 Okay, so let me remind you what I did yesterday. So, we have this basic invariant, we have a surface, k points fixed, p1 through pk on m, and we have these parameters, beta1 through beta k, and we're looking for conic metrics with cone angle 2 pi times 1 plus beta j, and so we have the fundamental invariant, chi of m plus the sum of beta j, and this is, I can write this as chi of m beta, sort of vector beta here, so this is like an orbifold, an Euler characteristic or conic Euler characteristic, and so in particular the sign of this determines if I'm going to have a chance of finding a constant curvature metric, it's going to determine the sign of the curvature. So, and this came up, and this naturally appeared when I applied Gauss-Benet on the complement of small disks and took a limit. Okay, so yesterday we talked about the case where chi of m, so first we talked about the case where chi of m beta equals zero, and then we have a linear problem, purely linear problem, and we found that the solution, so remember that I, so there I started with let's say g tilde which is equal to a smooth metric on m, and I'm trying to solve, so the way I formulated it was Laplacian g tilde view is equal to k g tilde, but then I added the sum beta j times delta pj. Okay, so I thought of the curvature as being concentrated at a delta function, so the curvature of the conic metric as being concentrated at the delta functions, and this formulation of it gave me a good insight that u should be written as the sum of beta j times, and I guess I wanted it 2 pi here, sorry, beta j times the greens function, so this is just a green function for g tilde, j goes from 1 to k plus u tilde where u tilde is smooth. Okay, so this green function has asymptotics and local coordinates, so g of z pj, so in coordinates where pj is equal to zero, this looks like log of mod z. Okay, so altogether when I look at e to the 2u times g tilde, this up to some smooth multiple is going to look like mod z to the 2 beta j times mod dz squared, so this was the other form of conic metric, so conformally in two dimensions it's just conformally smooth, and all the action happens in some sort of conformal, you know, in some sort of extra smooth factor, so smooth positive factor there, but this captures the conic singularity completely. Okay, so it's a linear problem, and the, you know, we solved this using more or less garden variety elliptic analysis just using the fact that the right-hand side is in some negative order so-bluff space, and then we had a good guess what the solution could be, and then applied elliptic regularity to say that the remainder term was smooth. Okay, so that was the linear, the linear case when chi is zero, then the next case when chi of m beta is negative, we have to solve a genuinely nonlinear equation, so we let g not be conic with the right cone angles, so we have a good guess for how to write down the cone angles, we just choose a metric that looks like this, knew each of the conic points, and we perturb off of it, so I'm trying to look for g which is e to the 2u g not, and so then we have to solve Laplace and g not u minus k not minus e to the 2u is equal to zero, and so then we followed exactly the same procedure as in the smooth case, namely we used the method of barriers, we could find sub and super solutions to this equation, okay, so we use the method of barriers, and the new ingredient here was that we're solving the Laplacian or Laplacian minus lambda for a conic metric, for a metric of conic singularities, and I stated the basic Fredholm theorem here, okay, now I'm going to be talking about that in greater detail tomorrow, so I want to pursue sort of further geometric aspects today, but the key to this was to understand the mapping properties of Laplacian g not or Laplacian g not minus lambda, so we need to understand this, this was a singular operator which remember if I wrote it in polar coordinates near z equals zero, if I used polar coordinates in a, well, geodesic polar coordinates for the metric g not, this thing looked like, so Laplacian g not looks like d by dr squared plus 1 over r d by dr plus 1 over beta, let me see, 1 plus beta, j squared r squared d by d theta squared plus higher order terms, okay, so it's sort of a regular singular type operator, operators of this type are called conic, I explained sort of the larger context of general nth order conic operators, and I hinted at the fact that they're sort of a general strategy, general methodology for dealing with these that parallels what one does for operators in smooth manifold to serve a certain change, a turn of mind, which is that you have a calculus of suited differential operators, which are just operators which generalize the green function, okay, so they're integral operators that have certain singularities along the diagonal, and there's a very well understood and classical picture that goes back not quite centuries, but decades, many decades, and there's an analog of all of that for, for conic operators, and as I suggested also for operators with more complicated types of singularities, okay, so we have some strategy for understanding this, and we also had a regularity theorem, so that was sort of the other big point, and that told me how when I solved the successive stages of the iteration, so if I solved a plus and g naught minus lambda of u j plus 1 is equal to whatever the right hand side was, f of x u j, so I had to solve equations like this iteratively, I start off with u naught being something, you know, maybe the initial guy is some u bar, the sub solution, this was smooth, and then that actually told me exactly how regular u 1 was, and then u 2, and then u 3, so we had precise regularity statements all along the way, so each of these u j's look like a constant, so a 0, this depends on j, plus a 11 j times cosine theta plus a 12 j sine theta times r to the 1 over 1 plus beta j plus, oh excuse me, different j's, I apologize, sorry, I'll call this L, so this is the term of the sequence, and then I have the jth cone angle, I'm sorry, plus big O of r squared, okay, so I had a very precise regularity statement at every stage of the