 We looked at our whole problem g equals e to the 2u g0. We want to find this constant curvature. So find this, which we'd set up as a PDE. And I've discussed how the various techniques to solve this in various cases. So the fundamental invariant involved the cone angles, chi of m beta. And so we solved the two cases. One is when chi of m beta is less than or equal to 0 and all beta. And so this was just directly attacking. So there were some linear theory involved, and which I'll be discussing at a greater length, starting in about a half an hour. And then I discussed the case where chi of m beta is positive, but with two restrictions, so in fact, two different cases. So this is greater than 0. So one case was m is not equal to the sphere. And so this was using variational arguments. So this was this whole thing of looking at this energy functional. And we saw there that the energy was unbounded below, but there was a way of finding saddle points. So that was sort of the second variational argument I sketched. But so that gives a fairly complete understanding. There are various borderline cases there which are not perfectly well understood, but that's sort of where I'm going to leave it. I, of course, just sketch those arguments. There's quite a lot that goes into them, but I'm trying to give an overall feel. And then the other case I talked about yesterday was when m is equal to s2, but in the case where all of the beta i's are less than 0. So I talked about this in the general case for any m, but I then extended it to general cone angles when m was not equal to s2. And so here, the origins of this variational argument, this was Troinov. And so we had these Troinov constraints, which we saw in two different guises. So it says that beta j is greater than the sum i not equal to j beta i for all j. And this is, again, just in the case where all beta l's are less than 0 for all l. And we saw this appearing in two different ways. So one was the geometric argument I gave using the Gauss-Benet theorem. And the other was the purely analytic formulation, which is coming from the best constants in the most returning inequality. So there are many other ways to understand this. And this is an interesting area of what these all mean. So what I want to, yeah, so forgive me. So the angle is 2 pi times 1 plus beta j is equal to theta j is equal to the cone angle. So beta j less than 0 means precisely that the angle is less than 2 pi. And then that has many nice analytic and geometric consequences. So I've talked about some of those, and I'll talk about a few more about how it helps you, both of the analysis of the geometry. OK? OK, good. So today I want to outline, I mean, give some ideas in a very nice extension of this paper that came out just a few years ago by Gabriele Mondello and Dmitry Panov that significantly extended our understanding. So now we can focus entirely on the case of S2. So we're looking at spherical surfaces. So we're looking at m is equal to S2. We're always looking at the case k is equal to plus 1, because we've settled all these other cases, and general beta. So their theorem is the following. So the theorem proved in this paper. So they say that a necessary condition for the existence of a solution is that you have a following set of inequalities. Now it's kind of looks rather different than the other thing. But what I do is the following. So I take the vector beta, beta 1 through beta k, and I take the lattice zk odd. So I take the set of all integer points, and I just take the odd sub lattice, the set of things whose the sum of the entries is an odd number. So it's an index 2 subgroup. And I want the l1 distance of this to be greater than or equal to 1. Amazing. OK, so it's kind of a remarkable condition. We can look a little bit what this looks like. The point is this actually reduces. So it's not obvious. It takes a little bit of footwork. But this reduces to this Trinoff condition in the case. Well, I mean, at least the open condition reduces to this in the case where the betas are negative. So roughly speaking, here's sort of what this looks like. So suppose I were to look in the beta negative case. So here's 0, 0, 0. Let's suppose k equals 3. So here's sort of the odd sub lattice. And then I'd have, so these things are one away. And then I have this as one away. So let me see if I get this right. I was sort of screw up drawing this. But I think I take, sorry, I always forget which way this picture goes. Yeah, so I go like this. So I draw these diagonals. And then I do the same thing on the other side. And I sort of fill in a solid, the solid region, which is cut out by. So I have that diagonal down there, and then the corresponding diagonal over here and back. Excuse me? So these are beta 1, beta 2, and beta 3. So this is the k equals 3 axis. And I'm looking at a region where the betas are negative. So I'm just kind of illustrating what this condition would look like there. The points. So this is the space of betas, right? The x's correspond to, yeah. So this is the origin. This is sort of the piece of the odd lattice. The odd sub lattice. I mean, it's not the set of points whose integers add up to a, whose entries add up to an odd number. That's what it means, yeah. OK, so you have this solid region. And in general, what you're doing is you're taking out a certain generalized octahedron. You're taking around some polytope around every point of this odd lattice. Now, so a necessary condition for the existence is this. They say a number of new things about this thing. So the interior of this region is connected. That's very useful. The L1 norm, yeah. Is connected for k greater or equal to 4. So k equals 2. It actually, the Troinov condition degenerates. And the only things you get are beta 1 can equal beta 2. So k equals 2 is kind of a degenerate case. It doesn't quite fit into this. It doesn't really fit, OK. Even worse. Yeah, so low k's is bad. k equals 3 is a little bit odd, because this region, this general region is not connected. But in higher k's, it's connected. OK, so this does. So it takes a little bit of work, but you can check that. OK, so right. OK, so what do I want to say? So they prove this is a necessary condition. Furthermore, this is sufficient for betas in the interior of this region. So in other words, they're