 Thank you. Can you listen to me? Yeah. Okay. Okay, so I want to talk to you about complex differential equations It might seem at the first site like a little bit off Subject, but in fact when you think of the things the top the topics that the trend was Talking to us about in his course he talked about one career and the uniformization agreement surfaces and Complex differential equations are both a motivation and the tool to study these these things I will I will talk about other sorts of interactions between or the same probably between complex differential equations and structures geometric structures and curves and the things that interest me Are some special features that only some very special differential equations have So I won't be describing Sort of I will be far from the generic setting and I'm interested in knowing Understanding these very very particular differential equations. Okay, so let me begin so for for me at the French a complex differential equation will be some complex manifold and Some holomorphic vector field on this manifold Vector field is given once and for all so we will be talking in fact about autonomous complex differential equations. Okay, so Typically you have the local theorem that guarantees you so this is a complex manifold and this is a complex vector field so you have locally the You have your vector field. I will draw it like this although Okay, so you have An initial condition and the theorem on the existence and uniqueness of solutions tells you that here in C there is is some neighborhood of zero and some way to parametrize a small piece of small disc around your initial condition and This will be this will solve the differential equation meaning that if you take the vector field Here and you push it by this solution of the differential equation Then what you get is just the restriction of your vector field X to the image here. Okay, so that's the local Solution the local theorem on the existence and uniqueness of solutions. It tells you that Such a domain exists such map exists and that the germ at this point is unique. Okay so if when you do this in fact you can Start gluing all the local solutions and you can start pasting these local solutions and you will get when you take this maximal connected sets Given by the images of these local solutions. You will get the foliation of your manifold Foliation by complex one-dimensional leaves and there's lots of questions. You can ask about these leaves You can when wonder how do they get up get messed up in in your ambient manifold but See they come from a differential equation and they are not just It's not just a naked foliation. There's a natural structure going That exists along the leaves of the foliation See I told you that We had a mapping here That is a solution to the differential equation So I can probably I can look at the inverse of this mapping that will be defined in neighborhood of some point within the leaf Not within the ambient manifold But within the leaf of the foliation and this will give you a map into C into the complex numbers and when you think of it the fact of being an autonomous differential equation says that If I have a solution to the differential equation say phi and if I shift My difference my solution by some small complex parameter. I will parameterize the same equation the same the same leaf of the solutions and Because of unicity. This is essentially the only way to do it. The only thing the only way you can Define your differential modify your solution. The only you can change you Can perturb your solution and be within the same leaf is by having a translation in time So in other words when I have that my leaf of my foliation, let me just draw it like a like a leaf It's a two-dimensional real surface a complex one dimensional one What I have is charts taking place in C such that when I have two different charts That are inverses of solutions. They will be linked By a translation Okay So we have these leaves that the solutions of our differential equation are naturally equipped with a translation structure They have coordinate charts into C and the changes of coordinates are given by translations of C So this is one of the three geometries that we have in C We have translation This is the group of maps of the form Okay, it's just translations by complex numbers. He's a complex number We have also we have three geometries in three complex geometry geometries. We have also the geometry of a fine maps and we have projective geometry one dimensional projective geometry the geometry of the group of Möbius transformations homography and These are the complex geometries. These are the complex groups acting olomorphically transitively on a curve and we have our favorite real geometries all fall within these Geometry so here in projective. We have spherical and hyperbolic geometry and here within your fine. We have Euclidean and so long so complex differential equations sit within this Geometry translation geometry and in fact our equivalence to it If you have a complex curve and it has charts taking values in C and changes of coordinates given by translations You can just pull back Your vector field be over DC DC over C and since this vector field is invariant by Translations you will get a vector field in your complex curve. Okay, so the geometry of Vector fields is the geometry of translations. You can also think of affine One-dimensional complex geometry as related to two to vector fields if you have So what is so here you will have charts taking values in C with changes of coordinates given by affine maps and the fact is that affine maps for the translational part will preserve this vector field and the Homothetic part will change it up to constant multiple so if you have some some charts for a curve and You put in each Chart a vector field such that the difference between one and the neighboring one two charts The ratio is a constant then you have an affine structure. It's another way to say it So we'll come back to that later Okay, so the vector field is not zero then yes, you have you have this Then you see a structure and if the vector field is zero