problem, so this is just a linear statement about regularity, and then I used this to derive a maximum principle, so remember the problem was that the maximum or minimum might occur at the cone points, and we used this regularity statement to prevent that, and so we finally got a sequence of functions, so that u bar is equal to u naught less than u 1 is less or equal to u upper bar, and that allowed us to take a convergent subsequence, and finally we were going to get the same, so we get a solution to this equation, that's sort of the limit of all of these iterative equations, so if I take the limit as L goes to infinity in these guys, and then finally the thing which I'd stated, and I sort of conflated the two things, but I have exactly the same sort of regularity statement for the actual solution of the nonlinear problem, okay, so these were all the steps that we did for the, for the negative case, and the thing I want to stress is it's basically linear analysis, I mean of course it's a little bit trumped up, so to speak, but it's basically linear analysis, and it's not very sophisticated PDE, okay, so then the last thing I did yesterday was starting to look at the case chi of m beta is positive, and I made the assumption that all of the beta j's are negative, which corresponds to the cone angles less than 2 pi, and this corresponds to the really important geometric property that if I take a minimizing geodesic from any point to any other point, it never goes through a cone point in its interior, okay, so it's a very geometric property, and that follows directly from this hypothesis here, so and then what I concluded with it was this beautiful fact that, so let me use this as my starting point for the next discussion, so the beautiful fact is that if there exists a spherical cone metric g with, so spherical just means k is equal to plus one, so if there exists a spherical cone metric g with all cone angles less than 2 pi, that corresponds to all beta j is negative, then you have the following inequalities that beta j is greater than the sum i not equal to j of beta i, and this is true for all j, okay, and these are called the Troinov constraints, and the way I derived it was kind of this very geometric way, I sort of talked about the Dirichlet polygon for the spherical cone surface, so I took the spherical cone surface and developed it onto the football, so I could sort of think about the whole surface as living as being sort of obtained by edge identifications from some sort of polygon, which lives in the football, and then the defect between this and this is exactly that area, which is positive, okay, so it's a very geometric idea. Now Troinov when he proved the theorem of existence, so what I stated, and we'll sketch the proof of today, is that this turns out to be necessary and sufficient for existence in the case where the cone angles are less than 2 pi, so the proof is extremely different than anything I've done so far, and he came across this condition from a very different point of view. Okay, so I'm now going to talk about existence, and I'm first going to talk about a strategy that one can use even in the negative case, it turns out to be much easier, but suppose I have two metrics, so again I'm going to have g which is equal to e to the 2 u g naught, same sort of thing, and suppose I want g to have some constant curvature as a specified function, it may or may not be constant, so this is an old problem, the Nirenberg problem, very closely related, and so the way that it's often been attacked in positive curvature is using the calculus of variations. So I'm going to write down an energy functional, so j sub k of u, and so this is going to assume that the areas are one, so there's different ways to write down this functional and you'd think that this is a trivial change, and it is, except for the non-trivial part, is keeping track of where the constants go, which is beyond me, right, and it turns out that the exact value of the constants at certain locations, certain places in this formula I'm going to write down are as deeply significant. Okay, so here's, with this normalization, here's the 2 k naught u, and then I'm going to have minus, let me get, make sure I have this sign right, again I have to, okay so it's minus k, sorry I'm just trying to think if I want to put this in. Well, let me do it this way, so minus, excuse me, minus k times log of integral u to the 2 u, I guess I want to do it that way. Yeah, so I'm going to assume here that k is constant but k naught is not, so let's just suppose that we're trying to solve uniformization on the sphere, right, with k is equal to constant, k naught is, you know, just whatever it is, we're on a smooth metric, and you write down this energy function, okay. So, there's a slightly modified form of this if you're trying to prescribe if k is a function, then k obviously has to go inside the integral in a certain way, but I will, I'll admit that. Okay, so let's just check that this has the right property, namely, I want to compute its Euler Lagrange equation. So, suppose that, suppose we can prove that j sub k view is greater than or equal to some constant for all u, and I'll be a little bit fuzzy about which u's I'm taking but obviously I need the gradient of u to be an L2, I need this e to the 2 u to be integrable and so on, okay. So, suppose that that's true, and suppose we're even luckier and find out that we can find a minimum of this function. So, we need this to be bounded below and we also need a minimum, okay. Suppose further that some given function, maybe I'll call it u naught minimizes j sub k. So, then, you know, you do the standard trick which is I take j sub k of u naught plus epsilon v. This is always greater than or equal to j sub k of u naught. So, this is just a standard way of calculating the Euler Lagrange equation. So, this tells me that it's necessary that d by d epsilon to d epsilon equals 0 of j sub k of u naught plus epsilon v is equal to 0. And we can just go through and see what this derivative is. Here it is. It's going to be twice the integral of grad u naught dot grad v. That's from the first term. Minus 2 k naught