 Thank you very much. It's a it's a great pleasure to be back in Maryland. I have such fond memories here appreciate the invitation from from the organizers Can view this on anyway, so This is joint work. Let's see if I got this working. Nope Okay, joint work with Andrea Saad, Jan Svoboda, and Richard Wentworth And so what I'm going to do structurally in the talk just as a piece of theater art is I have I have a certain amount of background that I want to review and I will do that on the slides the slides that let me rapidly go back through some material Fairly impressed and the idea is to maybe do it so that you're left with some basic notation and some impressions And when I want to really start saying things That Did I want to convey that I'll move to the chalkboard and so this just saves me some time and so mostly what I want to do to begin with is Is is take Steve's very eloquent treatment of Higgs bundles and completely butcher them so and so that it's or butcher it the treatment and That will form enough backgrounds so I can go on and explain the relationship between the objects that I care about Okay, so wrong way So as usual and for about two-thirds of this conference S is a differentiable surface of genus 2 in which I'm going to put various structures I'll be what I'm interested in is The carriage of variety that Steve talked about other people have talked about representations of the fundamental group of the surface into psl2c up to conjugations of that representation of my psl2c so I'm interested in this topological object and There's a number of ways of understanding that topological object. I want to relate them To emphasize here the representations may or may not be discreet. I don't care. There's no quotient manifold in the setting so I'll first Remind you or present a gauge theory perspective perspective on the asymptotics of this object Let me restrict. I'm interested in the asymptotics of this character variety so I start with a gauge theory perspective and then the objects that arise called are called limiting configurations and But It psl2c has an is the Roughly the isometry group of hyperbolic space. I want to interpret those gauge theoretic objects in terms of hyperbolic geometry That's sort of the plan So let me recall Higgs bundles So I'm going to do a rapid reminder of Higgs bundles in some I'm going to do it in psl2c Maybe I'll refer to pslNc Steve already did this very well and So it'll be some sort of When it's easy to say the more general Nc version I'll do that and sometimes I'll just stick with the psl2c first step and in this I want to highlight this step because this is Part of our motivation and particularly my maybe my motivation I said I'm interested in understanding the asymptotics of this character variety, which is a topological object What's my first step pick a Riemann surface? So in the culture, I was raised you you don't do that if you want to study a topological object You stay on the topological object. So this first ad hoc choice is Is something we'll focus on and So it's ad hoc choice We have to interpret and so the as Steve pointed out the representation into psl2c determines a flat rank-to bundle that's just topology and Hence if I have a rank-to bundle then I have a flat connection. So I've now trans translated my topological representation into a connection language type of flat connection Now the basic formula that Steve highlighted is governed by Hitchens equations and so what I want to point out in these equations And this is the impressionistic part is that there's three terms There's the flat connection on the left. So that has to do with topology There's and this connection is going to be broken up into two bits and this bit is going to be sort of like a deramish It should have to do with a metric and this part here is going to have to do with like a double bow theory like with complex analysis, so I'm Shoot I'm not used to this yet wrong way Here we go so D is the flat connection and To find a what we do is on this flat on this rank-to bundle e I look for Hermitian metric on the bundle. So that's a metric on each fiber and And What's that? This is Finding a metric on each fiber is the same thing is finding a map to the space of metrics So from each point I need a metric so in each point I look for a map to the space of metrics and then I pick out one point so that's equivalent to a to a Equivariant map on the universal cover to the space of metrics on the Hermitian bundle in the space of metrics is SL2C mod su2 And so here I'm going to reprise what I just said the target's the space of Hermitian metrics on the fiber and Then so now that I'm looking for a map to this space of metrics I recognize that the space of metrics is a symmetric space has a geometry so since it has a geometry I can look for a very good map so our best map so I'm going to look for a harmonic map to that space and Once I find the harmonic map this there's a uniqueness theorem involved because this is has done positive curvature Then I get the induced connection used unitary connection on the bundle and I write that