Then what you have is just some constant solutions I won't be saying anything about about these and it may be very complicated and of course there's there's some So the good thing is that there's some problems that are interesting to people studying differential equations That can be phrased just in terms of these geometric structure There are other ones like the dynamics or everything that cannot be directly read, but the two are not unrelated Okay, so let me just get me site one of these properties so One property that a geometric structure can have is that of completeness So here I'm in the context of gx structures like those we have We have some space and a group acting on that space And we can take charts in the space X With changes of coordinates within the group G. I probably change this vector field to Z so So one property of a structure is completeness and completeness means that your manifold is just a quotient of your model Under some group of transformations. Okay, so you can have complete translation structures for example on Torai when you portion by two really two independent translations and this is completeness and What does it mean for? Such a structure coming from a vector field to be complete. Well, it means that There's some solution So this is in vector fields This means that if you have your manifold and with your vector field, there is some solution That will be parametrized by holomorphically by See all of see you have a global Globally defined solution. Sometimes we call this an entire solution. Okay, so entire solutions are in correspondence with those leaves that carry a complete structure in this sense okay, so There's some theorems Concerning entire solutions of vector fields. Let me cite quote the first one is by manquist 1913 more than a hundred years old and it says the following it deals with Algebraic differential equations Suppose you have some function W of t and suppose that the derivative with respect to t. This is a come a holomorphic function Satisfies this differential equation. It's just some rational function p and q will be polynomials W and t and suppose that you have a Function so you have a differential equation of this form and a solution that Is not algebraic whose graph is not within an algebraic curve then the theorem of malchist says that in fact your equation is In fact a Riccati equation this means that it has a form W squared times some function of t plus w CFT so you have From the fact that you have one entire solution So this is entire you have an entire if you have an entire solution If you have in such a differential equation one leave that carries a complete structure Then you know a lot about the differential equation What? Then they see no, but oh so if you have this differential equation if He has the solution is an entire one and it's not algebraic then The ratio of your your rational function is is of this form. This is a Riccati equation And there's another theorem. There's many theorems. There's theorems by Yosida and Malmquist and Ilpolene and many other people But let me just jump 100 years in time there and quote this theorem by Marco Brunella says the following suppose you have a differential equation of this form again p and q polynomial and Imagine that all of the solutions are come our entire that all of the solutions all of the leaves of the Foliation have a complete structure in this sense then This is called a complete vector field and then what Brunella gives us a classification of these Such things is these are very rare in general. You will not find this these are very very rare and If everything if all the solutions are complete then Brunella gives a classification up to a polynomial automorphism and in fact An important part of this classification is that in fact You have some variables You have some some function of z and w that satisfies a differential equation autonomously Okay, so let me put it like this. There's there exists some function some algebraic function that is there's like this or you can See that if you project onto the Y variable then what you have is that? In your man in your surface here C2 you would have some projection that is But your differential equation your vector field your flow will preserve serve some vibration given by the fibers of this Y Yes Yes, it's a z and w why you have you'll have some projection into a curve called With time because this is Y of t Yeah, they're both functions of t then you can't you're okay, so so this is a theorem by Brunella and then a complement to this theorem by well, that is more than a complement by boosting do we and Heraldo and I think this is 13 says that it's up. It's it's sufficient to have Realist theorem for one leaf to be complete So if you if you have polynomial vector field like this you have a complete Leaf such that it's translation structure is complete Then you have a structure here Like this you can you have a vibration like this. Okay, so that's for completeness. That's an interesting What? No, no, it's a global one It's very important that these are polynomial. I said it but I didn't write it and it's it's it's about Completeness of solutions you have to look at the whole Solution, it's not something that you can grasp locally, you know that we want entire solutions complete complete So this is something that happens It's not local on the on the leaf. It's it's a global If M is compact and your vector field is holomorphic then you have the same theorem We have in the real case that any complex vector field will be complete Any holomorphic vector field in a complex manifold will be complete But the thing is there are very very few holomorphic vector fields in compact complex man So here you have this vector field PQ and then if you want to compactify it to CP2 Like in romance lecture, you will find some poles. You have some it doesn't have a nice Compactification as holomorphic vector field will be have some nice one as a metamorphic vector field And then you won't be able to say anything if M this the theorem is not for CP2 is for polynomial vector fields on C2 so Or every holomorphic vector field in CP2 is linear and