Thank you. Thank you for the introductions. Also, I want to thank the organizer of this meeting for giving me the honor to talk in this conference in honor of Marcel Berger. So Marcel Berger was my grandfather, my father was here, was Sylvain Gallaud. And as mathematically, I want to say, and I'm very... And I'm very proud to belong to the Marcel Berger schools because they have found the schools. He opened many doors in mathematics for us to enjoy our mathematics. So I'm very happy to be here. And also something that I want to mention. And in my mathematical education, I was impressed by the end of the book that Marcel brought with Berger Gostiot, with the book, Berger Gostiot on a curved surface. It contained two chapters that are devoid to the local and the global geometry of the surface. And it was a very nice survey of this geometry, both from the old results, classical result from surface, and also all the new areas that were open at those times. For instance, in this book, it was explained the result of the minimal volumes of R2, the result by Bavarin and Pierre-Ponsu. So it was really belonging... Geometry belongs to this long history. So it was very important for me to be sure that I want to do a geometry. Okay, so I'm going to talk about a joint project with Clara Aldana in Luxembourg. And my colleague, Samuel Tapie. So this is a very general question. Under which geometrical conditions do we get geometric compactness within a conformal class? So I'm... before, in 20 minutes, I will give the more precise questions that I'm not able to... we are not able to solve. But then before, I want to give some well-known results. So the first... these first questions arose in the questions of isospectral manifolds. So you consider a closed Romanian manifold. And so if f is a smooth function, I will denote gf is the conformal metric that is conformal to g0 with the factor exponential 2f. Okay, and if... this is the spectrum of the laparations associated to gf, then you get the following results. And if n is 2, this is due to Osgood, Philips and Sarnak. And if n is 3, so there are two groups of people. You have Brooks, Berry and Young. So I think this is in... and then you have Chang and Young in... okay, so they obtain the following results. If you take the set of f for all k integers, the eigenvalues, the case eigenvalues is equals to the k eigenvalues of g0. Okay, then this is compact. And this is not... we don't know if it is true in higher dimensions. Okay, in dimension 2, in fact, what could we get once we get this from the technique developed by Osgood, Philips and Sarnak, you get compactness of compact, and conformal class with a given spectrum. Okay, and then there is another result. If you take the following result, so assume you take p, a real number that is greater than n over 2, and you get v and lambda. So this is positive number. And you consider the set of functions such that the full Riemann curvature of gf to the power p is bounded by lambda, and the fact that the volumes say is fixed. Then what you obtain is that this set is precompact in c1 alpha for all alpha between 0 and 1. And there are examples of chain values. Not possible to remove the LP. Okay, so you can cook up examples where the conformal factor blew up, so go to 0. The exponential goes to plus infinity or minus infinity, where you keep the ln over 2, the full curvature tensor. Okay, then there are several... so this was in fact the starting point for the reflection. It was... this seems to be very strong, but on the other side you have these examples. And then we construct the literature, and there are other examples. So I want to say that it is... there is a result by Simon Brandl in 2003. So it is on the round sphere. Okay, so you take a real number, and you take the number f, where you get the following. So you assume that the q curvature, as a metric to the square, is bounded by e. And the fact that the full curvature is bigger than a quantity that is the one... So the q curvature is a curvature that generalizes the Gaussian curvature in higher dimensions that satisfies also some nice conformal transformations. Okay, and perhaps with volumes. Here it's 2. Yeah, it's power 2, yes. Okay, and the volume is the same. So you obtain that the set of f is bounded in hn. Okay, you get n derivative in L2. And then from sublif in bedding you get precompactness. Okay, and... okay, and... okay, and now there is also another story where this kind of compactness within a conformal class is important is about the Obain conjecture. Okay, it's about... you take the... Can you raise the middle board for me? Is this better? Yes. Thank you. Okay, you take the set of f, so that the scalar curvature of Gf is constant positive, and the volumes is one. Okay, the question was about the compactness of this set. Okay, and in fact the result is quite strange. So if