v. That's from this term. And then here I'm going to have minus, sorry, I'm splitting it onto two lines, minus k times 2 times, and so I'm going to have 1 over the integral of e to the 2u because of the log and then times e to the 2u times v. And that's an integral here. So, this is supposed to equal 0 for every choice of v. And if we isolate the u naught, so, you know, this is the same as saying that the integral, so I have a factor of 2 that's common to everybody. And, oh dear, did I get my signs right? I certainly hope I got my signs right. I think I needed this to be a plus. Yeah, forgive me. Okay, so that's going to say the following, that I'm going to have minus Laplace in u naught plus 2k naught minus 2k naught plus this, excuse me, minus k naught plus k divided by this number times e to the 2u, all multiplying v is equal to 0 for all v. And that implies that this expression is equal to 0. That's exactly the equation I want. This one. Hold on for a second. So this should be a plus. That should be a plus. That should be a plus and that's what I want. Yes? Okay. Now let me see. So maybe I had the signs right. So, now I've lost it completely. Okay, so let's see. So I want these signs to be opposite. Plus gets when I integrate by parts, I want that to be a minus. And I want this to have that when I integrate by parts that to have the same sign as that. So, oh right, so this is a minus because that was a minus there. Saved. Okay. Okay, so we get this equation and so the point is if we can find a minimizer, then we're in good shape. Okay. Now, what's hard about this functional is the following. That you have this term. Please, yeah. So the area form with respect to G0. G0. So that's why I'm assuming that. So there's extra terms which are sort of makes these integrals averages if the area is non-zero. If the area is not equal to one. That's dividing by the area of the new metric. So the integral of e to the 2u is the integral of the new metric. That's exactly right. Right, so right now I'm on the smooth surface. So right now I'm going to assume that G0 is smooth and then we're going to modify this. Yeah, so then no boundary terms. And in fact there won't be, and that has to be justified in the conic case. Yeah, but that's a good point. Okay, so it turns out that the crucial thing is that you have these two competing quantities. And this is, you know, if you can prove this, then this is sort of, I won't say easy, but it follows some more standard functional analysis. So the question is why is this bound to blow? And so you have, this is a nice positive term. This is not very serious, right? But this is a possibly very unbounded negative term. Okay, so it turns out that there is a fundamental inequality called the Moser-Trödinger inequality, which says the following that, so the logarithm of the integral of e to the 2 times u minus its average is bounded by 1 over 4 pi times the integral of grad u squared plus constant. Now, I'm not going to describe the proof of this. I'll take this as a black box. This is written in many places. There's a nice book by Alice Chang that describes this in great detail. And I think I have it here. It's called, it's a little book by the European Math Society. And I don't have the title off the top of my head. But this is a very standard fact. Some version of it is in Gilbarg and Schrödinger and so on. Okay, so this is, you know, interesting inequality. It follows from, and is related to various other standard things. But the important thing here is this constant, 1 over 4 pi. So what you do, so first of all, let me just sort of point out that when we combine these two things here. So let's see, where do we, where do I start here? So suppose that if u bar, which is equal to the average of u divided by the area, is equal to 0. So just for simplicity, let's just consider functions whose average is 0. So then that's exactly the term that appears right here, okay? And so then what that's going to tell me is that the integral of grad u squared minus k times log of integral of e to the 2u, which is the thing that appears here, is going to be greater than or equal to. So I'm going to have a 1 minus k over 4 pi times the times, yeah, times the integral of grad u squared, sorry. And then minus a constant, okay? So the point is here that you need k to be small. If k is 4 pi, it already looks kind of bad. If k is bigger than 4 pi, you're really in trouble, okay? It's going to have an unbounded blow. Okay, so somehow or another, the role of sharp constants has a lot to do with this. Now, what is k supposed to be here? Now, I'm going to remind you, we have the area is equal to 1 on a smooth surface. So the integral of k is equal to k times the area, which is equal to k because the area is 1. That implies that this integral is 2 pi times the Euler characteristic, so that's 4 pi. The k I have here is, in fact, exactly critical. So 4 pi is exactly critical. Okay, so what do we do? Well, now, two things. Suppose that we arrange things so that this constant is positive, right? Then I claim that you get this lower bound here. So the other term and the linear term in J sub k is, as I say, very easy to control. This Moser-Trudinger inequality implies that we have a lower bound for this thing for all u and h1, so one derivative square integrable. And then we apply various compactness theorems, so it's syllable of embedding and various versions of that to say that if I have a minimizing sequence, then we can extract a convergent subsequence. Okay, so really the whole key is this, what's called coercivity, trying to get a lower bound on this functional. So, now, how do I even do this in a smooth case where k is equal to 4 pi? So this leads people to sort of worry about, can I make the constant better? Okay, can I make this constant better? Now, it turns out that there are various improvements. So this is exactly the right constant on Euclidean space, and on domains in Euclidean space, and then you can sort of try to see what happens on surfaces, and it turns out that on the sphere, for example, you can actually get a better constant. So here's a fact. On S2, for actually a slightly more general