 So thank you, it's a pleasure to be talking here today. So I'm going to talk about the renormalized volume of quasi-flexion manifolds, and especially the connection between the renormalized volume and the volume of the convex core, and data, you know, connected to the renormalized volume that's leave at infinity of quasi-flexion manifolds, connect in the way they are connected to data that live on the boundary of the convex core. And the reason I'm talking about this is that, I think there are a number of questions that are still, you know, to be answered in this area. So I should, you know, start by recording some very basic things. I'm going to talk about quasi-flexion manifolds. So the definition is there. A quasi-flexion manifold, you know, it's a manifold which is a hyperbolic, a complete hyperbolic manifold, which is homeomorphic to a surface cross R, where S is a closed oriented surface of genus at least two. And we want this manifold to contain a compact convex and non-empty subset. And this means we are looking at something that looks like this. It's a manifold that has this structure of a product. And, you know, it goes to infinity on both sides. And it contains this compact convex non-empty subset. And here, you know, again, we have this surface. I mean, it's a basically simple simplified picture, but that's the idea. So when we are in this situation, there is a number of things that we know. So what is that? Because it's a complete hyperbolic manifold. It is a quotient of hyperbolic free space by a group acting by isometries. And another thing is that, I mean, this is more connected to the quasi-flexion situation. So we can define the limit set of the action. And this limit set is defined by taking any point in the hyperbolic free space, taking the orbit of this point under the action of the group, and looking at the intersection of the closure of this orbit with the boundary at infinity. And if we start with from a quasi-flexion manifold, we get something which is a Jordan curve. It's actually better than this, but let's say it's a Jordan curve. In the boundary at infinity of H3, which is CP1. And we have a properly discontinuous action of the final group of S of, you know, this group on hyperbolic free space, but also on the boundary at infinity of hyperbolic free space outside this Jordan curve. So in other terms, you know, we have this boundary of the manifold and the boundary of the manifold, of the quasi-flexion manifold here is made of those two surfaces S plus and S minus. And we'll find this here. The, this, you know, upper boundary, upper component of the complement of the Jordan curve is the universal cover of S plus. And the boundary of the lower, the lower boundary will be found on the lower side of the Jordan curve, right? So that's all I will say, I guess, at this point on quasi-flexion manifold. But this leads to interesting structures on the boundary, the boundary S plus and S minus. And the reason is that because we have this action of the group, of the final group of the manifold or of the surface on the boundary at infinity by isometries of hyperbolic free space, you know, isometries of hyperbolic free space act on the boundary by Cp1 by actions of PSL2C, so by Mobius transformations. We're going to get at infinity a structure which is called a Cp1 structure. So it's a structure which is locally modeled on Cp1 with changes of charts which are given by Mobius transformations, elements of PSL2C. So we get this pretty rich structure on the boundary at infinity. We get this Cp1 structure, sigma plus and sigma minus here. And this is a bit too rich in a way, you know, basically the space of Cp1 structures is a space of real dimension 12 G minus 12. And if you think of it, there are reasons to believe by, you know, counting dimensions that the space of all quasi-fection structures on this manifold also has dimension 12 G minus 12. So if you know the data sigma plus, you already know the whole manifold, right? In fact, there is also sort of another structure which is a weaker structure on the boundary at infinity which we're going to call C plus and C minus. And C plus and C minus contains a lot less information. It's given by the complex structure underlying the Cp1 structure, sigma plus. So C plus and C minus are in the Tachron space of the surface, right? They are, you know, complex structures. And a key point in this, you know, CRI is a CRM of BERS, the BERS, double uniformization CRM that tells you that if you know C plus and C minus, you can uniquely reconstruct the manifold quasi-fection manifold M, right? That's, you know, a very, very, very, very basic situation of what physicists here called ADS-CFT correspondence. I mean, you have some data at infinity. There is a unique way to fill this by, you know, a complete hyperbolic metric. Okay. Oh, sorry. Oh, okay. Yeah. So this basically gives you the, you know, starting point of the data at infinity and we're going to see more data at infinity later. But we can also play a different game. And instead of looking at what happens at infinity, we're going to use this convex subset that we have here. And, you know, a basic remark is that if you take any convex non-empty closed convex subsets of quasi-fection manifold, for instance, the intersection will still be a convex subset. And for this reason, there is the smallest non-empty convex subset in M, which is going to be called C of M, its convex core. And this has very interesting properties, I think discovered by Thurston, I guess. One, you know, one first basic remark is that, you know, it is, except in very simple cases called function cases, when, when, you know, this, you know, in those simple cases, the subspace, this convex subspace is a total digit of the surface, right? This can happen, but we're going to forget about this situation