n is below the compact, so this is a series of results. You have in lower dimension, this is due to several people depending on the dimensions. Okay, so I think this is hanign, xanign, zong, qirin, zong, qiri. Chun, this is compact and this is due to Brindel and then to Marques. And this is 2008. And this result was also in 2008. OK, and finally, the last result that I want to talk about, perhaps here. This is a result by Mathias Sehn in 2016, publishing a nice journals, the Annale de la Faculté des Sciences de Toulouse. And it's a very nice argument. So something is able to get the same conclusions as Gershky's papers, but with replacements. So he only assumes some things on the scalar curvature, but he adds some things about on the spectrums. The volumes of the Rohn's figure. So this is a normalization. OK, he assumes that the scalar curvature, again, p is larger than m over 2, and this is larger than 1. And he assumes the fact that the first eigenvalues of m is greater than n plus 1. So the first eigenvalue is not equals to the one of the sphere. Then your time is precompact in c1 alpha. So he's able to replace all the hypotheses on the full curvature tensor by only a spectral hypothesis. OK, so now if you go to the literature on the precompactness of Riemannian metric, there are a lot of other results. And then you can ask, if you make one of these hypotheses, what is the constraint you get on the conformal class? OK, so now you get if you get the set F, the set of. We've set bounded on the diameter and a lower bound on the Ritchie curvature. OK, this set is precompact in the Gromov of the d'Austopologie. OK, and then you have the works of Chigur, Codings, Chans, that describes this limit space. But it's for in general. So the question is, if you stay in a conformal class, did you get a better understanding or perhaps better results? OK, because here the precompactness is as a metric space. So it says that it converges to a metric space. It does not say that the limit is a Riemannian manifold or a manifold, even. And distance is not necessarily, even if it's a manifold, it's not necessarily related to the ground matrix. It's not necessarily conformal. OK, so I'm going to make a remark. OK, so volumes. This is given by this quantity. But now if you look at what is gradient for the metric, G naught. The distance function, so the distance, if you assume f is smooth, this also a bilib sheet. So this is differentiable almost everywhere. And in fact, what you get, you get that this is equals to almost everywhere. So this is a lip sheet function. But what you get for free is a kind of sub-of regularity from the volume estimates. OK. Volume implies W1n estimates on the distance. But it's very weak. And moreover, there is some troubles because this sub-olive space is precisely the one that do not embed in the continuous functions. It is the critical ones. But if you add a little bit more, you take a p here that is bigger than n, and you assume a fix, then you get W1p a priori results for the distance. And you get a C alpha estimate. And from this, you can extract subsequence that converge and give you convergence in the Gromov-Hosdorff sense. And the distance estimates. And the distance with respect to the originals, d0. And then from this, by Asculi, you can extract subsequence that converge to the distance and get something. So let me give you a guess. Shkari is not very geometrical. So the game is now how you can get these assumptions from geometry. And this is the first result. OK. So n is larger than 3. So you fix a compact closed manifolds with a grommetrix, and you look at conformal metrics. And the volumes is fixed. And moreover, the fact that the total, you take the scalar curvature of this metric, you only take its positive part to the critical exponent. And what you assume? You assume it's far from being the one of the sphere. So this is n, the one of the sphere minus delta. And delta is strictly positive. OK. So this is precompact of the chronov. So in fact, what we get? You get a distance converge. OK. So what we get? What is the limit space? So this is the metric you have. You have a pseudo distance. OK, and you identify points where the distance are 0. So drawbacks, they can have such a point. So the topology can change dramatically. So problems. OK. It makes it possible to build example where you collapse the hyper surface to a point. OK. And first, this is also possible that f, if you take a sequence, it can be unbounded. And there are another problem. The limit could be not conformal to j0. Because, a priori, you say, well, perhaps the function converge in some weak sense. You are conformal to a strange factor that oscillate, can vanish somewhere. But this is not conformal to j0. And even you have example, and even not remanion. Because I'm going to give an example, the well-known example that gives you