class of things, so I can actually put in a variable function right here that would correspond to prescribing the scalar curvature instead of looking for something, you would actually, you can get a slightly better thing under a certain assumption. So if k of x is equal to its value at the antipodal point, then the constant is 8 pi. Okay, so you actually get a better inequality, you get a smaller right-hand side, and that allows you to put in 1 over 8 pi here and you're fine. Now, when k is constant, this is certainly true, but this theorem is applied in a more general context when you're trying to prescribe Gaussian curvature. Okay, so this just shows that you take the basic inequality where you already have to work fairly hard to get the constant 4 pi, and then you have to work even harder to improve the constant. Okay, so this is the strategy. You get a lower bound, then you apply various compactness theorems and you get a minimizer and you've solved your problem of the new guy, of the thing you're trying to find. Okay, so we want something that has curvature 4 pi because that's going to be the standard constant curvature on a sphere of area 1. Okay, so suppose we do the same thing on the sphere, so I'm going to now immediately assume that my manifold is the sphere, so this turns out to be the interesting case, so I assume, as I was doing here, but now with conic points we didn't need to make this case, so I assume the m is equal to s2. I'll explain why I can reduce to this case. And I'm going to look at the conic situation. So I assume that all beta j's are negative and I want to do exactly the same thing. So I'm going to write down the same j sub k view, the same energy. And we could follow the same strategy. You have to worry a little bit about sobel of embedding theorems and, well, first of all, you have to worry about is there an inequality like this? And there is, and in fact there's an even better one as the point. Okay, or worse, depending on how you look at it. But anyway, there's an inequality like this, and then all the theorems that lead to sort of compactness and being able to extract a convergent subsequence from a minimizing sequence go through. I mean, you have to do a little bit of work, but it's not very exciting. It's pretty straightforward. Okay, so the big thing is, is the functional bounded from below? Okay, so what Troinov discovered, so this is a paper in the late 80s. It was his thesis. And what he discovered was that there is actually a better, was returning inequality, and let me write it down for you. So his, so I'm going to define a quantity, tau, and this is a function of beta, and it's going to be the minimum of 2 and 2 plus twice the minimum of the beta j's, beta i's. Okay, so there's something local about this because I'm not looking at every, so this is the minimum of these two things. So I take the quantity, so all of the betas are negative. So just by definition, so he was trying to consider the case where some of the betas could be positive. Some of the betas are negative, then this is automatically the guy that wins, right? And so this is equal to 2 plus twice the minimum, 2 times 1 plus the minimum of the beta i's. This is strictly less than 2. And so his big theorem was the following that, so his real theorem was that you have, there is a motor, motor Trüdinger inequality, 4 pi replaced by 2 pi times this tau. Okay, so in other words, it's worse. That's 4 pi times tau is less than 4 pi in our case because this thing is less than 2, so 1 over it is bigger. So it's worse inequality, but there's sort of different amount of room. Okay, so what happens if we try to do the same thing, if we try to understand the coercivity of this functional? So you run through exactly the same argument. So suppose you have unit area. What is the integral of k? Well, it's no longer 4 pi, but it's 2 pi times the conic Euler characteristic. So what we need, so what we need for coercivity for lower boundedness for j sub k is greater or equal to minus c, that, well, it's going to be 1 minus, and then I'm going to have the conic Euler characteristic times 2 pi divided by 2 pi times tau of beta is strictly greater than 0. So, namely, this is the new constant that comes out in front. So this is the replacement for this right here. So we no longer have any, we can't fool around with sort of talking about even functions or anything like that because I'm on a conic surface, and this is the quantity that we need. And so that's, of course, exactly the statement that we're talking about is less than tau of beta. And let's write that out. So that's, well, chi of m is 2 because we're on the sphere. So I have 2 plus the sum beta i, i goes from 1 to k, and this is supposed to be strictly less than 2 plus the minimum of beta i. Okay, the 2's cancel. And let's suppose that this is a cane that's some fixed index j. So I'm going to have the sum i goes from 1 to k of beta i is less than 2 times beta j for some particular j. That's the guy that realizes the minimum. And then I take it away from this side. What I'm going to get is that beta j is greater than the sum i goes from 1 to k, i is not equal to j of beta i, and that is exactly the Troinoff constraint. So here we've seen it exactly analytically. So something which we saw geometrically before appears as getting this analytic condition here. Okay? Yeah, so in fact there's some negative beta i. And so his theorem applied, right? No, so in fact, yeah, so basically that's right. That's exactly right. But this is the condition that we really need. Okay, so when we're not the sphere, I'm going to actually show you a different method to get existence no matter what. But anyway, so this is the condition, and it turns out to be exactly the same as that geometric condition I got before. So I think this is a very beautiful correspondence because something that we saw from a Gauss-Bene calculation appears here. Okay, now I'm not going to get into the guts of that proof because it's one of those sort of lengthy things and I'd spend the rest of the hour doing that. So just to sort of sketch that that's how the existence goes. So in the case where all the beta