able to say exactly which sets of cone angles can arise, at least in the interiors of this region. So there's possible issues on the boundary. I mean, they didn't say anything about what happens when beta lies on the boundary of that region. But you're in the interior. You can always realize any k-tuple of cone angles by some arrangement. So what that's really saying is that there's some arrangement of points on the sphere so that the cone angles of those points are in. So you're not specifying the marked conformal type of the sphere. You're only specifying the set of cone angles. Now I want to say a little bit about their proof. So the necessity is kind of an interesting argument, which I'll give some more details of, and then just say a few words about the existence. So one way that you can start building spherical cone surfaces is the following. You take a spherical triangle, and then you take another one, and then another one, and you start pasting them together. And so it's very much like in Alex's lecture, you start pasting these things together, and you can sort of do this around at various cone angles. And there's a real, you can make this sort of a combinatorial problem. And so in some sense, that's how you can sort of build up. I mean, when you look at something with cone angles bigger than 2 pi, this thing is sort of wrapping around, but it still sort of intersects up. So there's a certain amount of combinatorics about exactly how you can paste these things together. And this paper actually sort of does that, especially when k is equal to 3 or 4, they really construct all possible, this existence term, they sort of analyze by studying all possible ways of pasting together triangles and showing that you can attain all possible betas in the interior of this region. So they do that quite explicitly, and then they have sort of this interesting inductive procedure for generalizing this from k equals 3 and 4 up to general k. So I'll say a few words about that. Now I should say that the k equals 3 case, there was a couple of papers on this previously. So Aramenco had studied this problem at great length using complex analysis, and then Umerhara and Yamada. So these are using very different techniques, but this synthetic approach was very quite powerful. OK, so I want to say a few words about where this condition comes from. Excuse me? It's just greater than 1, yeah. So they say a necessary condition is that this is greater than equal to 1, and it's sufficient to be strictly greater than 1. OK, so the beginnings of the proof and it introduced sort of a very important idea, which many of you will have thought of as sort of the starting point for any discussion of constant curvature metrics, which is the developing map. So since I haven't mentioned it, let me sort of say it very explicitly now. So if we have the sphere with these various points on it, so there's going to be a developing map. So let me call this m star. This is equal to s2 minus p1 through pk. So it's the punctured surface. And the developing map is a mapping that takes pi1 of m star, well, in fact, let me say explicitly the whole anomaly map. So it may sort of row from into s03. So how do I define this? Well, so I start off with the developing map, which is that I start with a point. This is just identified with a point on the sphere. So I sort of map it over. And then if I follow along any path, I sort of piece together these neighborhoods, this sort of the standard continuation process. And so this whole anomaly map does the following thing. So suppose I take any loop. I start off with a point and a vector. So I have a point p and v. So that's in the unit tangent space to s2. So then I start following this around. And at the end, so I can sort of continuously develop it. And at the end, it's going to get mapped to some other point. So if I take p and v, then this is going to eventually get mapped to some other point on the sphere and some other tangent vector. So this is going to get mapped to some q and w. And there exists a unique a in SO3 such that a of pv equals qw. So that's the whole anomaly representation. So that's the first basic thing. And one can try to study this whole theory of surfaces by just understanding this whole anomaly representation. The fundamental group of this surface is, well, very explicit. Bit complicated, of course, but very explicit. So they make the observation that's not immediately obvious that this map lifts to su2. So claim is not a three-dimensional vector space, but that's the group of symmetries of the unit tangent bundle of s2. Yeah, I mean this depends on the spherical metric, the spherical cone metric, absolutely. So if I take, for example, if I take a path that just goes around one of these things, that's explicitly going to be a rotation by angle, in fact, beta j or e to the i theta j. So you explicitly see the cone angle when you do the whole on the way around a cone angle, OK? Excuse me? So2, because it's unit tangent model of s2 is s03, right? So you have the isometries of s2 and the action. It's not on the tangent space, it's the unit tangent bundle, right? So think about it that way. If you take any two points and any two tangent vectors, then you can rotate along one axis to get from p to q and then rotate to get the corresponding vector from v to w, OK? So there's a way of composing two rotations to get that. OK, good. OK, so what's the lifting to? So to s2, OK, so here's sort of the trick to see this. So suppose I choose a point which is not one of the conic points and I stereographically project, OK? So I've identified the sphere with just the plane. I have some point at infinity and I can take sort of this circle that goes all around all of these points, OK? Now, if I start looking at this representation, obviously for any particular loop, I can lift any one of these guys to s2. And so the real question is, why is it so? Let me see. I don't want to say this. I'm not going to say this very fluently. So if you start off looking at the developing map as you go along here, well, you're just developing into, well, using that picture, you can sort of delve into a family of rotations. Each one of those rotations, you can lift to s2 and you sort of finally come around. Now, the point is, so let me see. I'm not seeing this