as a So start with a flat bundle Look for a good Hermitian metric that gives another connection on the bundle d plus a Well now I have two connections on the bundle I have the original flat connection that I have this sort of harmonic connection Well the difference between two connections is some other sort of tensor and that tensor We can call psi in some language that's the difference between the flat and harmonic connection and then with respect to The Riemann surface structure and now the whole Morphic structure on the bundle We can look at the 1-0 part of that difference. That's and that will be the the Higgs field phi Okay, so Yeah, no Right, oh, yeah, yeah Yes, so I It's a turn connection Okay, oops Hey, yeah, it has it has a 1-0 part the The 1-0 part of the d bar for the whole Morphic structure, and then there's the 1-0 is the turn connection all together No No, I mean, it's the unitary connection Yeah, yeah Yeah Okay Let's see so this problem with going well Let's see so I said so as I said the harmonic matrix and I'll produce the whole Morphic structure on e and in that structure The harmonicity says that this 1-0 part of the difference of these of the flat connection the unitary connection is whole Morphic So these Hitchin equations the self-duality equations summarize the decomposition in this way that the d as I said is flat the H the Harmonic metric is harmonic and the Higgs field is whole Morphic so And the equation say that if you take any one of these three as input maybe with a slight extra structure You can find the other two It's a summary of the equations and a Higgs bundle as Steve told us is a is a pair of Of a whole Morphic vector bundle and such a Higgs field Which satisfies some sort of stability condition the stability condition is enough to guarantee the solution to the Hitchin equations great so In the SL to C setting Phi has this form So if I have the square root of the canonical bundle plus the Inverse square root of the canonical bundle the Phi has this form in good coordinates and the Coefficient to these characteristic polynomials are whole Morphic k forms in this case whole Morphic two forms for quadratic differentials So for rank two the determinant one of the invariant polynomials are the invariant polynomial for the Quadratic case and equals to case is a whole Morphic quadratic differential and from the point of view of Ilana's talk, it's the hop differential of the harmonic map to the symmetric space Roughly this psi if you remember Phi plus Phi star H is the Tangent map to the harmonic map Okay, so Make sure right, and so Hitchin then describes this space of Representations in terms of this vibration which Steve talked about maybe I'll draw a picture in the n equals to case so For n equals to we have a base which is the quadratic differentials on On On the surface X and then over each one of these points. There's a Prim variety of differentials There's some sort of Taurus and over some of them. I guess I better do this right. There's some sort of degenerate Prim variety And so this is just a beautiful picture of the space of representations that There is a certain there's a detail I'm going to suppress throughout the talk going on throughout this to see whether I decide to mention it There's a there's a double cover issue. I'm gonna more or less suppress that and so the the the prim variety refers to sort of just refers to Differentials or line buttles that have a certain oddness with respect to this so right Maybe I'll just suppress that suppress all mention of that Okay, I want to go forward and that's not forward forward is down. Okay, so there's double cover issues Which we're going to suppress So the the question on the table for the afternoon is what happens when we leave all compact sets in this character variety? So what happens when we? Go to infinity in this space, so here I have some quadratic differential Q in here, and it is the determinant of one of these Fees and as I go to infinity, there's lots of ways of going to infinity. I mean so for example, maybe I'll look at points in this Character variety For which, you know t times q zero goes to infinity so t goes to infinity I look maybe at a ray of quadratic Differentials and then over that ray pick out some points in the Torah I and they just leave all compact sets in the character variety and the question is what's the geometry? So this happens for example when quadratic whether the Higgs field leaves all compact sets or when to determine into the Higgs field Which is this point in the base goes to infinity All right, so now Restrict a bit further We are going these quadratic differentials earlier whole morphic forms on a Riemann surface They can generically all so what whole morphic form on this Hormorphic object is more or less determined by its divisor more or less determined by its zeros up to a phase so Generically all the zeros are simple The in some sense the most interesting cases are when they're not but we're going to focus we're going to restrict ourselves to the case where somehow all of the zeros so here The Q inverse of zero are