i's are negative and this condition is satisfied, you have a lower bound from this and then you apply the standard method of the calculus of variations to get an infim and there's your solution. So last bit about last fact about all beta i negative, let me say last facts about all beta i less than zero. So when does you have uniqueness in all cases? Okay, well, I'm lying because I told you that in the flat case we have no way to normalize so you can take any flat conic metric and scale it. So let's forget about that. But in the hyperbolic case, in the constant curvature minus one case we can get uniqueness using the maximum principle. And in the spherical case, well it's not at all obvious. I told you the maximum principle just doesn't work here and in fact it's sort of a harder theorem. So in k equals plus one, this is due to Luo and Tian in the late 90s. Okay, and they were adapting ideas from higher dimensional scalar geometry, sort of uniqueness there for scalar Einstein metrics that they made work in this one complex dimensional singular context. Okay, and again it's somewhat long proof. I suspect that there's probably a much shorter geometric proof and actually I was just talking about that with Alex yesterday, but I don't know one right off the top of my head. Okay, so that's one fact. The second thing you might ask is what about the moduli of these metrics. So I've sort of given you an existence theorem but I haven't told you how they fit together. So there's a theorem that the moduli space, so let me just sort of state very roughly there is a good Teichner theory in the sense that I can define a space so I'm going to call it M-conic and maybe I should call it M-conic less than 2 pi. Okay, and in fact I want to throw all the possible parameters in here. So the point is that one thing that's interesting here is you have the genus of the surface and that constrains things in the normal smooth case. Here you can vary the cone angles and if you just think about chi of M-beta, for a given surface even if the Euler characteristic is negative I can choose cone angles even less than 2 pi. I can add a whole, let me see, so when I'm on the sphere I can make this positive or zero or negative. So in principle if I look at this whole space here and I don't normalize the curvature to be one minus one or zero, I can sort of get a whole continuum and so the claim is this is a smooth manifold of dimension. Okay, so what should be its dimension? So 6g minus 6 for the underlying moduli. So let me talk about it when you have chi. Genus, so 6g minus 6. Then I'm going to have k points wandering around and then each point has a cone angle, so plus 3k. And in fact you can say much more it sort of fibers over and the sort of point is that something happens in this cone angle less than 2 pi case. So namely it looks perfectly nice the one sort of nice observation about this so this is something I proved with apartment advice. Because 2k, because you have k points with two parameters, excuse me, yes I'm doing real dimension, two points but then each one has a cone angle. So if I just say chronic but without specifying what the cone angle is so I'm allowing any curvature or any cone angle so the point is if I fix the area to be something, I can either sort of fix the area, fix the curvature, fix neither but this is sort of the right dimension count when I am. k is the number of points so I should have put that maybe chronic k, too many indices but I'm letting beta be anything less than 2 but less than 0. That's exactly right. So the one interesting observation about this so as I say it's not a well it's an interesting space but it's more interesting than the Teichner space itself is or a modular space itself I can mod out by mapping class group but the interesting thing is that this interpolates between the smooth case that's all beta j are equal to 0 and the standard marked Teichner space so the marked case that's all beta j equal to minus 1. So I didn't explicitly talk about the minus 1 limit but what's happening is to a cone angle so here's our cone angle so you have two possible limits of this when beta j is less than 0. One is that this converges to just a cone over a circle radius circumference 2 pi that's a flat surface I mean it's perfectly smooth the other thing it can do is this cone angle can go to 0 and in fact what will happen is that this will then go to a hyperbolic cusp okay so suppose that the Euler characteristic of m is negative so we know that there are in fact even on the sphere and you have enough points so you're in the stable range where you have a hyperbolic matrix so these cone metrics sort of naturally interpolate between smooth metrics and these things so buried in this statement is sort of an analytic proof of the fact that the space of a complete matrix is a nice modular space of course well known but it actually bizarrely didn't seem to have been done in the sort of using methods of geometric analysis rather than complex analysis okay well so obstructions to the affirmation theory right so what's sort of behind this theorem here is I just look at the local deformations and you know basically it should be an implicit function theorem argument so I have a bunch of parameters so think about what you might want to do so suppose I fix so I have m and I fix the points p1 through pk beta 1 through beta k right and then I look at sort of nearby arrangements so I want to vary them now one thing that I can do is I can just sort of choose local variations to p's so I can let them vary so they're sort of k disks that they wander around in and I can let the cone angles vary in intervals so I have this sort of extra 3k dimensional deformation space now let's suppose that g0 is a solution for this given arrangement so this has constant curvature and the right cone angles at the right positions okay so then I vary things a little bit so I try to find so yeah so but that's an open condition that's sort of buried in here yeah so I mean this is subject to you know it's a smooth manifold but you know all the beta's have to satisfy Troinov but the point I should have said this also the Troinov condition is