you know everything about linear differential equations So the holomorphic cases is there are very little holomorphic vector fields on complex man Okay, so this is this property completeness and there's another property of gx structures Probably put it right here That's on C2 In C2 if you have a polynomial vector field Such that one of the leaves carries a complete translation structure then what you have is that There is a vibration by a rational function in C2 over a curve This vibration and such that the vector field The flow of the vector field will preserve this vibration in other words There is some variable y of your some variables of the original ones that will satisfy a differential equation You can isolate one variable in terms of differential equations Geometric terms you have a vibration that's preserved by the flow of the vector We have a structure theorem out of Existence of No, the vibration. This is C and then this is C2 This is where the vector field is the polynomial vector field is and this is C also You have some some or CP1 it's maybe rational. Okay, so That's that's some and all you need is for some Leave to be complete. So there's another notion related to completeness. Let us call it uniformizability so This notion in general for JG structures means the following means that So the complete ones are quotient of the whole space by the action of some group the uniformizable ones will be the quotient of some open subset of Your space by a group Okay For example one example of a uniformizable projective structure will be the Schottky uniformization No, so example of uniformization What? Because this is a general notion for J structures. This is also a general notion for J No, this is I know M is the manifold that will carry a vector field and here I'm just saying you take a G structure you get what take one of these you take G will be one of your groups X will be one of your spaces You have some geometry locally modeled on the space Which changes coordinates are given by the group and then this notion makes sense. So the uniformizable ones are Those that can be a quotient of some domain of the space by the action of some group and of course there's a More general notion so let me just give you an example of how uniformizable structures may be These are the Schottky ones so for projective structures okay, so So we saw that if you take in this we've been talking about Schottky groups for a while now so we have these transformations acting on P1 projective transformations and We know that there's will be some limit set here a canter set and With that when you remove the canter set and you act by this Schottky group you will get a compact Curve more over a compact compact curve endowed with a projective structures Since they change the group that we use that we are using it to act here is a projective one Okay, so this is not every and this Projective structure is very fragile in the curve if you fix the structure of the curve and you move a little bit the structure Then it won't be Uniformizable if you move the group you will find another if you move the Schottky group it will remain a Schottky group it will be It will give you a uniformizable structure on another curve You will move the curve with the group But if you just fix the curve and move the projective structure Then it won't be uniformizable In this sense so as you see is it's related to some sort of injectivity of some developing map But it's not quite that just being the quotient of an open subset of the model by the group okay, and There's no differential equation is just I'm illustrating this notion that's valid for all G structures of uniformizability that Okay, so So what's what's what's the notion? What's a what's the notion in this side of the for vector fields if I have m and I have my vector field Z on M, and if I have a leaf carrying Uniformizable translation structure. What what can I say what what does it mean? So let me discuss a little bit in order to arrive here. Let me discuss a bit more this property of Uniformizability okay, so suppose I mean some j structure in some manifold and Suppose the following suppose so so this is some manifold s Here's a universal cover of s and I have some developing map for the structure over my model x, okay So that has some equivariant destroyer presentation and some equivariant properties Take place. So suppose that in s I Have two different points and a path Oh, let me say something about this uniformizable thing structures is that They can be restricted if you have a structure and uniformizable structure on a manifold Then the structure induced in any open subset of the manifold is still uniformize Because you're just if your original one was a portion of some open subset Then a portion that the smaller subset will be will give you your so you can restrict this and they will continue to be uniformizable So if you have a uniform structure in your surface s and you look at what happens In the neighborhood of a curve that you can lift To the reverse of cover if by any chance It happens that The developing map of this curve Maps the endpoints to the same place Then it cannot be uniformized. So if you find such a curve Then you cannot recover s as a quotient of some open subset of the model by the action of some group of translations and this is because Well, this wants to be the open set that parameterizes After quotient this set, but you see here we have some Overlapping so we won't be able to resolve the parametrization unit in the uniformization unit and in fact, this is the only obstruction for uniformizability if you don't find such a pair of points and a curve then automatically your Structure is uniformizable. Okay. So what does it mean? Here, so I'll do the same. So this is for general This is for general gx structures. And let me tell you what what is what is What happens when you have a vector field in the process of the vector field for this translation structure? Well, so I'll do the same drawing Here I am on M with my vector field z and I have one leaf that has is one leaf that has is its translation structure and suppose that I can find