all simple so or yeah Zeroes are all simple So and we're sort of interested in what happens out here in that stratum Okay, so now we come to a recent paper of rave and his co-authors Svoboda vice and vitt and they parametrize an end of this character variety in terms of Objects they call limiting configurations. So this is phi infinity a infinity. So now rave has to sit on his tongue while I Do this Or he can take the fruit or he can take the chalk so the so let me explain and so That's really intriguing right this sort of analytic description. So let me just say what it is So here phi infinity is some sort of Higgs is a Higgs field and a infinity is again a unitary connection as in the the d plus a plus 5 plus 5 star But it has some singularities at the zeros of q infinity. These are limiting configurations It's sort of they're sitting out here at infinity. So something has to be degenerate and the degeneration is in the singularities of of a infinity singular at the interesting places of the of the differential at the zeros of q infinity and these this pair Satisfies a limiting decoupled version of Hitchens equations plug in T times phi infinity into the equations, but T go to that's T grow large Take a limit look at the sort of equations you get and I mean I'm Now I'm being obnoxious the And just still out of that with quite a bit of thought what the what the relevant equation should be and this pair Satisfies that that set of equations. So these are the limiting equation the limiting version of the equations. I didn't show you Okay, one thing to notice is there's no longer a harmonic map Right there's for for every one of the of the representations or the Higgs bundles corresponding to representation in the variety where we can think about the Geometry in terms of this harmonic map to SL to C mud SU2 as we leave all compact sets the Representations to generate there won't be a map left over and so this is sort of one of the extraordinary parts So what gets left out? There is still something which survives and that's this a infinity Okay So the I should back that up. I can't seem to back that up. Okay, so So this data seems to have aims to capture the families of degenerating harmonic maps So what is the modular space of such pairs and they I they carefully identify the modular space? well the Q infinity This isn't quite right Well, okay Sorry and Q infinity I guess did well determines the yes determines the phi infinity excuse me that is right but But there's a modular space of a infinities the same way. There's sort of a torus of a's over here over the Q There's a there's a primitive variety of of a infinities here And so maybe We can pick a base one. Maybe we pick the base one for instance that Has to do with limits in PSL to are sitting inside PSL to see if hyperbolic surfaces Inside this space of representations and we can find a new limiting configuration Maybe by looking at the difference between the some base one and the new one by adding something an alpha And then what is alpha? Alpha is going to be some one form that has some conditions So alpha is a one form on the punctured on the surface punctured at the zeros with values in some Some line bundle built out of the phi infinity and I won't go into the details Rafe will be happy to explain all the details to you. So Ellen find me. It's a real line bundle Just to get a definition It's the one forms with values and little s u2 that commute with this phi infinity That's a bunch of matrices So it's closed form values that commute we have to worry about gauge equivalence If we demand that alpha is not only one of these Deformations in the in the fiber direction is not only closed but co-closed at the harmonic It picks out some unique unitary gauge equivalence class And in the modularized space is the space of these harmonic forms up to a lattice of integral periods So dev lattice gives you a quotient, which is a torus So this m Infinity so now I'm way over here. Maybe I should draw myself another sector over here. And so here's my m infinity again we'll have Some phi infinity which you can think of in some sense as unit Higgs bundles or maybe the quadrat the determinant of these bundles has Has norm one in some sense and then over this there's some Prim variety of forms representing these with these singular Connections a infinity Okay, so I'll just summarize this has been on the slide for a while It's a very pretty picture. I think of the frontier Okay, so here's the questions I wanted. I guess yet though they're possible The alpha will be like differences of of Limits of fies Or difference or sorry differences of limits of A's So here's the questions. I want to talk about We're interested in this picture. Imagine this picture is a pick as the Description as the character variety written in some sort of coordinates Why do we have to pick a Riemann surface in the background in order to describe what the limits are? That seems ad hoc What happens to the description if I look at the limit configurations for x and I look at the Limiting configurations for x prime both are describing portions