very happily. Probably somebody can set me. s2 is the double cover here. So just a 2 to 1 cover, right? So there's really only a choice of sign at the end of the day, right? So here's sort of the point about all of this is that if you compose all of these loops, then that's going to be equivalent to this thing, right? And you want to know what's the winding number of this thing? What's the whole only around this thing? And the point is, it's winding number is 2. So if I just take sort of a parallel vector field here that crosses all these things, this winds around infinity twice, right? So that's basically the compatibility relationship you need. You know that just by this sort of individual liftings, you're going to hit it up to a plus or minus 1. But since the winding number around here is 2, it always closes up. And so this representation is well-defined, OK? So forgive me. I'm not saying that very carefully. And I better cut my losses and not try to explain it further. OK, so row hat lifts to a mapping to s2. So what that means is if we take each loop, so each one of these loops that goes around a puncture. So each loop, let's call these gamma j. It goes from 1 to k, leads to a matrix, aj, which is an sq2. And the relationship is that the product of these loops is trivial, because if I take the products and that's something I can contract around infinity. So what I get is a family of matrices whose product is equal to the identity. So they're not mutually commuting matrices. So it's a set of special unitary matrices whose product is equal to the identity. Now, what I'd said last time was that this corresponds to a geometric object on s3 itself. So sq2 is, of course, just s3. And any aj corresponds to an equivalence class of geodesic arcs. So here's s3. If I start with any point p, here is aj of p, just down. This is sitting on s3. And I just take the short geodesic arc here, OK? Now, so the loops here are on s2. And they correspond to matrices, sq2 matrices. And then I'm thinking of sq2 is acting on the 3-sphere. It's an element of the 3-sphere. So the holonomy is, like I say, it's acting on points and directions. Yeah, so the holonomy is an element of s03. Well, it's the holonomy of this spherical metric on m star. And the point is, is that it's, so when I call it the holonomy representation, I mean, it means when you go around the loop, it's sort of a macroscopic object. If I go around the loop, how much have you sort of rotated? And that rotates, but it's not, you're not, so the loop goes around, but if you look at the developing map, you start and finish at completely different places on s2. So if I take a loop here on m star and I lift it up via the developing map, the developing map is really only defined on the universal cover of m star. So very complicated, I think. So the developing map is really a mapping from m star tilde into s03, into isometries of the sphere. Yeah, because you're, well, I mean, let's take two punctures. So as I sort of go around, the developing map says you start going around, and I'm going to continue. But as I go around, I may not have sort of closed up. So if I take a closed loop here, each of these pieces or pieces on, they give me individual patches on s2, but going around a closed loop here does not necessarily go around a closed loop as I patch these things together. And in fact, it typically won't, because if I'm only going around sort of a partial, sort of less than two pi here, then I don't go around a full distance here. No, there's no identification of edges. It's just that, no, no. Well, this angle may be less than two pi, right? But so I mean, what the developing map is, is I take this neighborhood and I identify it with this. I take a nearby neighborhood and I identify it with the obvious patching map. And I keep on going around, and this is acting on s2. So this is on s2. This is on whatever cone surface. So as I patch these things around, there's a coherent wave continuing. But this map is not defined on m star. It's only defined on the universal cover. It's not defined on m star, because if I go around as a closed loop here, I will have not gone around sort of a full circuit on s2 necessarily. No, this is on s2. So sorry I didn't make that clear. The tangent model, the unit tangent model of s2. OK, so it actually sort of measures not only how the point travels, but how the unit vector that you start off with travels. Yeah, so forgive me I didn't make that very clear. OK, so this whole number representation goes into s03. And then there's this, really, it's a spin construction that sort of lifts this to s2. OK, that I can lift it? Yeah, but we're not on s2. We're not on, well, no. I mean, it has to do. It certainly is true, yes. No, but you can't lift. I mean, this is sort of equivalent to the existence of a spin structure. So it's making these individual that's coherent. And the point is, since the background, since this, yeah, right. That's not the issue, yeah. It's bigger. Yeah. It's bigger. Yeah, yeah, yeah. So I mean, if you're on a surface, if you do this on a more general surface, then there's a more serious relationship. That's right. OK, well anyway, sorry I'm just a stupid analyst. In any case, what we wind up with at the end of the day is k matrices whose product is the identity. Now, what does this look like on s3? So now I forget about that s2. I just say, I have these k matrices. So I just think of this as a new collection of objects. I have k matrices such that the product of the aj's is equal to the identity. What does that look like? Well, so what it looks like is each matrix corresponds to a g-desic arc. So I just take the g-desic arc that goes from here to here. Now, I can piece these together. So if I start at some point, I apply a1, then I go to a2, then I go to a3, and so on. I'm going to finally move around, and then the fact that these things product to the identity means at the end of the day I'm going to get a closed piecewise broken g-desic loop. So this purely algebraic object here, which came from this lifted holonomia representation corresponds to a broken g-desic loop on s3. What do I need to lift s2? Because s03 acts on s2, not s3. This is s3. I mean, this is sort of their construction. I mean, there may be some way of doing this on s2, but they want to sort of