independent of the location of the points right so it just looks like you know so the point is that the extra k dimensions from the angles it just is sort of a product okay and now it's just sort of a piece of the cube instead of the whole cube when you're in the spherical case okay so I first find a family of metrics so let's call it g0 q alpha so let's suppose that q's are the deformation of the p's and the alphas are the deformation of the betas right so I just find metrics like this which are let's say constant curvature out here but then I sort of screw them up a little bit in these discs okay so I just vary them a little bit and then I try to find a new fat you know a new conformal factor to make this a new solution with the new cone angles and new positions okay now what I do is I you know what you'd like to do is just sort of linearize the problem and apply the impressive function term so it turns out that are a number of difficulties which are kind of analytic and have to do with this theory of conic operators but here's the key point here's the key problem that the this deformation theorem so we can apply the implicit function theorem provided and there's this kind of weird condition that Laplacian g0 plus 2 of phi equals 0 has no solution namely 2 is not in the spectrum now I have to talk when I'm on a conic surface I need to still talk a little bit about what I mean about the Laplacian is a self-adjoining operator so in some sense that involves imposing boundary conditions at the conic points so I haven't told you that yet but there's a way to make it into a self-adjoining operator applying what's called the Friedrich's extension it has a discrete spectrum looks just the same as before and it turns out that this condition is necessary and sufficient for this deformation theory to go through now here's another very nice fact that actually relates to this Alexandrov geometry that I talked about before so if the curvature is equal to plus 1 and all beta j's are negative then the spectrum of the Laplacian of minus Laplacian g equals 0 lambda 1, lambda 2 etc so it's a discrete set what would this be in the smooth case well you have 0 the first non-zero eigenvalue is 2 exactly the bad guy it turns out that under these conditions lambda 1 is strictly greater than 2 okay so that's an eigenvalue estimate that actually well you know I won't discuss the proof of this but there's a way of estimating eigenvalues using using integration by parts of the eigenfunctions and precisely because in the case where beta j's are negative this bad condition never happens and so you always have an unobstructed deformation theory and that's why this turns out to be a nice smooth moduli space become what? well so we're looking in the case where beta j's are negative so that they never allow quite to become smooth so that only happens in the limit of course you could do that at the boundary of this space I'll be talking about that later but you can do that of course but that's not sort of strictly included in this yeah you could that's exactly right well so you sort of think about the following thing is that you know the fiber you have at each point so you sort of think about let's suppose that we're on a g greater than 1 surface and you let the cone angles vary from 0 to 2 pi so you can sort of think of you have a k-dimensional cube which is a cube of cone angles up here at this angle where all the betas are 0 in the ordinary smooth taichmuller space you don't have the 3k that's exactly right so it's a higher co-dimension corner this is a cube of dimension 3k and you've actually excuse me this is all of those points have disappeared so it's a corner there and then down here you see a lower co-dimension corner where just the angles have gone to 0 but you still have the locations of the points so it's kind of a funny interpolation and on the edges you can let some of the angles disappear to 2 pi or not so you know it's a space that kind of contains these two different corners so that's all I meant yeah that's exactly right okay so that's sort of what's behind this deformation theorem okay so it's exactly this eigenvalue estimate and this is sort of the thing that you need to make the implicit function theorem work okay good any other questions so no that's typically it's you can do this within a conformal class yeah so absolutely you can so that's sort of another thing that makes this simple you can always stay within a conformal class and the way you can stay within a conformal class because you're again in two dimensions you can sort of take z minus this qj to a power 2 alpha j this is sort of the extra factor but that's a conformal factor but the underlying conformal metric is smooth it's fixed that's exactly right that's a good point the next thing I want so okay so there's the theory for cone angles less than 2 pi sort of fully now the one part which I've omitted is sort of the representation theoretic point of view and I'm actually going to be getting to that either beginning of the lecture tomorrow or at the end of the lecture today okay so there's a whole group representation point of view which is quite interesting and important for this as I made my disclaimer at the beginning this is a very analytic take on this whole subject okay now I want to pursue the thought that I talked about for proving existence in the spherical case to tell you about another very beautiful theorem by Alessandro Carlotto and Andrea Marchiotti in the case where cone angles are bigger than 2 pi so what can be done so the case that's really open to us and this is really what I'm going to focus on now is some or all beta j's are greater than 0 okay now what that's actually going to mean is that you're not going to have this coercivity or lower bound so the real condition that you're worried about is when is it true that chi of m beta is greater than or greater than or equal to this tau beta so this is the condition suppose we drop this condition so then you can actually check by very explicit example you can just write down explicit functions jk of u is unbounded below and let me sort of explain to you what are the bad guys suppose that