of the frontier What happens to the description if we change from x to x prime? I'm presumably I get new limiting configurations. How do they relate to each other two sets? The other issue is that we're talking about SL2 C mod s u2 and surface groups acting on that this is That the our our holonomies are in Well are in this quotient that this is a And so this is a natural space for the Higgs bundle theory, but First for my culture We don't think of it. We recall this hyperbolic three space And there's quite a lot of sophisticated geometry has been developed for hyperbolic three space So how do I understand these limiting configurations in terms of that hyperbolic geometry? Okay, so let me switch to the blackboard now So so let me phrase an answer or try to answer these questions in terms of what Interpolates between the flat connection so the topology the character variety and the Higgs fields the fies and that's the harmonic maps the harmonic metrics in between so the the a's in between the D and the Phi so so so I don't So this stuff is as I said before has certain sophistication to it. I think it's very hard to To pick up what a Higgs bundle is from My treatment or maybe even Steve's very eloquent treatment if you've never seen it before What I So if you're if you're discouraged you can try one more time I'm sort of gonna start again with some other aspects of a different sort of theory so I want to Recall just some basic aspects of harmonic maps of surfaces and Actually, you can get fairly far in this theory with just a couple examples So harmonic maps of surfaces, maybe I'll say these surfaces have targets in hyperbolic three space so So I just want to describe some examples And exact examples of maps from the complex plane to hyperbolic three space So the first example And I I suggest that you you really know a bunch of these examples So the first example is I want to take C so here I'll draw C and then I'll project this so C. I have a variable Z X plus I Y and then I'll go down to the real part of Z X and so now I have the projection onto the real line and This map is harmonic in any notion of harmonic that you can You can decide on and then if I take this real line and I map it to a geodesic In hyperbolic two space. That's a nice somatree. So this will be a harmonic map followed by an isometry That had better be harmonic in any definition you care to put in So that's a harmonic map to H2 And of course I can take H2 and I can stick H2 into H3 and that geodesic will come along and So this composition of maps will be a harmonic map from C to H3. So I think that's Perceive, I hope that's persuasive that that's harmonic and locally energy minimizing So We've had references to the hop differential In a number of geysers Ilana talked about the hop differential and I've talked about some quadratic differential the determinant of the Higgs field and I alluded to the fact that you could think of the Higgs field as the one zero part of the differential of the harmonic map So the hop differential is some sort of trace Of du squared or if you like the trace of Fee squared or maybe we can think of it as So I'll call it Q. So we can another way of saying is you can pull back Symmetric on The target H3 and you can look at the two zero part of that. That's That sort of follows from the formulas and that's supposed to be some quadratic differential great and In this case, so that's in general in this example What is it? Well, it's going to be so Q is going to be something times dz squared in the plane But how we can just look at this map and know that notice that if I translate up and down By C that doesn't affect the map at all And if I translate from left to right it only affects the map by translation along the geodesic so this quadratic differential If you think about in terms of pulling back the metric along this geodesic is unaffected by translations in any either direction So that quadratic differential had better just be some multiple of dz squared now Alex has has Highlighted the importance and when looking at quadratic differentials of horizontal and vertical Foldations so this So the full Asian structure Of Q well, that's just going to have horizontal lines that go horizontally and vertical lines Which go vertically and you can see how those how the structure of those Foliations of those leaves relates to the map the vertical lines are being collapsed to points Because that's just the projection and the horizontal lines are going into Hyperbolic three space just isometrically to the along the geodesic So once we know this example, it's easy to make other examples So for instance now we work backwards and Suppose I have a quadratic differential, which is zd z squared. This is the hop differential for some harmonic map, which one? well Now I have a single zero at the origin and if we work out What are the horizontal and vertical leaves? We get some picture that looks like this So here one of these leaves is the real part How do I want to say this? So Q with respect to some direction V is positive this is the real leaves of the quadratic