for the rest of the talk. And otherwise, the boundary of C of M is the disjoint union of two surfaces, which are both copies homomorphic to S. So as for the boundary at infinity, we have two surfaces here and here, but now they are locally convex surfaces. And they're locally convex surfaces of a very specific type. You know, if you think of it, they are, because they are, you know, boundaries of smallest convex subsets, they cannot have any extreme point. You know, they cannot have any vertex. Otherwise, you could cut off the vertex and get something smaller, which would be a contradiction. So what you must have is something that is locally a pleated surface, right? It's like, you know, something like a polyhedron. It has faces, it has vertices, edges, sorry, but it has no vertex, yeah? So if you think of the, you know, simpler situation you can think of, what you'll have is a surface which is made of totally geosic pieces that are glued along edges, right? And the sort of, you know, richness of this situation discovered by Thurston, I guess, is that it could be a lot more interesting. And what can happen is that usually you have a bending which is along a measured lamination. So instead of having a bending along a finite number of edges, you could have a bending along geodesic lamination. And along each leaf of this bending, you could have a very small bending. Right, but the consequence of this is, you know, basically you can think of it in the case of polyhedra. If you think of a line where there is some bending, there will be no singularity of the induced metric, you know? That's something that happens already in Euclidean free space. If you glue two health planes along their common boundary, what you get is something which is isometric to a full plane. And this is going to be true in hypolic free space too. If you glue two hypolic planes, health planes into their common boundaries, there will be no singularity of the induced metric. And the same happens for those, you know, metrics. They will be induced metrics on those surfaces. We'll have no singularity along the bending lamination. There will be hypolic metrics. So you find immediately that on the boundary of the convex core, you have a rich, you know, geometric situation, rich geometric data. You have two hypolic metrics and you have two measured laminations. Yeah. So I didn't tell you exactly what a measured lamination is and I'm not going to tell you exactly. But what I should say is that you can define a space of measured laminations by taking the space of, you know, weighted multi-curves on the surface. So in order to, you know, you take a surface, you know, a weighted multi-curve is going to be a disjoint union of simple closed curves with a weight for each of them. And then you're going to say that two weighted multi-curves are closed, sorry, they're closed if they are intersection with any closed curve on the surface is almost the same. So this gives you a topology on the space of multi-curves, weighted multi-curves on the surface and taking the completion of the space of weighted multi-curves with respect to this topology gives you the space of measured laminations. Okay, now another thing that I need to understand this is the notion of lengths. So obviously, if you have a weighted multi-curve, you can define its lengths for any reference hyperbolic metric, right? You take the, you know, you realize every curve as a geodesic, take the lengths of this curve and multiply it by the weight. And the sum of those, you know, weighted lengths is the length of the weighted multi-curve. And now a theorem of Thurston is that the length function that you have just defined extends by continuity to length function defined on measured laminations. Yeah? Okay, so this is the background and I can tell you more about open questions. Okay, there are, you know, two things that we, I'd like to know, you know, things that were constructed by Thurston and that we don't know how to prove at the moment. You know, again, you know, if we come back to this picture here, on the boundary of the convex core, we have this hyperbolic metric M plus, M minus is here, and we have the upper binding lamination and the lower binding lamination L plus and L minus. So first question, is it true that exactly like in the situation of the best double uniformization, any two hyperbolic metrics can be uniquely realized on the boundary of the convex core? Yeah? So if you think of this in terms of the data at infinity of C plus instead of M plus, this is exactly the statement of the best double uniformization theorem. But in this case, this is not completely known. Actually, it's known that the existence holds and this is a consequence of a work of Epstein and Marden and also work of Labourry in the early 90s. But uniqueness is not known. And uniqueness is not known basically because the local rigidity statement is not known. And there is another conjecture of Lawson, which is that if you choose any reasonable couple of minus and L plus, meaning any two laminations that satisfy some natural conditions. So one is that they must feel. So if you take any closed curve on the surface, it must intersect either of them basically. And moreover, it's clear that any closed curve in those laminations must have weight at most pi, I mean less than pi actually. The otherwise this cannot work. And those are natural conditions. And the conjecture that Lawson made is that if you take any two laminations, measured laminations, satisfying those conditions, then you can realize them uniquely as the measured banding lamination on the boundary of the convex core of some quantification manifold. And again, you know, this is not known. So it's true. I mean, it's a theorem of Bonin and Nothal that it's true if you take two measured banding laminations that are of the simplest