where this picture occurs. And so for the moment we are not, then I will give another result that is kind of reinforcement of this one, where we remove this condition. So the topology then change. You have really a distance on the manifold end. But let me give an example to illustrate the difficulty. So it is not an example where you have these conditions. But it is an example where you have compactness, and you understand what is the limit. OK, so on the torus, you take the following matrix. Take a matrix that is conformal. So the flat matrix. So it's very nice. The associate distance is bounded from the Euclidean matrix, or the flat matrix. It's easy to find this one. This conformal factor is bounded between some constant and another constant. So it really bilips sheets at every point. And what happens is that. So this is a topology. This is a to a finseler metric. And the finseler metric is associated to the stable norms of the periodic on Rn, that is given by. This is the periodic matrix. You take at large distance the geometry look like a finseler metric. So the geodesic distance for this metric at large scales is at bound distance, if you want, from the finseler norms. And finseler norm is the one that you will get by scaling on these manifolds. So at the end, you have a sequence of smooth metrics that are all conformal to a given one. They are all bilips sheets. And it converts to the grommet-fossed-dorf to a finseler metric. So it's not conformal. It's not Riemannian. So this is a phenomenon that for the moment, we are not able to rule out by geometry hypothesis. OK, so what we find quite convenient to circumvent the first problems is a concept that came from harmonic analysis. And that is due to Rn to David and Sims, that we can easily adapt on any metric space. And this is a concept of strong r-infinity weighted. OK, so this has come from. Since in 82. So in this context, what does it mean? It means that, in fact, it's something on the measure to be a strong r-infinity with constant d. If you get two conditions, it's first that this measure is doubling with respect to the ball of your original metrics. It means that the volumes, you take the ball for the metric G0 is controlled by B times the volumes of the metric. OK, so the diameter for G0, you get this. So the volume of ball, so it's something that compares the geometry for GF from the geometry for G0. But it only compares, in fact, it's a geometry that is associated to the volumes, in fact. So this is the first conditions. And the second condition is the fact that, so I will explain in a minute what is this. This is uniformly comparable to the distance for F. So if you get two points on your manifold, what you can take is the ball center at x and of radius, the distance. So B x, y is the geodesic ball for the metric D0 center at x. So in fact, by the doubling hypothesis, you can put either here, it's not symmetric in x and y. But the ball center at x and y with this is at the same volumes up to a fixed scale. So this is these two conditions. And so what David seems to show is if you have such a volume forms on RN that satisfies all these vertices, that means it shows that then you have all very nice properties. You have the conformal metric that satisfies the Poincare inequalities. You have the volume growth, and also you get the isoparametric inequalities, the uniform isoparametric inequalities from these assumptions for the conformal metrics. OK, and what it is helpful for is that if you get this, it is classical to get an improvement. That is, from the volumes, you get these higher integrability. For some p, that is bigger than n, and p, it depends on g0 and on b. So from this, as soon as you get this, you get Grammophos-Dorff convergence. But also because you have these two sides, estimates, you don't have collapsing. So it truly converge toward a distance that is, in fact, a builder equivalence to the ground, the given one. So from this hypothesis. So now the question is, under which geometrical hypothesis you are able to show that a conformal metric satisfies these kind of strong infinity conditions. So this is all terms. Again, with n, g0. Then a constant, epsilon, that depends only on. We can give an explicit description of these numbers in terms of a lower bound on the Ricci curvature for g0, volume of g0, and the diameter of g0. So something that came from estimates, sub-oref inequalities, and so on, that you can get under some geometrical hypothesis on g0. So we can make it explicit. So we have two metrics. We have first same volumes. And the fact that we do not have concentrations in ln over two norms for the scalar curvature. So you take a r0 positive, and the fact that's. So this is for the metric g0, the walls. You do not get concentrations in the ln over two norms for the scalar curvature. Once you get that dfk, d infinity, and more over, is