a path Within the leaf like the one I described there and suppose that the developing map Maps this band The end points of this band to get overlapped Okay, well, you know what we recall that the structure was given by the solutions to the differential equation. So This inverse of the developing map will be parametrizing a solution. So If your solution if your structure is not uniformizable Then you will find this curve and then your solution will necessarily be multi-value so the equivalent notion for vector fields When you have some domain Some that parametrizes that that will be the domain omega which parametrizes which Uniformizes your structures you have in some sense a maximum single-valued okay, so completeness entire solutions and uniformizability means No multi-valued solutions Okay, so locally you have uniqueness of solutions. That's just the fact that you have charts But this is something about the global geometry of the structure You cannot see that locally you need to travel along some distance in order to notice some lack of uniformizability You're all locally. You're always uniformizable locally. You have inverses of your choice. This is some global thing that happens okay, so Question is the following for algebraic differential equations for back for vector fields and manifold algebraic vector fields and manifolds for a uniformizable leaf can be said about for uniformizable leaf say Omega a subset of C quotient that by the group what one can be said about omega so This will be very very rare In general when we have a vector field it won't be complete In general it none of the leaves will have this uniformizability property some very special vector fields We'll have one leaf which carries a structure that is uniformizable And we want to know more things about these vector fields We want to know more things about these differential equations these algebraic differential equations having single-value solutions and about the functions they define so So this is so what can we say about omega and there's this notion of the central boundaries for for differential equations So let me talk to you about Family of differential equations. That's definitely an algebraic one It's a Variation of some things by Alfvén Brioche Brioche you pronounce Yes, she Brieuski Brieuski Brieuski, Shazie, Ramanujan and some differential equations attached to the names and then this is just a this example is just a variation Okay, so we have a differential equation. That's a polynomial differential equation in C3 that depends upon a parameter alpha alpha takes values in C as well and There is some so again, these are very very special Differential equations is as far from Jerry. This has a very particular solution solutions of these equations Have a very important invariance property. That's the following Suppose we have a solution local global say local solution x of t y of t z of t and that you find some Matrix ABCD on SL2C. So I forgot say that there Then you can use this this element of SL2C to modify Your solution in the way that's written over there And then when you do so you will find another solution of the same equation So let me explain you what's happening. Okay, so how does it change? Well? You can say that first I multiply y and c by some by some by some factor and X by the square of that factor Over there, and then I add something to X Okay, so when you look at it in C3 you get the following you have some So if you look at this this changes of coordinates This this modifications you see that the ratio of y over z isn't changed That is the image as functions the image y over c isn't changed. So if you project onto y over z you're in and P1 and the You have some hyperplanes in C3 Then we just draw one above each one of these points we will have some hyperplane I Draw it like this and the thing is that in almost everywhere This pencil of hyperplanes will be transverse to the solutions of the equation So there's a whole family here. So you have your solutions of the equation and You can read this formula in the following way if so remember we have plenty of other hyperplanes two planes here So these hyperplanes Allow us to identify one solution to another just by pushing along the same hyperplane every point what this formula says us is that If we want to recover the parametrization of these curves as solutions of the equation we need to Modify the parametrization, but we need that we need to modify the parametrization Projectively we are in a vector field here every leaf carries a translation structure But there's no harm in thinking of these translation structures as projective structures And what this formula says is that under the holonomy of these planes under the Identification that this plane makes this projective structure is invariant by the holon. Okay. This is a very very particular vector field okay, so In particular this vector field So this is happens almost everywhere and they will be there will be four Bad points points where we don't have this transverse picture, but outside of these points what I just told you that there's some natural projective structure in the space of leaves of This function tells you that we have some projective structure naturally attached To each alpha so we fix alpha you have this drawing and this alpha will give you a projective structure in p1 with four mark mark points That will depend upon alpha and in fact is related to the solutions of the equation because if I take a local solution if this is C and I have a Solution of my equation taking values one of these curves and then if I project Then the inverse will be a chart of this projective structure and this is all in that formula see Okay, so let me tell you what flavors of projective structures we get as alphabars Okay, so we have a projective structure And I'm describing just this very very particular Time of equations which will Not answer the question, but at least show us the kind of What's in the zoo will tell us what that's what what are the things that we can find? Okay, so So we will have a projective structure that depends upon alpha and there's fear minus four points and In