think of these g-desic loops on s3, not broken g-desic loops on s3, not s2. So now the question is, what can you say about the space of broken g-desic loops? So you have a collection L1 through Lk, which are the edge lengths of these things, which of course can be determined from the AJs. And what can you say about inequalities between these? So this is a problem that's been studied for many other reasons. So they found elsewhere in the literature. So the precise form that they used was a theorem of Bezois. And the theorem that they quote is that, so there exists such a loop with edge lengths L1 through Lk, if and only if. So you have the following funny condition. So I take the sum of pi minus Lj for all j in a subset, capital J, plus the sum LiI not equal to j. This has to be greater than or equal to pi. And this is for all j inside 1 through k, which is odd. So it's kind of a bizarre relationship. And I can say a few words about it, but not very much. But it's kind of a generalized triangle inequality. So it's saying you take all odd subsets, take an odd number of edges, add up pi minus their lengths, and add up just the rest of the lengths. And that always has to be greater or equal to pi. So what this is doing is extending the following thing, that if I were to try to construct such a broken g-desic loop, suppose for the sake of argument that you have a loop with k arcs where k is odd. And suppose that you try to construct these all with length exactly pi. So suppose you're on S3, and you try to do something with length exactly pi. Well, if I take capital J being the whole thing, so that's an allowable thing of k is odd. And if all the arcs are exactly pi, this would clearly be violated, because this sum would be empty and all of these would be 0. So what that's saying is that if I try to construct a g-desic loop with k odd, I'm just going back and forth between poles and odd number of times, and it's not closed. Now, if I perturb that a little bit, by not going exactly pi, but going close to pi. So in other words, if I look at a special case where k is odd, and each lj looks like pi minus a small correction. So let's call that correction delta j. So then this is going to equal delta j. And if I take the special case of this where capital J is everything from 1 to k, so I have an odd number of arcs. And if the sum of the delta j's is strictly less than pi, then that's impossible. So in other words, I said the obvious thing is if I go back and forth an odd number of times, I don't close up. But if I don't go quite back and forth, but the defect adds up to less than pi, then I still can't close up. So this is some sort of generalization of that. So you can think of it as some sort of glorified triangle inequality. OK, so their observation is that quoting this, well, I need to now finally relate these l i's to the cone angles. And the last point here is simply that each a j is conjugate to e to the i beta j, e to the minus i beta j. So how do you compute that? Well, remember, these a j's come from this lifted holonomia representation, so I'm going around a loop. So I first take the holonomia representation, then I lift that to s u 2. Well, I can take the loop of a very special sort. I can just go very near the cone point. So I start here, go around here, go very near here, and then go back. So that's the conjugation, going here and back. And then you can just very explicitly compute what happens as you go around a small loop. And that's really just a computation on the football. So this is a picture on s 2, or s 2 minus these points. So I'm computing the holonomia representation here. So I go around the loop. I might as well just go very close, go around a small thing, and then I very explicitly compute what happens as I go around the loop of a football, and it's just that matrix. So what does that mean? It means that these lengths are exactly beta j. OK, so that's the forward direction. So they sort of reduce it to this geometric problem, which had been solved for completely different reasons. And if you decode this condition, so it's not sort of trivial, but if you decode what this condition means, it corresponds exactly to this sort of funny L1 distance from the odd sublattice is greater or equal to 1. So it's kind of a remarkable thing, and it actually takes a few page. It's not a trivial computation, but they decode it. And why that looks like that, and why that looks like the Troinov inequalities is non-obvious. I think that's one of the very nice things about the paper that they wrote this in this form. OK, so what about this efficiency? So there's an interesting idea, which is a synthetic geometric idea. And what we're trying to do, and what I'll be describing as soon as, yes, so that's exactly what I was saying, that if you have a matrix that looks like this, then it rotates by essentially an angle beta i. So the LIs are essentially exactly beta i. Excuse me, maybe I should have said 2 pi beta j. So yeah, that's probably 2 pi. Excuse me, that's a fact. I think this is a pi times. So if we do this right, this is pi times this. And the L is pi times. The Lj is pi times beta j. OK, forgive me. OK, so the sufficiency is based on two ideas. So if I have a certain cone surface with a given arrangement of angles, so the first fact they use is that if a vector beta is attained, by which I mean there exists a cone surface with those cone angles, then if and beta is in the interior of this region of the constraint set, then all points, all, let's say, beta primes, which are near to beta, also attained. OK, so this is a fact due to low. And this is a purely, I feel like algebraic. I mean, he works in the character variety, and he's just able to sort of check this openness that if you're in the interior of this polyhedral region that you have an openness condition. We'll be able to recover that analytically later, but the original proof of that was algebraic, as it were. OK, so that's one thing. And then the second thing is that if I have a given cone surface, then what I do, if I want to get an inductive argument, so if I want to say that there's a certain set of angles which are attained, then I can sort of take this cone angle and I can split it into two. So it's kind of an odd picture, but cone points can be split into things with smaller cone