possible type, meaning they have a long closed curves. They basically weighted multi-curves in the sense I talked about here. So in this case, Bonin and Nothal proved that the conjecture is true. There is a unique, you can realize them uniquely as the measured banding lamination on the boundary of a quasi-flexion manifold. And in general, they proved that existence holds, but they couldn't prove uniqueness. And it's exactly the same sort of situation as in this conjecture for the induced metric on the boundary. And what's funny is that in this case, you're also, what is lacking is this local rigidity statement. And what Bonin proved is that the rigidity statement for the measured lamination and for the induced metric, they are actually equivalent. So if you can prove one, you can prove the other one. But we don't know how to prove either. So that's an interesting situation where some very basic statement is still unknown. And now I'd like to say a little bit more about the way those things are connected, right? The measured lamination, the banding lamination, and the induced metric. So the way they're connected is by the Schleff-Liff-Hormola. I'm going to go through a little bit of polyhedral geometry and then come back to convex cores. So what is the Schleff-Liff-Hormola? So this is all mathematics from the 19th century. Now, instead of taking some complicated thing like a quasi-flexon manifold, I'm going to take a simple object like a polyhedron in hyperbolic free space. So I take this polyhedron and I take a first order deformation of this polyhedron. I call it p dot. So p is a polyhedron. p dot is the first order of deformation. And I suppose that p dot doesn't change the combinatorics of p. So I'm just moving the vertices in such a way that the combinatorics doesn't change. And Schleff-Liff, in 1850, I guess, discovered that there is a beautiful formula relating the volume of p, v is the volume of p, and actually the variation of the volume and the variation of the angles, the interior angle, the hydrolangles of the polyhedron. The formula is exactly the following. Twice the variation of v is the sum over the edges of the length of the edge times the variation of the interior angle. Or if you want to state it in a way that's going to be closer to what we do, twice the variation is equal to the sum of the length times the variation of the exterior angles, which are pi minus the interior angles. There is a nicer way to look at this simple formula. So the nicer way is you can define this dual volume of this polyhedron. And I'm just going to state it, to define it as the dual volume being the volume minus the sum, half the sum of the length times the exterior angles. And then clearly from this theorem, it's obvious that you have this formula that twice the variation of the dual volume is minus the sum of the variation of the length times the angles. Yeah? Yes? That's all you have to doubt about. Is it me that I have? I don't know. Am I wrong in the formula? No? Okay. Well, anyway. I hope this is correct. It could be wrong in the coefficients, but I hope this is correct. Okay, so what is the connection to what we do? You know, basically those polyhedron, again, they look like convex cores in some sense. And actually there is a Schleff-Lieff-Lieff-Lieff-Lieff-Lieff-Lieff-Lieff for convex cores of Quasifax and Manifold. And it's called the Bonhomme-Schleff-Lieff-Lieff-Lieff formula. And Bonhomme proved this, you know, I mean, first, you know, worked a lot to understand what the formula means. And then he proved the formula. So now we look at the volume of the convex core of M. And we call this VC of M. And we want to understand how this is going to vary in the deformation of a Quasifax and Manifold. So the Bonnet-Wong-Schleffli formula tells you the following. It's exactly the same as the Schleffli formula that we saw before, except that you have to understand what we are talking about. But it tells you that if you take a variation of the deformation of the quasi-fix and manifold, twice the variation of the volume of the convex core is going to be the length for the induced metric on the boundary of the convex core of the variation of the banding measured lamination. So this plays the role of the angles, of the x-ray angles. And basically, from the variation of the pleating angles, you make up something which is what Bonnet-Wong calls a holder-cocycle. And he proved that you can make sense of the length of a holder-cocycle. And when you actually make sense of this, you actually get twice the variation of the volume of the convex core. So now there is a notion of dual volume also for quasi-fix and manifold. It's the volume minus 1 half the length of the measured banding lamination, exactly as in a previous situation. Here we had the sum of the length time x-ray angle. And here you have to close over to this machine. The batteries are off. What am I doing? OK. So this is exactly the same formula in terms of the dual volume. The dual volume is the volume minus 1 half the sum of the length time banding angle of the banding curve but in this measured lamination situation. And then you have a dual Bonnet-Wong-Schleifli formula, which we're going to refer to in the future. Maybe I'm going to write it again, which is that the variation of the dual volume of the convex core is equal to minus the variation of the length of the lamination L on the variation of the metric m dot. So if you think of this, this is a much simpler formula. You just take the length of the measured lamination L as a function of a Taj-Mal space. And it happens to be a smooth function. I think Steve proved this, right, analytic function. And so it is a smooth function. You can take its differential and you evaluate the differential on the variation of the induced metric, which is a tangent vector to the Taj-Mal