differential and So and now we try to work backwards from this. What does the map look like? Well? I'm this quadratic differential is on C So very far away from the origin if I stand Way away from the origin I become unaware of the zero the zero is becomes less prominent is very far away so an observer Here in the complex plane Thinks that say quadratic differential is not a zd z squared, but d z squared One times d z squared. There's good local coordinates for that and so Is that so what does the harmonic map look like? Well, it looks like the original example so way out so if I want to know say mapping to h2 What does this map look like well very very far away? These leaves map to something which is nearly a geodesic because that quadratic differential is nearly d z squared and very far over here it maps to An arc which is nearly a geodesic and way down here the same argument happens and it maps to something Which is nearly a geodesic so all together this harmonic map with sorry the harmonic map with this hop differential Maps to the interior of an ideal triangle and that's the harmonic map here Okay, and of course I can take that ideal triangle ideal triangles are rigid and If I'm mapping to hyperbolic three space, I'll just have an ideal triangle sitting in some plane sorry drawer Sitting in some plane in Hyperbolic three space So that's now Well, we can do two zeros. So that was example one. Let's do Two zeros I promise. I'll stop with two zeros the rest of the talk won't be I'll get to five zeros in that run at a time So if I have two zeros, well now my Foliations Maybe look like this So I'm thinking here about z minus a z minus b Dz squared and what does where does this map to well again very far away. I start looking Like a geodesic so you can imagine that mapping to hyperbolic two space I'll get something that's roughly For geodesics if I do this right Now the problem is that when I put that into h3 I Have a different Experience of when I put in the ideal triangle ideal triangles are rigid. There's only one way to put them into hyperbolic three space Into a Hyperbolic quadrilateral Well that let me draw hyperbolic three space in the upper half space model I can put one of my punctures at infinity and then I don't know if this is coming across I have four distinguished points and there's a flexibility to four distinguished points So I don't actually know this map all I know is that it goes into the convex hull of those four geodesics However We were interested in asymptotics I'm interested in when this quadratic differential is very large so So I'm not really thinking of z minus a times z minus a Times e minus b d z squared well t times it with t is very large So what happens when t is very large? Well these zeros then if I start with z minus a times e minus b they start in to look like they are very very far apart T time they'll be roughly distance t away from each other So when t is really really large an observer here in the middle Is going to think that their quadratic differential is just plain old d z squared so the image of of these Leaves is going to be very close to The geodesic along the diagonal and again harmonic maps have nice convexity properties They map into the convex hull of their boundary values. So since I know that the middle here goes to a geodesic Then what or goes near a geodesic then what happens is? The image of the harmonic map maps are very close to Two ideal triangles joined along a seam just sort of bent Okay There's a digression that I have so I want to come back to the next thread picks up on this two triangles bent along a seam so So I'll write it to Ideal triangles then so long as seen I Can't resist a digression and for the digression. I might as well come back to Plain old z d z squared like this and I told you So this is out of order. I probably should have done this digression at this point and You know, so we've with I hope I've convinced you this goes into the an ideal triangle And the argument I made was way over here. You're a geodesic way over here. You're geodesic and I drew those two geodesics like this but The problem is how about when I traverse from here to here And I'm supposed to be very far away from the origin So this goes to a geodesic this goes to a geodesic presumably the same geodesic some sense that goes to a geodesic that goes to a geodesic so So what has happened is that very far away? I have flipped a solution to this Harmonic map equation for z d z start to the geodesics it for a long time It was very close to this one and then at some moment it jumped to another Solution to that same harmonic map problem the original example just took to a Geod just took had image a geodesic but here we have to have some sort of flipping phenomenon and so this is reminiscent of a phenomenon for odys all the stokes phenomenon and So what's happening here is that as you transition from from here to here? The asymptotic solution jumps And all it is really true one thing I lied about is that all is really true is that the forward asymptotics Could be prescribed, but the backwards asymptotics couldn't in