the original metrics. Yes? Yeah, it's epsilon. Excuse me. I'm sorry, the rate of universal constant or. No, it's not universal. It depends on g0. So it depends on the lower bound on Ricci, upper bound on diameter, and lower bound on volumes. V is the number. So it assumes that's all you have a fixed V. You have sequence of conformal metrics with fixed volumes, and scalar curvature do not concentrate too much in ln over two. OK, so the proof is to show that, in fact, such metrics satisfy those conditions. OK, so before finishing, I'm going to give only two words at the proof how we get these results. OK, so the first steps is analytic steps, is to have a better understanding of the conformal factor. That f is u plus v. OK, it can be decomposed, the conformal factor, the metric that satisfies these conditions here, where the first one is bounded in w2n over 2. That is to say u has two derivatives in ln over 2. It means that if it was true for f, it means that f has certain curvature that are bound in ln over 2. So it is a part, OK, where you improve the scalar curvature estimate to estimate on the full curvature tensor. But then you have some dust V, and V is not, is older continuous. OK, so this is bad because this u can be unbounded, but this one is quite nice, it's only older. And everything can be quantized. So you have bound on the wn2n over 2 norms of u, and you get an estimate on the older continuity exponent for V and the older norms of V. OK, and from this, so this is the odd part, and then we adapt, it's not the difficulty, the result of bonk. What they show, they show that's on ln. If you take ln, the big measure is a strong infinity weight when you get such a bound on ln, then you get the strong infinity conditions. And then from this, we know that it's older, so it's bounded. So you get two metric of Billy sheets, and then you use the adaptation of these results to get the b-older equivalence between the two distance. OK, so I want to stop here. Thank you. Yeah, it's only to, so the air is to say that at this scale, so air is fixed. But it doesn't matter what it is. Not matter what it is. OK. But it's fixed within the sequence. Yeah, OK. Any other questions? OK, if not, thank you very much. Thank you.
 Thank you. Thank you for the introductions. Also, I want to thank the organizer of this  meeting for giving me the honor to talk in this conference in honor of Marcel Berger.  So Marcel Berger was my grandfather, my father was here, was Sylvain Gallaud. And as mathematically,  I want to say, and I'm very... And I'm very proud to belong to the Marcel Berger schools  because they have found the schools. He opened many doors in mathematics for us to enjoy our  mathematics. So I'm very happy to be here. And also something that I want to mention. And in my  mathematical education, I was impressed by the end of the book that Marcel brought with Berger  Gostiot, with the book, Berger Gostiot on a curved surface. It contained two chapters that are  devoid to the local and the global geometry of the surface. And it was a very nice survey of  this geometry, both from the old results, classical result from surface, and also all the new areas  that were open at those times. For instance, in this book, it was explained the result of the  minimal volumes of R2, the result by Bavarin and Pierre-Ponsu. So it was really belonging... Geometry  belongs to this long history. So it was very important for me to be sure that I want to do a geometry.  Okay, so I'm going to talk about a joint project with Clara Aldana in Luxembourg.  And my colleague, Samuel Tapie. So this is a very general question.  Under which geometrical conditions do we get geometric compactness within a conformal class?  So I'm... before, in 20 minutes, I will give the more precise questions that I'm not able to...  we are not able to solve. But then before, I want to give some well-known results. So the first...  these first questions arose in the questions of isospectral manifolds. So you consider a closed  Romanian manifold. And so if f is a smooth function, I will denote gf is the conformal  metric that is conformal to g0 with the factor exponential 2f. Okay, and if...  this is the spectrum of the laparations associated to gf, then you get the following results.  