space of the boundary of m. And OK. And this is the dual Bonnet-Wong-Schleifli formula, basically. You can see it as a consequence of this Bonnet-Wong-Schleifli formula, but it's also much simpler. And actually, you don't need this theory of hold-up or cycle to make sense of it. OK. Now I'm going to jump to infinity and talk about renormalized volume. And we're going to see later what the connection to what we said on the boundary of the convex core is. And the connection is very close if we think of those things. So the renormalized volume of quantum manifold is something that has two different origins. You can look at it in terms of high public metrics. And then in this setting, if you think of it in terms of renormalized metric, you can extend it. And it was actually defined first, I guess, for things that are called Poincare Einstein metrics, which are Einstein metrics in higher dimension that behave at infinity, basically, like the Poincare model. That's a very simple way of explaining this. But basically, in this setting, you can define a notion of renormalized volume. And this was done first by physicists, like Hamington's Candaris. And then it was translated into mathematically understandable things by Witten. And then there were a work of Raham and Witten about this. So that's the first way to look at this. You think of Riemannian metrics. And then there is another way to look at it in terms of complex analysis. And in that setting, what you do is you define what is called the UVIL functional for representations. And this was done by Tartagin and Zograph. So there are two completely different ways to look at this object. And if you look at the intersection, somehow you find a renormalized volume of hyperbolic three dimensional manifolds. So I'm going to give you a definition that we worked on with Kirill Krasnoff, who is a physicist, not in gamma norm. And the definition is fairly simple, but you have to follow the details. And basically, the idea is of following. So you take this subset n of our manifold m. Maybe I'm going to do another drawing so that we follow more easily what's going on. What's happening, right? So we take this quasi-function manifold m. It's always the same. And in m, I'm going to take some convex subset n. And it doesn't really have to be, it should not be the convex core. I mean, that's a sort of confusion that some people make. You really should think, forget about the convex core in this situation. Now you define this, you chose this convex subset. And then you can define a sort of volume, which is simply the volume of n minus some boundary term. But there are reasons to think that this is the right boundary term, but anyway, just take this as a definition. And then basically what you do is you take the surfaces equidistant from n. And yeah? Oh yeah, it's the mean curvature of the boundary. Actually, yeah, it actually should be twice the mean curvature of the boundary, depending on the definition. So h is the trace of b, which is a shape operator of the surface. It's a sum of the principal curvatures, yeah. OK, so now you take this subset n. And you take the surfaces which are equidistant to n. And you call them SR, right? R being the distance to n. And you look at the induced metric, 1R, IR, 1R, big 1R, on SR. And then you look at the limit. Oh, there is some typo here. You take the limit of those metrics, renormalized in such a way that it converges. So the limit, when R goes to infinity, of exponential minus 2R, 1R. There is a missing term in this expression. And in this way, you get two metrics. I mean, you can prove that those things converge. Basically, in this hyperbolic situation, the surfaces become bigger and bigger when you go to infinity. And the coefficient that you need to have a converging limit is exactly this exponential minus 2R. So you get the limit. You get 1 minus star and 1 plus star. And those are compatible with C plus minus. In a sense that the complex factor on C plus minus rotates the vectors for those metrics. So they are in the correct conformal class. And then what we can prove, and what I'm going to say is slightly false, but not completely false, is that if you know n, you know 1 star. But if you know 1 star, you know n, too. So you can think of this quantity, w of n, as a function of 1 star. And in this way, you forget about n. And you just look at this as a function on metrics in the conformal class at infinity. And seen in this way, it has interesting property. So the key property is that you fix the area of 1 star. You look at all the metrics in the conformal class at infinity, a fixed area. Then this w of 1 star is maximal exactly when 1 star has constant curvature. So 1w is a way to uniformize metrics in the boundary at infinity of m, in the conformal class at infinity. And then you can define the vr, so the renormalized volume. It's going to be exactly w of 1 star when 1 star is the hyperbolic metric in the conformal class at infinity. So that's the definition. It's a somehow indirect definition, but it's reasonably simple. Maximizing, you get a function which only depends on c plus and c minus. And therefore, because of the Bers-W uniformization theorem, it's a function from the space of square effects and manifold to r. Yeah? It has interesting properties. So one key property is that it's the same as the UV functional that was defined by Sograf and Tartajan and studied by those people. It's not completely obvious to me why it is the same function. The reason it is the same function is because they have the same variation formula. It leads to an obvious proof. But otherwise, I don't really understand why it is the same function. But maybe as a consequence of this, or maybe you can prove it directly, this function is actually if you fix c minus and you vary c plus. So you change, you vary the