angles. Now, you may wonder what that looks like. It looks kind of like that. How about an analytic expression? Well, that's easy to write down. Suppose we go to flat metrics instead of spherical metrics. And remember, spherical metrics are just a small perturbation locally of flat metrics, so it shouldn't matter very much. We can think about flat metrics. So suppose I have two points, p and q. And I want to have a flat metric which makes cone angles of both p and q. We had a very explicit expression for this, up to smooth factors, which was times, let's say, delta. This is the Euclidean metric. So u sub pq was equal to just beta 1 log of mod z minus p plus beta 2 times log of mod z minus q. So in other words, if I exponentiate this, this is z minus p to the 2 beta 1 mod z minus q to the 2 beta 2 times dz squared. So there is a flat metric which has two cone points, and I can let p and q converge to one another. And what happens is the beta's add. Now remember, the betas are not the cone angles. It's beta plus 1, 2 pi times beta plus 1. So the formula for how cone angles add is a little bit weird. So cone angle 2 pi beta 1 plus 1 combined with, I don't know, union 2 pi times beta 2 plus 1 is going to result in a limit in 2 pi times beta 1 plus beta 2 plus 1. So in other words, you subtract off a 2 pi when these things come together. This is a flat metric, because that was what we'd done before. This is an explicit solution to the curvature equation nearby with explicit cone angles. This isn't global, but this is a nice one. You mean for the spherical metric? Yeah, well, it doesn't matter. So there's a smooth factor times this locally, and that's all I was caring about. So the point is this is, cone points can coalesce, and yeah, so basically that's the local surgery in the flat case. And there's a corresponding local surgery in the spherical case. So you can find spherical metrics, which sort of look like this and converge to this. Now, the point is you build local models. Are they what I mean? No, of course not. No, they're not. Outside of it. No, they're not. And so there's some sort of surgery. Now, you need to do something globally. That's exactly right. And so they're actually combining 1 and 2. And so what they show is that if I replace this by this, then if you compensate by changing all of the other cone angles, you can make this match up. So you have this region down here and this region down here. So there's the rest of the surface with all this other cone angles. And if I open this up to two different things here, and I compensate by changing these cone angles in a small, do this by a small amount, I can glue these matched smoothly. So that's really the point of their proof. Long before. Can you about this constraint? No. The constraint set that? So he proved a fact that looks quite different, but they interpreted it in this context. So as I say, we can recover this analytically, but his fact looks to theorem about something else. But they. Well, what's it about? It's about the representations of pi 1. So he was looking at spherical cone metrics, but he was looking at it in a very, you're just looking at maps of the fundamental group. OK. OK, so this is a very schematic picture of what they did. So what you get here is that you have something with k singularities. You generate something with k plus 1 singularities. And then you have to do further combinatorics to say that you have this constraint set. So let me call it a, excuse me? Yeah, so it'll still be spherical. Yeah, so I mean this new thing that you generate will still be a spherical metric. Oh yeah, yeah, they still add up, because this is sort of an infinitesimal computation, that the only difference between the flat and the spherical case is some sort of smooth correction. So it doesn't affect the principle singularity. That's exactly right. So you have this constraint set. Let's call it p sub k. This is the constraint set. And you pick a beta, which is in the interior of the constraint set. And you generate, let's say, a beta primed. And this is, well, you've constructed a new spherical metric. We knew that that was a necessary condition, so that has to be in pk plus 1. And so they actually show that there's a combination of a continuity method plus the dimension counting and various other things that you can actually recover everything in the interior here by regenerating things here. So there's a lot of work to be done here. It's a lengthy paper. And there's a lot of sort of intricate combinatorial arguments. But the idea is that you can generate all possible cone angles of k plus 1 type from things of k type by this regeneration. Now they're not trying to reduce them to be less than 2 pi. They're just trying to reduce them to a case they already understood. So fewer numbers of points. You added one point. Because they understood this set inductively. And so they're getting to this set. So they understood k equals 3 really explicitly. They pasted together spherical triangles. And then k equals 4, they did it similarly. Then k equals 5, you're reduced to the k equals 4 case and so on. OK, so I gave a not the most coherent description of this, but hopefully give some sort of flavor of what they do. OK, so what I'd like to spend the last time today and then all of tomorrow talking about is how we can recover and extend these facts analytically. So the question that I would like to really describe is, so how do we recover this analytically? And if it were just a matter of recovering, it wouldn't be so interesting because their proof works perfectly well. But the hope is to extend it. So the more general question is which sets p1 through pk, beta 1 through beta k, are attained by spherical cone surfaces? And essentially what they've done is you have some set, let's call this script s, which is sitting inside the set of all k tuples of points and cone angles. So I can ask which sets actually correspond to spherical cone surfaces. What they describe is the projection of this set, called this s sub k. So Mandela and Pano describe the image of the projection, s sub k to p sub k. This is really to rk. And they say that the image is exactly p sub k. OK, so in other words, what are the conformal markings of the points? So what's the fiber? So given a pair of cone angles, which are