so namely you can write down a function so here I have this conic surface it's doing whatever and suppose that here's a beta and this is sort of the thing that makes this condition violate okay so it's the minimum of the beta i's okay well what you can do is you can actually choose a u u sub lambda I'm going to call it and I can write down an explicit formula for you here um no I'm sorry I cannot write down an explicit formula uh okay so what it is is there's going to be a phenomenon of bubbling so you're going to find a u sub lambda and these things are sort of concentrating right here so you're going to find functions u sub lambda which are very small on the rest of the surface in fact I just transplant them to be supported near this particular cone point they're getting very big and then they go off to zero here and in the limit so there's an explicit formula and you know this formula is just something like so log of uh one plus uh lambda plus the distance from uh pj to z or something like that so this is not the exact formula but it has this kind of flavor okay so lambda appears in a very explicit way and it's a very elementary formula in terms of the distance on these surfaces so u sub lambda concentrates near this point pj and the limit e to the two u lambda times g naught is equal to well geometrically what's happening is that u lambda is going to zero everywhere else so everywhere everything else on the surface gets suppressed all I'm going to see is sort of the neighborhood of this and it's going to just correspond to a football so here's two pi times one plus beta j so in other words what's happening is that you just bubble off a football at that point lambda is going to infinity so this is sort of the positive curvature case very much so that's so we've solved the problem completely in the case of negative or zero curvature we found unique solutions so that can never happen so we're always in the positive curvature case okay so this is sort of the bad guy and that's exactly corresponding to this the energy of these u sub lambdas are going to minus infinity and you have bubbling okay so the theorem I want to explain briefly of Carlotto and Machiotti it's known and that was the theorem I already proved so if the chi of m beta so okay let me state this theorem and then you'll see how it's known through this theorem okay so the theorem of Carlotto and Machiotti is suppose that m is not equal to s2 so it exactly does not work in this case then jk of u has critical points which can be obtained by min max method okay so remember that to get solutions out of j sub k all I cared about was it was a critical point it was a minimum, great, that's certainly a critical point but as long as you find a critical point you're in good shape so what they needed to do is to find so you have a function that's unbounded above certainly it always is, an unbounded below and you have to find saddle points and they did this by a very interesting topological argument so there's a topological argument which is essentially more theoretic, so topological argument so we look at the space h1 of the surface okay and this is sort of where j is defined okay so j is defined in h1 so j maps h1 to r and it covers all of r it's unbounded below as well as above so what you're going to do is you're going to look at two level sets you're going to look at j less than or equal to some constant c so you're going to call this lambda sub c so you're going to look at the sub level set and you're going to look at two limiting cases so the first case is when c is very positive okay and then you can sort of explicitly show that the set is connected contractable so then lambda sub c is contractable okay and you just do that very explicitly you show that you can sort of deform one thing to another but you can always do it making the energy stay bounded below the two constants so the interesting thing is what happens when c is very negative so so if u is in j sub c where c is very negative then in fact you can get a profile for what u looks like so things that have very negative energy basically look like this okay well maybe not single bubbles but they look like multiple bubbles in fact then u is concentrated in l small regions that may or may not be clustered around conic points so small regions and so you know basically here's the surface without showing where the conic points are they're sort of clustered like this so in other words what's happening is roughly speaking as c goes to minus infinity the elements of the sub level set are getting very close to sort of formal sums of delta functions on the surface so now we want to understand what's the topology of the set j sub c and basically what they first prove is this lemma that u is up to sort of extremely small errors it can only be big in these sort of well separated regions and so then you can ask what's the topology of the space of formal barycenters and the number l is sort of predicted by c and the genus and so on so if I look at the set of a j times delta q j goes from one to l I just look at the set of formal points on the surface and I can ask does it have topology and the answer precisely when m is not equal to s2 is that you can actually calculate what the homotopy groups at that are so this is sort of this is non-trivial topologically okay so so the set of these things which is a very good approximation to elements of the sub level set have topology okay so you have the situation that the sub level set down here you can actually calculate some of its homotopy groups and they're non-trivial if I go up here right so suppose that you know pi sub 3 is non-zero pi sub 2 is non-zero okay if I go up here I have a contractable space so somewhere here there must be something happening to kill the topology and how you can detect that is you take two parameter family so if pi 2 is non-zero I take two parameter families of solutions so I take U of let me call my parameters T1 and T2 two parameter families of solutions down here and I know that they're contractable if I allow the energy to go up so I take the minimum on each element like this I take the maximum energy on each one of