quasi-effects and manifold, keeping the same conformance factor at minus infinity and varying what happens at plus infinity. You get a kilo potential for the valid-velocity metric on Tashmore space. It has interesting analytic properties. Any questions on this? OK, so now what I want to do is really make precise the analogy between the boundary of the convex core and the boundary at infinity. And for this, I need to understand better the data at infinity. So far, what we have, the data at infinity that we have is this complex structure. But I want to have something which is the analog of the measured banding lamination at infinity. And I think there is a very natural candidate, and I'm going to try to convince you that this is the right candidate. So what you can do is the following. So you have this one R, which is the induced metric on this surface at constant distance R from n. And you can write explicitly the induced metric on SR, 1R, in terms of R. There is an asymptotic expansion, which in this case, because we are in high-poly geometry, is actually exact. So there are exactly three terms. So one is the one we already know, because this is the definition of 1R, 1 star, basically. And then there is a constant term, and then there is an exponentially decreasing term. And that's it. And this is the term we're going to be interested in. And actually, we are going to be interested in not only this term, but the trace-less part of this term. And the reason is the following. So if 1 star is a high-poly metric, so you have to make this choice of a high-poly metric in the conformal class at infinity, then the trace-less part of this second fundamental form of infinity, 2 star, is codazzi. Codazzi means that if nabla is the levitivita connection of 1 star, then d nabla of 2 star 0 is equal to 0. Maybe I can write what this means. This means this d nabla 2 star 0. If you know, it's something that can be evaluated on two vectors. It's a two-form with values in z. That's going to be simpler. So this is going to be nabla x 2 star 0 of y z minus nabla y of 2 star 0 of x z. And this happens to be equal to 0. So it's what we call a codazzi, codazzi tensor. And as a consequence, because it's a trace-less symmetric 2 tensor, which is codazzi, it's known by very general arguments, old arguments, due to HOPF, I guess, that it is the real part of a holomorphic quadratic differential, 2. So you get this data at infinity. And there is a way to understand this data at infinity in terms of the Cp1 structure at infinity sigma. The way you do this is maybe I shouldn't get into this. But you have a Cp1 structure at infinity. You also have a complex structure, which you can uniformize. So there is a map from the disk to the disk from the usual disk into this, which is the Riemann uniformization map. And this is a map from a surface with a Cp1 structure, the disk, to another surface with a Cp1 structure, which is a sphere. And well, you can take what is called the Schwarzson derivative. Oops, sorry, what is called the Schwarzson derivative of this uniformization map. And you also get a holomorphic quadratic differential. And it happens to be the same as this Q. So the data, I mean, there is a connection between the representation point of view and the Riemann metric point of view, which is that this second term is connected to this Schwarzson uniformization map. Now, the key thing is that this variation formula, there is a variation formula for the renormalized volume, which is very simple. So it might remind you of the Schleff-Leformula, but at this point, it's not going to be very close. The variation, the first sort of variation of the renormalized volume is given by this scalar product between this second form at infinity, a tracest part, and the variation of the hyperbolic metric at infinity, 1 star. Well, we're going to get a closer interpretation. And to get this closer interpretation, I need to say a few things. Oh, sorry. Yeah. Yeah, I need to say a few things on measured foliation and on extremal lengths. Yeah, so far, we've been talking about measured laminations and hyperbolic lengths. And this was on the boundary of the convex core, but at infinity, we'll have those different, slightly different notions, but related notions. So what is a measured foliation on the surface? So it's a foliation which has singular points. It can have singular points, along with a transverse measure. And we're going to call this space of measured foliation Mf. And there is a very close connection between Mf and the space of measured laminations, which I'm not going to explain, but there is a 1 to 1 correspondence. And you can think of those two things as basically two ways of looking at the same object. Now, if you have a holomorphic quadratic differential Q, define the horizontal measured foliations, which we can call hfq. And again, I'm not going to give the proper definition, but basically locally at a non-singular point, at a generic point, you can write the holomorphic quadratic differential as dc squared. And then the foliation will be by the curve, which are the curve imaginary part of z is equal to constant. So the parallel to the x axis. And the transverse measure is going to be given by how much you move in the direction of the z axis. Those things, they can be glued nicely to get this measured foliation. Now, from this, I want to define the measured foliation at infinity as the horizontal foliation of Q plus minus. This Q plus minus, it's this holomorphic quadratic differential at infinity. And it's pretty reasonable to look at the measured foliation associated to it. And we get this data. Now, there is a notion of length defined for measured foliation, not in terms of hyperbolic metrics, but in terms of conformal structures. And the definition is, well, we know it's the following. So you take