the four points we will have some parabolic structure. This will look like the quotient of some half plane By a translation locally, so we will have parabolic the structure will be parabolic in this The structure will be parabolic and that's the only restriction for the projective structure We will find we will find every projective structure in this in actually this in this fear This is a harmonic for punctured sphere and we will find every projective structure in it such that the Structure on the ends has this parabolic property. This is a vector space of dimensional one So let me draw this parameter plane this alpha plane and there's some point that's alpha equals zero Where what we will have actually here is the uniformization of the four punctured sphere So for alpha equals zero We will have that this projective structure can be obtained by taking Very symmetric ideal rectangle and then multi Reflecting have some group and when you mod out by this group we will find this four punctured sphere so in this case What we find when we look at the differential equation? It says that in fact Our solution so I told you this is our solution So this this is our chart of the chart of our projective structure and in fact Well, the inverse will be naturally a developing map for a projective structure So if we take the disc we're portioned by the group we obtain the four punctured sphere It means that the image of the developing map is this domain this disc in size C so in this case this domain Will be the natural domain of parametrization of the solutions of our equation that happens for alpha equals zero we have a natural boundary for the for the Solution of the differential equation of the algebraic differential equation So as we start moving alpha so that happens for alpha equals zero There's an extra symmetry that allows us to say that this happens so when we start moving so We start moving we go from a From this group from our foxy and group to some quasi foxy and group as Karen series explained to us yesterday We have some quasi foxy and group so close to alpha equals zero. We have some quasi foxy and group and then there's some Set here some cauliflower some that's called the bear supplies Whose interior? Okay, so I'll just show you the picture So so I saw this I stole this picture from the internet and so I stole so the following one So this is the burst slice so you can have in the center. There's alpha equals zero This is the plane where alpha lives. This is the alpha the plane where alpha is so In the interior of this bear slice You will find quasi foxy and groups groups like the one like the ones we had here like this one so in this group What I'm saying and the same thing happens. So Since we are the quotient since So the group will now act on this domain It will produce you a four puncture sphere the same conformal structure with a different projective structure Again the inverse of the so again this developing map Will map on to the domain where the solutions are defined so for some alpha inside this bear slice This is a picture in the purple region You will find three holomorphic functions x y and z and if you take the derivatives with the respect to To time I didn't want to show you this You will find that they satisfy the algebraic differential equation and see now the natural boundary is no longer a circle It's a fractal curve The boundary of these quasi-fucsion this quest quasi circle Okay, so this is the burst slice and yesterday Caroline series told us about what happens at the boundary So let me just repeat it or misquote it briefly so What you have is you have your quest-fucsion group you have a four puncture sphere in the top at the beginning you find The same four puncture sphere up to orientation as you start moving alpha on top The quotient will be the same four puncture sphere the lower one will become different a different four puncture sphere Did you Okay, so So at the end so what happens in the boundary there's essentially two things that may happen so one of the curves So we have some quasi circle here So on one side when you take the quotient we have the very same four puncture sphere the harmonic four puncture sphere And on the other side as we start degenerating two things may happen. So this is One thing and this is another thing that may happen is that we might be for the hyperbolic structure we might be Pinching one curve and the other thing that we happen is that this open set starts becoming smaller and smaller smaller like the pseudonosis of Scenario we heard about yesterday and then we meet just This part will just essentially cover up a Big set a set with whose complement has empty interior Okay, so all of these domains what I'm saying is all of these domains appear as domains were the solutions of some Algebraic differential equations are defined. We don't only we not only have Circles as natural boundaries. We have also fractal curves fractal Jordan curves And also in the boundary here, we find some nasty sets as boundaries of our sets were the solutions of our algebraic differential equations are defined our single-valued solutions of algebra differential equations beyond so beyond this set beyond this Very slice we will find multi-valuedness of the solutions. So For the differential equation I Showed you for the parameters This is a set of parameters with single-valued solutions. This is a set of parameters where the Translation structures on the leaves will be uniformized Okay, so This is the only way I know in which we can produce natural boundaries for three-dimensional algebra differential equations And I would like to say that this is in fact the only way, but I don't know how to prove it because So I think This is the only way to do it if you find a three-dimensional algebra differential equation, and you have some single-valued solution Defined in some subset some subset with whose exterior has interior say that it's really there's some natural boundary Name I'm pretty sure the same mechanism is behind. I Don't know how to prove it. Of course, I would be telling you that But I know that this doesn't happen in dimension two you