the set of k's which are allowable. And more than that, you'd like to understand how many solutions there are and various subtleties like that. So this is maybe not the most central question, but it's one of these things that seems to be, it's an ancient problem. And it seems to be sort of still, the full answer is still beyond our hope. So at the end of the day, the story that we have right now is a bit complicated. So we don't have a full description here, but we know the s sub k is mostly a smooth set, but this sort of surprises that it has a stratified structure. So it has definite singularities that come about from an analytic obstruction. Now what these analytic obstructions relate to, geometrically, we don't understand. So I think that that's one of the big outstanding questions. So in other words, there's going to be some analytic condition on a solution that satisfies, lies in this set. And it disallows for a sort of smooth deformability. So there's not going to be a smooth module I pay the space near certain points. If we had a clean geometric description of what that would look like, I'd be very happy, but we just don't have that. So the theorem was that if the beta j satisfies some, they'd live in some. If they're in the interior, yes. There exists some collection of points, but we don't know where they are. You don't know where they are. The extended question is, given betas which set of points, what's the fibers of this mapping? Can we describe that? Yeah, so that actually is something I was going to get to. So here's an example where you see some sort of constraint. So there's actually a, you do not get a full space of deformation. So take two footballs. So this can have beta 1 and beta 2. This has to be beta 1 up here. And this has to be beta 2 again. So you have four points. And I take any short geodesic here and here of the same length, and I sew them together along that. So what I'm now going to have is something with two, four, six cone points. Two of the cone points are of angle 4 pi. So when I join the usual thing, join this to this side and this thing to this side, however we do it. I'm going to get two cone points of angle 4 pi, and then two with angle beta 1 and two with angle beta 2. So you can see what the deformations are here. And the things that you can explicitly do where you can change the length and you can change the direction at which geodesic it is, you can change this football only in a certain way. You can change the beta 1 at both sides, and you can change the beta 2 at both sides, right? But the points have to be sort of specified. So if you do a full dimension count, you just get too small of a dimension. So that's a very explicit construction. It is a closed set. So you can show that under general principles, it'll be probably can show by other methods. It's an analytic variety, an algebraic variety, maybe something like that. But it's certainly a closed set. And actually you can do that by sort of a geometric limiting argument too. If you have a sequence of spherical cone surfaces, then except for when the points sort of come together. I mean there's sort of compactness thermals which aren't too hard to obtain. OK, so the set PK is, well, so I didn't say one thing about it, so that you have the set PK and we describe what happens in the interior. So one result, which I forgot to mention, is that you can ask what happens when the betas lie in the boundary of the set, OK? So there were two results that came after the Manila Ponoff paper. So there was Kappavitch, Misha Kappavitch, and a student of his day, they wrote a couple of, they wrote a first paper then Kappavitch followed up, where they analyzed successfully not all the boundaries, but they did the principal boundaries and the vertices. So I think there's still some edges left unchecked. And the only possible thing that you can get there are footballs. So in other words, what that's saying is if you have a spherical cone surface, and this is sort of vindicating evidence, if you have a spherical cone surface or a sequence of these things, which is getting closer and closer to the boundary, or the betas are getting closer to the boundary, then the points have to be collapsing conformally to antipodal points, OK? So that's sort of a further indication that these points are going to be constrained. OK, so this is sort of what's in the literature now about the problem. Now I want to shift gears a little bit and talk for the last little bit today in the beginning of the lecture tomorrow about some of the analytic underpinnings. And the reason I have to do this is I have to sort of introduce a set of concepts so that before we can even describe the approach of how I attack this problem more generally, I need a certain framework of geometric language and analytic language that I have to develop a bit. So this is going to look quite different than what I talked about here. Let me go back to a story that I started two days ago, which was the theory of conic operators. So remember that if I have a cone point, I introduce polar coordinates, r and theta. And if I sort of explicitly replace the cone point by the surface, the circle, r equals 0, namely I'm sort of opening up that cone point. And I'll describe what I mean sort of more specifically, but this is sort of so I blow up the cone point. So right now this is just kind of an artifice, but we're going to be developing a more sophisticated version of this picture in a little bit. So we blow up this cone point, which is just replace the cone point by its unit normal bundle. So this is the surface r equals 0. And all the different points on here just correspond to all the different directions of approach to the cone point. So as I come in in various directions to the cone point, back in the original surface, they all go to the same cone point. Here they get separated out and they go to different points in the circle. So it's a very easy geometric instruction. This is a real blow up. Now this sort of signifies already that polar coordinates are going to be particularly useful. They're not lines. So this is different than a complex blow up in many ways. And it's also not real lines because, of course, if I have a line that goes across the cone point, I don't quite know how to continue it on the other side. So