any curve on a surface. But now our surface is going to be a Riemann surface. There is no hyperbolic metric. Just a Riemann surface notion. And then what you do is you take a curve. And what you look at is the biggest annulus that you can embed in this surface. So I'm going to make a really bad drawing. I have no idea how to draw this properly. But I'm going to try to get a nice, big annulus to sit inside the surface in an embedded way. So I should do this on both sides, I guess. So maybe you have no idea what I'm trying to draw. But I'm trying to draw a big annulus, which is embedded in the surface in such a way that the core of the annulus is isotopic to the curve. And when you do this, you can look at the modulus of the annulus. And the definition is the following. Basically, you realize your annulus as something of by gluing the sides of a rectangle of h1. And then, basically, what you get in the other direction is the modulus of the annulus. Is that correct? Yeah. OK. So basically, this is a sort of notion of length. But that's defined purely in a conformal way. And again, like for the hyperbolic lengths, you know, kind of prove that this extremal length extends to measured foldations. It's not only defined for closed curve, but from a closed curve, you can get a measured foldation. And you can approximate measured foldations by things like closed curves. And eventually, you have this extension. OK. So I think we have everything in place to state the analogy that I'd like to explain. So the analogy will be the following. So there is an analogy between the renormalized volume of the manifold seen from infinity. And from the boundary of the convex core, we're going to look at not the volume of the convex core, but the dual volume of the convex core. This is a small difference, but it's going to make things a lot nicer. The induced metric on the boundary of the convex core is going to be analogous to the conformal metric at infinity, right? That's pretty natural. The measured lamination, the measured binding lamination on the boundary of the convex core is going to correspond to the measured foldation at infinity. That's also reasonable, given what we said. And instead of talking of the hyperbolic length of the measured binding lamination with respect to the hyperbolic metric, we're going to think of the extremal length of the measured foldation at infinity in terms of the conformal metric at infinity. I think I'm repeating this. So now I'm claiming that this is a pretty good analogy in the sense that, first, the same phenomena tend to occur in both situations, and also that the corresponding quantities are closed. And they're closed in a, OK. The point is that there are some things that I know and some things that I don't know. So the point is that there are quite a few interesting questions in this area. So one thing that works well is the following formula, which is a consequence of the variation formula that we gave earlier. We had this variation formula for the renormalized volume. So this was exactly the formula here. It was expressed in this slightly complicated way. But using what is called the Gardner formula, some kind of Gardner formula, you can reformulate it in the following way. First sort of variation of the renormalized volume in a deformation of a quasi-flexon manifold is exactly one-half the differential of the extremal length of the foliation at infinity evaluated on the variation of the conformal metric at infinity. And if you remember what we said before, this is exactly the same formula, unless I made a mistake in the coefficient. It's exactly the same as the dual Bono-Schleffly formula. The dual Bono-Schleffly formula was here. And it's exactly the same formula given the analogy that we have developed. Yeah. So again, there are two things. Again, one statement is that the things at infinity are close to things on the boundary of the convex core. And another thing is that they behave in the same way. So what do we mean by the fact that they're close? So there are things that are well-known. One thing is that, first, I should say, because it's going to simplify what I'm going to say later, that the volume and the dual volume, they're almost the same up to fixed constant that only depends on the genus. And this is a result of Benjamin and Canary. So what they proved is that the length, if you take any quasi-flexible manifold, the length of the measured binding lamination for the hyperbolic metric on the boundary of the convex core is bounded by some universal constant depending only on the genus. And as a consequence, because of the definition of the dual volume, the dual volume is more or less the same as the volume of the convex core up to this constant. Yeah, so then now we can compare the renormalized volume to the volume of the convex core or the dual volume. This is the same because of this statement. And what we have is that those things are the same up to explicit constant that depends only on the genus. Yeah? So that's one reason why VR and VCL close. And another reason is that you know that M and C they also close. And this is a result due to, I guess, the first estimate. This was a conjecture of Sullivan. There was an estimate due to Epstein and Amaden, which was refined by many people since then. Let's say that the induced metric on the boundary of the convex core and the conformal metric at infinity, they are uniformly quasi-conformal. So the data at infinity and the data on the boundary of the convex core in terms of the metrics, they are not very far. But again, the volumes, they are not also not very far up to some constant. Yeah. And OK. So I should say that basically this is true for quasi-flexion manifolds, but Bridgeman and Canary have proved that this is also true in some sense for more general convex or compact hypodic