really need to go into dimension three in order to Have this phenomenon of natural boundaries. So then we just in my that the time is left. Let me just cite quote the theorem that having a scholar the fact that There's no natural boundaries for algebraic differential equations in dimension two, okay, so Now what we'll have is the following suppose Let me just begin by stating a theorem So you have M a compact complex surface killer Z a Meromorphic vector field vector field having poles and suppose that in the complement of the lobes of poles of M We have a solution that's Uniformizable, but it but that is not complete So suppose that we have some domain Omega in C and we have a map. That's a solution that goes into M. That's a solution to a differential equation Phi I suppose that Phi does not extend so this is a really a Uniformization so I suppose does not extend we have your uniformization of one leaf of this Translation structurally suppose that Phi does not extend To see how that You're not missing anything. Okay, so Then what you have is the following is that your surface M fibers over a curve which is a p1 or an elliptic curve and This vibration will be invariant by the vector field the fibers will be as well either p1 or elliptic curves and The poles of the vector field will actually be here will be some some bad points in the fibers and Say suppose suppose for example that you have Elliptic curve a torus here. There will be some special points and then What will happen is you have here Here you have suppose that you have here a vector field this vector field allows you in the base since The vector this is the conclusion of the theorem the vector field will Preserve the fibers so it will induce one vector field in the base and the vector field in the base In this case will be complete say if this is just see over some lattice A vector field will be the image of This vector field that is translation invariant and we will have a way to parameterize the torus by the whole C But the thing is what will happen here at these points We will find some in possibility to lift the solution here here the solution will not be We will have will not be holomorphic We will have some strange things happenings and in fact what will happen is that as we approach this point We will be accumulating on one of these fibers Okay So this is the theorem supposing you just have one leaf that's uniformizable you get a structure theorem like the ones by like one by Brunella complemented by boosting doing and Hidaldo and And It says in particular since we will have some vibration over a curve and since vector fields in curves That are uniform that induce uniformizable structures are actually complete The bad points will be just these fibers, but these fibers will appear isolated in the domain where your solutions are defined So in have so in fact you can conclude from these descriptions that omega is the complement of Accountable set Okay, from destruction theorem you can so in particular this these these phenomena that phenomena that was we saw here of Essential boundaries does not appear here. Okay, so So this theorem very similar theorem and the proof intersects it a lot I proved with Julio rebello from to lose and We studied the case Where all the solutions? Uniformizable and in this version you only need one solution to be uniformizable for the vector field to For the structure theorem to take place Okay, so Let me just say Half a word about the proof of this theorem so they think so we have M our manifold that's fully ate it And there are some leaves that carry some uniformizable structure and some leaves that don't Suppose you have a leaf that doesn't so you have one of these paths such that the developing map Will overlap itself along this path. So this is a path This is a path within a leaf Who's developing map does this? So here the structure will not be uniformizable and in fact if you move this a little bit if You move this curve to a neighboring leaf You will be able to find another curve in the neighboring leaf That has the same phenomenon. This is robust. You can you can move it a little bit. It will remain with overlaps So we will find some closed set here where the solutions are uniformizable just as the Bears slice together with its with its boundary this sort of set May appear here So so this is So we need to understand this closed set and in fact You will need to approach the locus of poles of your vector field because if you stay away away from poles Then the same theorem that tells you that a holomorphic vector field in a compact manifold is complete Will tell you that if your solution stays away from the locus of poles then it will be automatically complete We are supposing that it isn't so so somehow in M if you have your locus of poles You will accumulate to the locus of poles so The thing is what happens in the locus of poles. You don't really know But it may happen for example that your vector field has this form This is that you have poles along Along this curve and that you can approach this curve by your solutions And what happens here is that at this curve of poles The parametrization doesn't make sense Because it has poles however There is some sort of renormalization trick that tells you don't look at the translation structure forget about it Look at the affine structure And in fact the affine structure induced by the translation structure will be defined in the locus of poles And now we have some nice affine structure in the curve of poles that has to be uniformized and Supported by a by an in a complex compact curve will have some singularities But there are very few uniformizable affine structures and compact curve of singularities Essentially, you will find they are uniformized by the Euclidean triangle groups and some generalizations of them and in fact, so you know you have some information about these special curves of poles and if you work a little bit harder and You do the tricks that the algebraic setting allows you you will find that they are actually fibers of a vibration And this will allow you to conclude the theory. So I'll stop here. Thank you very much