some sense the secondary solution to the differential equation here Can dominate the skill we can have a change in dominance from this solution to the asymptotic equation to this solution and It's a it's a subtle phenomenon. That's that's I think sort of cool That's a digression Remind you two zeros we get something with a seam Okay, so let me think about seams for a bit so This sort of bent so when you see this sort of bent pair this bent Join of two ideal triangles One is led to think about Or is reminded of the sort of theory of what are called pleated surfaces, which is a hyperbolic geometric object So so from Example two we are reminded of Pleated surfaces and this is a another fairly complicated object and so I don't sure I want it either I want to or it is helpful to Give careful definitions, so let me just approach this by example Okay, so on So if I have so if we have some quadratic differential q on X It's so it has some sort of foliation structure Which I can lift to x tilde on the universal cover that's complicated. I know something like that and If I draw that foliation on On This is supposed to be the on x tilde, maybe I think of that as hyperbolic to space Sorry I'm not paying for it. So one can you can imagine Taking this these you can imagine following these leaves as As they make their way past obstacles past other zeros and they have a Limit on the boundary of hyperbolic to space And what can imagine I don't see color chalk one can imagine straightening them so Between the end points of these leaves. There's A straightening to geodesic arcs and And when we straighten the geodesic arcs we arrive at an object Associated to the quadratic differential of foliation So this straightening Leads us to or produces So-called Geodesic lamination, so This may not be the best way to think of it. So And I want to emphasize some stuff that That Alex has been sort of emphasizing I think all week is that I drew this on the universal cover The these leaves as they wrap around a Riemann surface a very complicated dynamics so maybe I have a zero here on The Riemann surface, but if I follow this leaf a nearby leaf around it will wrap all over the Riemann surface and it may end up coming very Come back very close to the original zero So this leaf here may if I follow it on the Riemann surface It may find itself near a copy of this zero or you know a copy of This zero so the dynamics of these leaves is very very complicated. There are copies of this sort of picture all over The universal cover So and that's just something worth keeping in mind When you try to imagine harmonic maps based on this or anything so this So returning to the straightening process if I straighten these leaves that are very close to this Maybe to this tripod then the sort of picture I'll get will be sort of A number of ideal triangles with But because of the dynamics say the The transverse structure of these ideal triangles will have a canner set sort of structure Okay, you can imagine sort of you you You follow leaves along and then the leaves come near A zero and then they get split And then so if I follow a packet this packet goes along the packet comes through here It splits the split piece follows along it hits another zero it splits That's what a sort of canner set construction that gives One straightened gives geodesics which also have a sort of canner set transverse structure okay, so So what I'm trying to I'm suggesting is that there's a Geometry a hyperbolic geometric construction which parallels what we saw here that quadratic differentials become lead to a collection of geodesics in Hyperbolic two space or in this case hyperbolic three space with complimentary domains that are Hyperbolic Okay, so What is going so let me try to summarize this so I'm afraid I've That wasn't a clear explanation, but I'm right Right if it goes in to just if I know a priori that it's going into a hyperbolic two space Then yeah, there's no bending, but if all I know is that it's going into a hyperbolic three space and all I know is this Foliation structure then no I the I don't really know where it goes at any Finite time, but as t gets very very large as these zeros gets very separated I can predict that it ends up clinging to a pair of bent triangles Okay, so So what is a pleated surface? it's It's a collect it's a collection of data where I start with some hyperbolic surface s this will be a hyperbolic surface and I'll have some lamination That'll be distinguished that'll be like so my lamination here is this collection of leaves It'll be row-equivariant with respect to some representation So this is a lamination and There'll be some map F and the point with F is that it'll have two properties it will be So off of the lamination F is an isometry and on the lamination F is an isometry of the lamination of the leaves, so it's not isometries So so here I'm referring to an isometry it's ranked to it takes the tangent plane isometrically on the tangent plane Here I'm referring to so F is an isometry of the lamination the picture is this on the universal cover I have oops some some lamination