manifolds. Yeah? OK, so now this leads to a number of questions and statements. So there are some easy results. So basically what I said before is that the lengths of the measured lamination, so Bridgeman and Canary proved that the length of the measured bending lamination is bounded from above by your uniform constant. And what is easy to prove is that the same happens at infinity. So the extremal length of the measured radiation at infinity with respect to the conformal structure at infinity is bounded by a uniform constant depending only on the genus of the surface. Another statement that is clearly true, if you know M plus and L plus, you can uniquely reconstruct the quasi-flexion manifold. And this is the same in this situation from infinity. So if you know C plus and F plus, you can uniquely reconstruct the quasi-flexion manifold. This is completely, I mean, pretty easy, basically. I mean, consequence of things that have been proved before. And but then there are questions. So I mean, one basic question is, we said that L plus and L minus field, they intersect in a closed curve in a non-privileged way. And for F minus and F plus, that's something that should be true, but I don't know how to prove this. Another question that is, I think, interesting is, again, there is this contractural sustain that L plus and L minus uniquely determine the manifold. And this is completely open for F minus and F plus. What's funny, if you think of it, is that the sustain contractural on M plus and M minus, if you look at it from infinity, it's the fact that C plus and C minus uniquely determine the manifold. And this is completely true and well known in the terms of data at infinity, right? But for F plus and F minus, this is completely open. And it's not even clear what are the possible F minus and F plus. Given that for L minus and L plus, they are natural conditions that are necessary and sufficient. But the analog of those conditions for F minus and F plus, they are not completely obvious. Any questions on this? OK. OK, so I'm going to move on to some applications of renormalized volume and also connections to other areas. I'd like to talk a bit on bounds on the renormalized volume and the way you can use the things to obtain bounds on the volume of the convex core. So there is a theorem of Brock, proved in 2003, that the volume of the convex core is basically given by the distance between the conformance factor at infinity, seen as point in dash more space. Now, this is true in the quasi-azometry sense, right? Up to multiplying by some constant and adding some constant. Now, in terms of the renormalized volume, there is a much better estimate, which is that the renormalized volume is bounded by some very explicit constant time divided by the distance between them. And so where do we get this from? There is this variation formula that I gave you that tells you that the sort of gradient, I mean the differential of the renormalized volume is given by the real part of this homomorphic relative differential at infinity. And then it also comes from this Nehari estimate that tells you that Q, we said that Q is the Schwarzson derivative of the uniformization map from the disk to the upper boundary, upper component of the complement of the Jordan curve at infinity. So in particular, Q is the Schwarzson derivative of a univalent map. And there is an old result called the Nehari estimate that in this situation, it has to be bounded by some explicit constant, which is free health. So if you put this together with this variation formula, you immediately get that this renormalized volume is bounded by this quantity. Very simple argument. And now we said before that the renormalized volume is more or less the same as the volume of the convex core up to some constant. And as a consequence, you get that the volume of the convex core is bounded exactly in the same way by the right-possesson distance of the data at infinity. And this has been extended in different ways by Kojima and McShane and also MacMillan, and I'm not going to explain this. But this leads to applications. And I think maybe Jeff talked about some applications earlier in the first week of the school. But basically, there are important applications to the volume and entropy of mapping to write by Bruck and Brumbag and also by Kojima and McShane. And Bruck and Brumbag, they also gave applications to the systoles of the right-possesson metric on model X-PACE. And I'd like to point out that there are other types of applications. And typically, the sort of applications that you can have is that VR can be seen as a generating function on Taishman's face, the space of where the quasi-flex and manifold. And for instance, the sort of thing you can recover in a very simple way using this is what is called McMillan's quasi-flex and reciprocity. And so that's an interesting statement that you can see it in a very long way. So you look at the space of data at infinity obtained from quasi-flex and manifolds. Basically, we said that for every quasi-flex and manifold, we have this complex factor at infinity, C. And we have this holomorphic-quatic differential at infinity. But we know that C can be seen as a point in the Taishman space of the boundary at infinity, of the boundary of the manifold. And this Q is a cotangent vector to the Taishman space of the boundary. So what you get is a set of points in the cotangent bundle of the Taishman space of the boundary. Now, the dimension of this space is, so this has 6g minus 6 for the upper boundary, 6g minus 6 for the lower boundary. So the dimension of, I should write it here maybe, the dimension of the Taishman space of the boundary is 12g minus 12, but g is the genus of S. And therefore, the dimension of the cotangent space is 24g minus 24. And the dimension of B is 12g minus 12. This is equal to the