And if n is 2, this is due to Osgood, Philips and Sarnak. And if n is 3, so there are two  groups of people. You have Brooks, Berry and Young. So I think this is in... and then you  have Chang and Young in... okay, so they obtain the following results. If you take the set of f  for all k integers, the eigenvalues, the case eigenvalues is equals to the k eigenvalues of  g0. Okay, then this is compact. And this is not... we don't know if it is true in higher  dimensions. Okay, in dimension 2, in fact, what could we get once we get this from the  technique developed by Osgood, Philips and Sarnak, you get compactness of compact,  and conformal class with a given spectrum. Okay, and then there is another result. If  you take the following result, so assume you take p, a real number that is greater than  n over 2, and you get v and lambda. So this is positive number. And you consider the set  of functions such that the full Riemann curvature of gf to the power p is bounded by lambda,  and the fact that the volumes say is fixed. Then what you obtain is that this set is  precompact in c1 alpha for all alpha between 0 and 1. And there are examples of chain  values. Not possible to remove the LP. Okay, so you can cook up examples where the conformal  factor blew up, so go to 0. The exponential goes to plus infinity or minus infinity, where  you keep the ln over 2, the full curvature tensor. Okay, then there are several... so  this was in fact the starting point for the reflection. It was... this seems to be very  strong, but on the other side you have these examples. And then we construct the literature,  and there are other examples. So I want to say that it is... there is a result by Simon  Brandl in 2003. So it is on the round sphere. Okay, so you take a real number, and you take  the number f, where you get the following. So you assume that the q curvature, as a metric  to the square, is bounded by e. And the fact that the full curvature is bigger than a quantity  that is the one... So the q curvature is a curvature that generalizes the Gaussian curvature  in higher dimensions that satisfies also some nice conformal transformations. Okay, and  perhaps with volumes. Here it's 2. Yeah, it's power 2, yes. Okay, and the volume is the  same. So you obtain that the set of f is bounded in hn. Okay, you get n derivative in L2.  And then from sublif in bedding you get precompactness. Okay, and... okay, and... okay, and now there  is also another story where this kind of compactness within a conformal class is important is about  the Obain conjecture. Okay, it's about... you take the...  Can you raise the middle board for me? Is this better? Yes. Thank you. Okay, you take  the set of f, so that the scalar curvature of Gf is constant positive, and the volumes  is one. Okay, the question was about the compactness of this set. Okay, and in fact the result  is quite strange. So if n is below the compact, so this is a series of results. You have in  lower dimension, this is due to several people depending on the dimensions. Okay, so I think  this is hanign, xanign, zong, qirin, zong, qiri. Chun, this is  compact and this is due to Brindel and then to Marques.  And this is 2008.  And this result was also in 2008.  OK, and finally, the last result that I want to talk about,  perhaps here.  This is a result by Mathias Sehn in 2016,  publishing a nice journals, the Annale de la Faculté des Sciences  de Toulouse.  And it's a very nice argument.  So something is able to get the same conclusions  as Gershky's papers, but with replacements.  So he only assumes some things on the scalar curvature,  but he adds some things about on the spectrums.  The volumes of the Rohn's figure.  So this is a normalization.  OK, he assumes that the scalar curvature, again, p is larger  than m over 2, and this is larger than 1.  And he assumes the fact that the first eigenvalues of m  is greater than n plus 1.  So the first eigenvalue is not equals to the one of the sphere.  Then your time is precompact in c1 alpha.  So he's able to replace all the hypotheses  on the full curvature tensor by only a spectral hypothesis.  OK, so now if you go to the literature  on the precompactness of Riemannian metric,  there are a lot of other results.  And then you can ask, if you make one of these hypotheses,  what is the constraint you get on the conformal class?  OK, so now you get if you get the set F, the set of.  We've set bounded on the diameter and a lower bound  on the Ritchie curvature.  OK, this set is precompact in the Gromov of the d'Austopologie.  OK, and then you have the works of Chigur, Codings, Chans,  that describes this limit space.  But it's for in general.  So the question is, if you stay in a conformal class,  did you get a better understanding  or perhaps better results?  OK, because here the precompactness  is as a metric space.  So it says that it converges to a metric space.  It does not say that the limit is a Riemannian manifold  or a manifold, even.  And distance is not necessarily, even if it's a manifold,  it's not necessarily related to the ground matrix.  It's not necessarily conformal.  OK, so I'm going to make a remark.  OK, so volumes.  This is given by this quantity.  But now if you look at what is gradient for the metric,  G naught.  The distance function, so the distance,  if you assume f is smooth, this also a bilib sheet.  So this is differentiable almost everywhere.  And in fact, what you get, you get that this is equals to  almost everywhere.  