these families and then I take the minimum of all families so I take a non-trivial homotopy class down here so excuse me if this is an S2 right so I take a two parameter family right then I take a disc which fills it but it fills it in at the expense of going to very high energy okay so for every filling by a disc I take the maximum energy on that disc it's less than the big value this is plus C and this is minus C right and then I take the infimum over all such discs okay so this is a minimax strategy you have to worry about why should that point you found a sequence of functions why should they converge so you found a sequence of functions these correspond to you know the successive maxima of these various families and you know what I'm hoping is that there's some sort of mountain pass and I'm going to be hitting exactly that mountain pass okay so analysis yet again what they prove is that so for functions that look like this so basically they show that for functions that look like this these sort of well-centered clumps there's an even better constant in the Moser-Troetinger inequality so they get a so they improve the constant in Moser-Troetinger and what that's going to tell me so I mean that there's a lot that goes into here and so again I'm just giving you a very rough idea because of course these are the things with very negative energy these things do not have very negative energy but one dangerous situation that could happen is at the minimum of all of these maxima are still going to minus infinity are still getting very negative right but if they get very negative then they look like this and things that look like this have a better constant and you can get coercivity and you can usually can show that there's lower bound okay so it's a very intricate argument but basically they show that these minimaxes stay bounded below right so they can't get too high obviously and they can't go to minus infinity and then you have to do a little bit more work which is easier to say that they actually converge okay so it's a very clever minimax argument but it's using the fact that you have an improved constant that's really based on that's only valid for functions that are supported in these well well separated clumps okay so that's just an idea of what happens so after this theorem what this gives us is existence except for on the sphere okay doesn't say anything about uniqueness and that's sort of one of the big open questions but it just says I can choose the points anywhere I want I can run this you know this energy scheme and I find minimax solutions regardless of the points with a given cone angle okay doesn't say anything about uniqueness so what I'm going to be spending the rest of my lectures on is the last remaining case where I just have this sphere so let me just state this now so I have m is equal to s2 right arbitrary betas k is equal to plus 1 so obviously the betas have to satisfy the order characteristic constraint that 2 plus the sum of the betas is positive okay so the question is what can we say about existence or uniqueness so what I'll be talking about are two so let me just as a preview for what I'll be doing the next two days one is a quite elegant argument by Mandela and Ponov so this is a paper that really uses mostly synthetic geometry so just piecing together things and some sort of soft analytic arguments but really no real analysis that gives a gives a necessary and essentially sufficient condition on the set of cone angles so I can just ask forget about the points so forget about the points altogether and I just ask which k tuples of cone angles are necessary in order for there to be a solution now what I talked about yesterday was that there was this you know what I call a dual Gauss-Benet condition which turned into the Troinov condition so there were constraints so what are the generalizations of the Troinov constraints and they found those and they did this by converting to a problem of finding piecewise g desix on s3 so it turns out broken g desix on s3 and this turns out to be equivalent to this in a certain sense and that uses the Holonomi representation so I'll describe that tomorrow ok so that's the first step there's some modifications so this is a set of linear conditions and you have to understand what happens on the boundary they understood what happened in the interior and this was improved by Day and Kapovic so they gave some nice and very geometric arguments to say that what happens on the boundary is nothing interesting you don't get solutions everything degenerates to footballs ok and then the last thing and I'll stop here is can one specify so this is sort of part 2 can one specify the locations of the points and similarly can one count solutions ok so I'll give you what's known about this so this is sort of an ongoing project I'm doing with a so we have quite a lot of results about this so I'll describe these over the next two days it's sufficient yep yep not that I'm aware of they may well be I mean but the you know this thing about when you have sort of these multi bumps and you get sort of a better inequality is really based on sort of applying localized things and you get a constant that looks like if you have L points then the constant turns out to be something like 1 over 4 pi times L plus 1 or something like that so it's basically showing that all the error terms go away and that you really can just apply this L times over in a certain sense so but so the proof here about you know the constraints they give they're sort of complicated polyhedral constraints on the betas I don't know sort of the similar Gauss-Benet type argument for that but they transform it to something very geometric as I say you know they use the holonomia representation which I'll describe and they say that any spherical conic surface corresponds to you know a broken geodesic in S3 that closed polygon so I take you know a family of geodesics and it comes back and comes back to itself and then you can just study that problem abstractly about what are the closed polygonal paths in S3 and they're sort of generalized triangle inequalities for those and that's what these things are