dimension of B. So this is exactly the right situation where we can hope that this is going to be a grandian manifold of the cotangent space of the boundary for the cotangent simplex structure. And this happens to be true. So this is what MacMillan proved by different methods. But the proof is very simple if you use this tool. And the reason is that you have this variation formula. So the variation formula tells you that the variation of the renormal volume is given by the scalar product, I mean, the bracket with this cotangent vector. In other terms, the differential of the volume is given exactly by the UV form of the cotangent space. In other terms, this tells you that on this B, the UV form is the differential of the volume. But this means that the symsectic form, which is the differential of the UV form, is 0 on B. So you can obtain this result as a one line proof using the variation formula for the renormal volume. That's the first type of application. I mean, there are already several applications, but first type. We have other applications that I think are worth mentioning. So I'm sorry, I have to give a few more definitions. So I have to define the space of Cp1 for cross on S. So we mentioned this at the beginning. This is the space of geometric structures, locally modeled on Cp1, with changes of charts, which are mobius transformations in PSL2C. And so we have this space Cp. And there is a map, which was defined by Thorsten, which gives a parameterization of Cp by the product of Tashman's space of S by the space of measure limitations. And so how do you define it? Basically, I'm not going to define it, but it is defined in such a way that if you do the grafting, this map is called grafting, the grafting of the induced metric on the boundary of the convex core by this measured bending lamination, you get exactly this Cp1 factor at infinity. There is a unique definition that satisfies this property, and it more or less tells you how this works. And Thorsten proved that this map is a homeomorphism. So that's a nice way to describe the space of Cp1 structures in terms of things that you might or might not understand better. And a consequence of this renormalized volume thing is that this grafting map, it's symsectic, up to some factor, which is maybe 2 or 1 half. So it's not clear at all what this means if you think of it, because the space of measure lamination is not a C1 space. It's a piecewise linear space, but not C1. So if you actually want to understand this map as a C1 map, you have to remember that this space can be identified to the cotangent space of dash-mount space. So whenever you have a point in dash-mount space, that's a hyperlink metric and a measured lamination, you can associate to this point in dash-mount space with a cotangent vector. And in this way, you get a map from the cotangent space of dash-mount space into the space of Cp1 structures. And this map is C1. Although, again, it's not here where this map here cannot be C1 because it doesn't even make sense. But this map happens to be C1. And it is in tactic up to a constant factor. And the way to prove this, again, is by using the properties of variational properties of the renormalized volumes of not exactly quasi-flex and manifolds, but things that are made from quasi-flex and manifolds. Yes, I don't have time to say more. Any questions on this? But that's another type, probably. Yeah? Yeah, what is the same tactic? Yeah, that's a good point. So I think what you should take here is the, so I mean, there are different ways to look at it. But in this case, you should take the real, I have a doubt. I think you have to take the real part of the goldman syntactic structure, I mean, both goldman syntactic structure. But now, is it the real part of the imaginary part? I think it's the real part. I better check and tell you after that. I need to think of the proof to remember which one it is. But yeah. OK, so a third type of application is, again, a sort of symplectic application. And among those quantum effects and manifolds, they are really nicer ones. And the nicer ones, they are called almost-fucsion. A fucsion situation is when we have a totally geodynamic surface in the manifold. And almost-fucsion means that it's not that, that it's not that far, in the sense that there is a closed, minimal surface in the manifold with principal curvatures uniformly, I mean, strictly less than one. So this happens in some situations, and it doesn't happen in other situations. But there is a theorem of Unenbeck from many years ago that if m is almost-fucsion, so if there is a minimal surface with this property, then there is no other closed minimal surface. So then there is this minimal surface is unique among all closed minimal surfaces. And in this case, you can take the trace-less part of the second fundamental form of S. And it's, again, basically because we're dealing with a minimal surface, it's a coder at the tensor for the induced metric. And therefore, it's a real part of a holomorphic relative differential for the complex structure underlying the induced metric on the minimal surface. So we have this map from quantum effects and manifolds again to the cotangent space of dash-mall space, which sends a quasi-flexion manifold to the conformance structure of the induced metric on the minimal surface and the trace-less part of the second fundamental form. And there is another statement approved by Brisch-Lusteau in his thesis a few years ago that this map mean from the space of almost-flexion manifolds to the cotangent space of dash-mall space is, again, some tactic up to a constant factor. And again, I should think of which symplectic structure you take on this space of almost-flexion structure, but I think it must be the imaginary part of the Goldman form. And that's it. So thank you for your attention.