So this is a lip sheet function.  But what you get for free is a kind of sub-of regularity  from the volume estimates.  OK.  Volume implies W1n estimates on the distance.  But it's very weak.  And moreover, there is some troubles  because this sub-olive space is precisely the one that do not  embed in the continuous functions.  It is the critical ones.  But if you add a little bit more, you take a p here that  is bigger than n, and you assume a fix, then you get W1p  a priori results for the distance.  And you get a C alpha estimate.  And from this, you can extract subsequence that converge  and give you convergence in the Gromov-Hosdorff sense.  And the distance estimates.  And the distance with respect to the originals, d0.  And then from this, by Asculi, you can extract subsequence  that converge to the distance and get something.  So let me give you a guess.  Shkari is not very geometrical.  So the game is now how you can get these assumptions  from geometry.  And this is the first result.  OK.  So n is larger than 3.  So you fix a compact closed manifolds with a grommetrix,  and you look at conformal metrics.  And the volumes is fixed.  And moreover, the fact that the total,  you take the scalar curvature of this metric,  you only take its positive part to the critical exponent.  And what you assume?  You assume it's far from being the one of the sphere.  So this is n, the one of the sphere minus delta.  And delta is strictly positive.  OK.  So this is precompact of the chronov.  So in fact, what we get?  You get a distance converge.  OK.  So what we get? What is the limit space?  So this is the metric you have.  You have a pseudo distance.  OK, and you identify points where the distance are 0.  So drawbacks, they can have such a point.  So the topology can change dramatically.  So problems.  OK.  It makes it possible to build example where you collapse  the hyper surface to a point.  OK.  And first, this is also possible that f, if you take a  sequence, it can be unbounded.  And there are another problem.  The limit could be not conformal to j0.  Because, a priori, you say, well, perhaps the function  converge in some weak sense.  You are conformal to a strange factor that oscillate, can  vanish somewhere.  But this is not conformal to j0.  And even you have example, and even not remanion.  Because I'm going to give an example, the well-known  example that gives you where this picture occurs.  And so for the moment we are not, then I will give another  result that is kind of reinforcement of this one,  where we remove this condition.  So the topology then change.  You have really a distance on the manifold end.  But let me give an example to illustrate the difficulty.  So it is not an example where you have these conditions.  But it is an example where you have compactness, and you  understand what is the limit.  OK, so on the torus, you take the following matrix.  Take a matrix that is conformal.  So the flat matrix.  So it's very nice.  The associate distance is bounded from the Euclidean  matrix, or the flat matrix.  It's easy to find this one.  This conformal factor is bounded between some constant  and another constant.  So it really bilips sheets at every point.  And what happens is that.  So this is a topology.  This is a to a finseler metric.  And the finseler metric is associated  to the stable norms of the periodic on Rn,  that is given by.  This is the periodic matrix.  You take at large distance the geometry  look like a finseler metric.  So the geodesic distance for this metric at large scales  is at bound distance, if you want, from the finseler norms.  And finseler norm is the one that you will get by scaling  on these manifolds.  So at the end, you have a sequence of smooth metrics  that are all conformal to a given one.  They are all bilips sheets.  And it converts to the grommet-fossed-dorf to a finseler  metric.  So it's not conformal.  It's not Riemannian.  So this is a phenomenon that for the moment,  we are not able to rule out by geometry hypothesis.  OK, so what we find quite convenient  to circumvent the first problems is a concept that  came from harmonic analysis.  And that is due to Rn to David and Sims,  that we can easily adapt on any metric space.  And this is a concept of strong r-infinity weighted.  OK, so this has come from.  Since in 82.  So in this context, what does it mean?  It means that, in fact, it's something on the measure  to be a strong r-infinity with constant d.  If you get two conditions, it's first  that this measure is doubling with respect  to the ball of your original metrics.  It means that the volumes, you take  the ball for the metric G0 is controlled  by B times the volumes of the metric.  OK, so the diameter for G0, you get this.  So the volume of ball, so it's something  that compares the geometry for GF from the geometry for G0.  