that's sort of another difference. It's exactly right. So when I write down the Laplacian for a conic metric, remember, it looked like 1 over well, if I write it in this form, d by dr squared plus 1 over r d by dr was a 1 over 1 plus beta squared r squared d by d theta squared. There might be higher order terms, depending on correction terms to the metric. And I rewrote this a couple of days ago in the following way. I wrote it as 1 over r squared times r d by dr squared plus 1 over 1 plus beta squared d by d theta squared plus higher order terms. And I pointed out that this kind of emphasizes that, well, if I'm interested in studying the mapping properties or regularity properties of solutions of the Laplacian, well, this is the primary guide of study. This factor is pretty extraneous. I can just multiply or divide by it. So that's not so important. But it points to the primacy of these vector fields, r d by dr and d by d theta. And I called the span of these things. The span of these things is I called vb. So what I mean is any vector field v is in vb if v looks like some smooth function, a of r theta times r d by dr plus b of r theta d by d theta. And I can talk about differential operators, which are just sums of products of these vector fields. So I can talk about an operator which looks like a jl of r and theta r dr to the j d by d theta to the l. And the Laplacian up to this factor is one of those. OK, so yeah, that's exactly right. So the way to describe them invariantly is by blowing up all the cone points, I have a manifold with boundary. Now I have a surface with boundary. I can take the space of all smooth vector fields, which are tangent to the boundary. So in the interior, they're completely arbitrary. And at the boundary, they have to be tangent. And so that just forces the radial part to have a vanishing coefficient. OK, so the analytic question, which there's quite a complete solution, is what do inverses or partial inverses of conic elliptic operators look like? And when I say what do they look like, I mean precisely if I write this as an integral kernel, it's going to be a function of two sets of variables. So in other words, suppose I have the Laplacian. So suppose I take the Laplacian. And I'm trying to solve Laplacian equals f. Ideally, I'd write u is equal to g of f. This is just a symbolic way of writing that u of r and theta is equal to the integral g of r theta r tilde theta tilde f of r tilde theta tilde dr tilde, various normalizations, let's say dr tilde d theta tilde. So this is just like the Green's function. So what I'm asking is, what is the integral kernel of the Green's function for the Laplacian on an iconic manifold? So why am I asking this? Well, because I'm interested in things like the regularity theorem I wrote down, the Fredholm theorem I wrote down the other day. So all of those things sort of went into this business. And when I analyze the modular space of spherical cone metrics, I need to understand those questions and for more complicated operators, too. So what I want to spend a little bit of time talking about is what's known about this. And in some sense, this is understood completely. So first, a small review about what do you know in the smooth case. So first of all, just recall, so if I just have a smooth manifold and let's say the Laplacian on it, and I'm interested in understanding the Green's function for the Laplacian, it's really I'm trying to find some operator which satisfies this equation, or maybe this plus a projection onto the null space. This should be a projection onto the kernel or co-kernel. Say this would be the co-kernel in this case. This is an operator theoretic equation. I can think of this as an equation on the level of distributions. So if I think of the variables as being x, then I think of g as a distribution, values in x and x tilde, and this is going to be the delta function of x minus x tilde plus some smoothing kernels. So really, what I'm trying to solve is that equation. So if you've never seen something like this, this looks a little bit weird. But this is the integral kernel for the identity operator. And when I write that delta of x minus x tilde, I'm cheating a little bit because I'm using a local coordinate system. And there's a way of sort of turning this into an invariant distribution on m cross m. So the identity as an operator from L2 of m, let's say, to L2 of m corresponds to an integral kernel. And the integral kernel happens to be a distribution. It's just the delta function along the diagonal. Well, there's a nice general theorem that says any operator from functions on m to functions in m corresponds to a distribution on m cross m. And if you're an inverse to an elliptic differential operator, then this distribution, so here's a copy of m cross m, has some very special properties. Namely, it's going to be smooth away from the diagonal. This is the diagonal. This is the set of points x comma x in m cross m. And along m cross m, it has a very specific structure. It can blow up or it can decay depending on various things. But the example that you all know is the Newtonian potential, so something like x minus x tilde to the 2 minus m. So that's something that is smooth away from the diagonal. And along the diagonal, it doesn't make sense point-wise, but this is a well-defined distribution. So the whole one perspective of looking at suited differential operators is you want to say, what are the types of distributions that can occur as inverses of differential operators? How do we describe their singularities? And they're sort of a very classical theory that goes back 50 years on that. And roughly speaking, they all look kind of like that. They all look like powers of the distance functions to the diagonal, raised to powers which can be positive or negative. More generally, they look like asymptotic sums of those things. So on a conic space, you have the same story, but it's enhanced a little bit. So if m is conic, and I'm going to finish in just a couple of minutes, and I'll pick this up and describe it in a slightly more systematic way tomorrow. But if m is conic, and then I want to find, I want to solve this equation, this Laplacian has singularities. I've written it down here. So it has the sort of squashed singularity in the normal direction. What that