But it only compares, in fact, it's  a geometry that is associated to the volumes, in fact.  So this is the first conditions.  And the second condition is the fact that,  so I will explain in a minute what is this.  This is uniformly comparable to the distance for F.  So if you get two points on your manifold,  what you can take is the ball center at x and of radius,  the distance.  So B x, y is the geodesic ball for the metric D0 center  at x.  So in fact, by the doubling hypothesis,  you can put either here, it's not symmetric in x and y.  But the ball center at x and y with this  is at the same volumes up to a fixed scale.  So this is these two conditions.  And so what David seems to show is  if you have such a volume forms on RN that satisfies  all these vertices, that means it  shows that then you have all very nice properties.  You have the conformal metric that  satisfies the Poincare inequalities.  You have the volume growth, and also you get the isoparametric  inequalities, the uniform isoparametric inequalities  from these assumptions for the conformal metrics.  OK, and what it is helpful for is that if you get this,  it is classical to get an improvement.  That is, from the volumes, you get these higher  integrability.  For some p, that is bigger than n, and p,  it depends on g0 and on b.  So from this, as soon as you get this,  you get Grammophos-Dorff convergence.  But also because you have these two sides,  estimates, you don't have collapsing.  So it truly converge toward a distance that is, in fact,  a builder equivalence to the ground, the given one.  So from this hypothesis.  So now the question is, under which geometrical hypothesis  you are able to show that a conformal metric satisfies  these kind of strong infinity conditions.  So this is all terms.  Again, with n, g0.  Then a constant, epsilon, that depends only on.  We can give an explicit description of these numbers  in terms of a lower bound on the Ricci  curvature for g0, volume of g0, and the diameter of g0.  So something that came from estimates,  sub-oref inequalities, and so on,  that you can get under some geometrical hypothesis on g0.  So we can make it explicit.  So we have two metrics.  We have first same volumes.  And the fact that we do not have concentrations in ln  over two norms for the scalar curvature.  So you take a r0 positive, and the fact that's.  So this is for the metric g0, the walls.  You do not get concentrations in the ln over two norms  for the scalar curvature.  Once you get that dfk, d infinity, and more  over, is the original metrics.  Yes?  Yeah, it's epsilon.  Excuse me.  I'm sorry, the rate of universal constant or.  No, it's not universal.  It depends on g0.  So it depends on the lower bound on Ricci,  upper bound on diameter, and lower bound on volumes.  V is the number.  So it assumes that's all you have a fixed V.  You have sequence of conformal metrics with fixed volumes,  and scalar curvature do not concentrate too much in ln over two.  OK, so the proof is to show that, in fact, such metrics  satisfy those conditions.  OK, so before finishing, I'm going to give only two words  at the proof how we get these results.  OK, so the first steps is analytic steps,  is to have a better understanding of the conformal factor.  That f is u plus v. OK, it can be decomposed,  the conformal factor, the metric that satisfies these conditions  here, where the first one is bounded in w2n over 2.  That is to say u has two derivatives in ln over 2.  It means that if it was true for f,  it means that f has certain curvature that are bound in ln over 2.  So it is a part, OK, where you improve the scalar curvature  estimate to estimate on the full curvature tensor.  But then you have some dust V, and V is not, is older continuous.  OK, so this is bad because this u can be unbounded,  but this one is quite nice, it's only older.  And everything can be quantized.  So you have bound on the wn2n over 2 norms of u,  and you get an estimate on the older continuity  exponent for V and the older norms of V.  OK, and from this, so this is the odd part,  and then we adapt, it's not the difficulty,  the result of bonk.  What they show, they show that's on ln.  If you take ln, the big measure is a strong  infinity weight when you get such a bound on ln,  then you get the strong infinity conditions.  And then from this, we know that it's older, so it's bounded.  So you get two metric of Billy sheets,  and then you use the adaptation of these results  to get the b-older equivalence between the two distance.  OK, so I want to stop here.  Thank you.  Yeah, it's only to, so the air is to say that at this scale,  so air is fixed.  But it doesn't matter what it is.  Not matter what it is.  OK.  But it's fixed within the sequence.  Yeah, OK.  Any other questions?  OK, if not, thank you very much.  Thank you.
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