suggests is if I look at m cross m, so now I'm going to draw a picture of a quadrant. So this is r r tilde theta m theta tilde. So m cross m, in the smooth boundaryless case, m cross m is just another smooth compact band and pulled without any boundary. In this case, m has a boundary, this blow up of m has a boundary, m cross m has a corner. So r goes to 0, r tilde goes to 0, and this is the corner. So it has a boundary hypersurface, it has another boundary hypersurface, and they need it a corner. Where's the diagonal in this picture? It's something that comes down like this. So it's a, in this case, it's a two-dimensional sub-manifold of this overall four-dimensional manifold of the corners. This is the diagonal. And it comes down and it intersects the boundary of this whole thing exactly at the corner. And this is what's called a non-clean intersection. So this is kind of a terrible intersection here because it's not happening in a product type way. Okay, so the basic principle is that the green's function here is gonna be a function of r theta, r tilde, m theta tilde. It should be a distribution on this space and we'd like to understand what does it singularities look like? Where is it singular? So the question is where and how is g singular on m cross m? The answer is that there are three types of singularities. And so let me redraw this picture again. So there's a singularity along the diagonal. And that singularity is gonna look exactly like this. So there's no surprise, it just, it looks like as it would in the interior. There's gonna be a singularity along each boundary face. This corresponds to, so I won't write down g again, this corresponds to either r or r tilde going to zero independently, but not simultaneously. And then there's gonna be a worse singularity that happens as r and r tilde go to zero simultaneously. So as I go down to the corner from any direction, so I have this perfectly nice distribution, it's smooth in the interior away from the diagonal. And all I'm asking is, what does it look like as I go to these various boundary faces of the diagonal? And the answer is it's a bit of a mess. So how do I resolve that? And the answer is surprisingly simple. So the answer is, suppose I take this manifold of corner, so there's r, r tilde, theta and theta tilde. So everybody see what this is? This is just a sector. And I introduce polar coordinates around the corner. So I blow up the corner in exactly the same way that I blew up the conic point. So I replace this by a picture that looks a bit more complicated. So now I have something that has one more boundary component. The diagonal goes off like this, but it intersects in the middle of the corner. So this is the important picture to understand. So I've taken each point of this corner and I've replaced it by its inward-pointing spherical normal bundle. So in this case, they're just a quarter circle. Every point in this corner gets replaced by a quarter circle. These things fit together to a boundary. And it attaches to these other two boundaries and I have a new space. So it looks like a bizarre picture. Why would I do that? It'll get worse tomorrow, so don't worry. And so the theorem, and this is an old theorem, it goes back 30 years now, is that if the Green's function of a conical plosion is, well, as smooth as it could be on this space. So what it means is that in this sort of these funny polar coordinates for this space, it sort of has a good asymptotic expansion at this boundary and at this front face and at the diagonal it looks just like the Newtonian potential. So in other words, in this picture, there was sort of this weird squashed up singularities that went into the corner and I've resolved it by passing to a more complicated space. So I have slightly more complicated geometry, but it simplifies analytic description on these lines. So, okay, so what I have when I say there's a smooth expansion here, a smooth expansion at this space, they sort of are independent of one another and at the corner it's just like a product expansion. And that's sort of how you know when you've met with success in this kind of story is you have this a priori complicated object and lifted to this space, it looks as simple as it possibly could be. Okay, so it has product type expansions at the corners and just smooth expansion, well, smooth like expansions at the boundary phases. Okay, and I've accomplished this by blow up. Okay, so this is the theme that I'll be talking about tomorrow is that I have a priori very complicated analytic objects and I simplify them by this geometric blow up construction. Okay, and in the last, so sorry, I've gone over a couple of minutes. Let me just say two more words. What I'm aiming for is the following. And this sort of justifies introducing this language of blow ups. So I'm looking at the collection of points and betas and I wanna know which of these occur. So in fact, the way I study this is by looking at the configuration space of all points. So this is really gonna be some large surf, some large space which looks like M to the K, but I actually have to remove all the partial diagonals. So I remove all partial diagonals where any of the P's equal one another. Okay, so I've taken the space M raised to the power K, that's the K tuples. Of course, I have to divide by the symmetric group because they're unordered. I remove all the cases where any of the P's are equal to one another. And above each point, I have a curve. I have the sphere minus exactly those points. So this is just sort of the universal curve over the simple part of the configuration space. So the picture that we develop is that there's a way of compactifying this open set and correspondingly compactify this universal family of curves to, well, something that looks a lot worse than that, but it's a massive blow up of this space. And the main theorem is that these families of spherical metrics, conic metrics, exist as sort of regular objects on these spaces. So in other words, what this does is it gives very precise meaning to if I have two conic, if I have a family of conic metrics where these points are coalescing or if I have any collection of points which are coalescing, it's actually happening in a very smooth way once I lift it up to this resolved picture. So I'll describe that tomorrow.