which has Was complementary domains, maybe look like ideal triangles this gets mapped So this is equivalent to H2 this gets mapped into H3 in a way that bends along the lamination off of the lamination is an isometry so It's its image on each one of these pieces is somehow straight It's pieces of hyperbolic two space and then but on the but on the leaves It's only an isometry along the leaf. There's no transverse Isometry so this can bend and this is a very hard picture to draw Because I'm trying to draw a bending along the canner set But it looks something like that and you can see it's sort of pleated. So it's a pleated surface Okay Yes That's all to see Goes to H3. Oh, I'm sorry So F tilde Takes me from the universal cover if you like H2 to H3. Thank you That's the bending was one feature to notice about this which is that So, how do you make a pleated surface? so I can take Maybe I'll just think about taking a pair of ideal triangles and mapping them in and I have a couple parameters to keep track of One is I can keep track of the bending angle So they can go in along some angle the other thing you can do is Remember along the geodesics along along this arc I only need to take this this geodesic to a geodesic parameterized by arc length So one can also shear So I could take this ideal triangle I have a pair of once I've bent I could take that one ideal triangle another ideal triangle and shear one against the other That maintains the bending angle But it is a somewhat different picture There is in some sense there are natural feet Here and so one can very much Either keep the feet at the same relative distance or displace them So there's some amount of shearing and shearing and bending are the basic operations you can do to a pleated surface Okay, so That's the data. Here's our results. What do I have? Thank you. I can see it here. It's perfectly good Let's try Okay, so what I'm going to do now is tell you our results. There we go, which I put on slides because they're so long to state So the first result is this again with my collaborators at Spoboda and should went worth So we let the representations tend to a limiting configuration remember limiting configurations from Missas Bobo device and Vita earlier in the talk and let UT be the corresponding harmonic map So it's root is UT is roti equivariant Suppose our lamination let lambda Phi infinity be the straightening of this horizontal Foliation so I move from foliations to laminations by some natural geometric process, which I did a poor job explaining Okay Then there is a pleated surface in space which is Row equivariant for some row some other row So that if I want to know what the image of that harmonic map is I take This pleated surface and I shear it exactly by by roughly t to the half So and one can track this family of representations heading towards that limiting configuration by building a pleated surface out of Phi infinity and then Shearing that surface with t and the as t gets very very large You get closer at the image of the harmonic map gets closer and closer to the sheared pleated surface So it's very much like this picture here where the harmonic map gets very close to some sort of bent picture That's structural result number one. So to a limiting configuration. We have a Pleated surface or more precisely a sort of shearing class of pleated surfaces. I don't actually know which one we're starting with Energy the equivariant energy of the harmonic map Yes, yeah, yeah, yes I really need well remember I said here that as When t is very large Then the point here start thinking they are mapping to geodesics But the key words are when t is very large or distance is very large So I'm really when I said this I was thinking of a quadratic differential That's t times z minus a times z minus b. So something that's That's very far out in quadratic differential space Okay, yeah We're I got more results. Okay. So first is there's a pleated surface Okay, so in short harmonic maps are well approximated by pleated serve by shearing's of pleated surfaces along some lamination you build out of out of and out of the hop differential, but But but not so much the complex the complex structure is going to go away We're really interested in that in a topological feature of the hop differential interested in that foliation So the Riemann surface is not so relevant Yeah, you get a different quadratic differential. We're coming to this Well, let me let me give me give you partial answer. I'll give you a partial answer Okay, my looking at my watch says if I care about time so that the creates the illusion of concern for the audience that the The quadratic differential gives a the horizontal foliation is a measured foliation. It's a topological object. So The other When I change x to x prime I can find a quadratic differential Q prime whose horizontal measured foliation is the same So what is the point is what is relevant here is the topological aspect of the quadratic differential that relates to the Riemann surface and not so much else We're coming okay now. I'm gonna come I'm gonna move on just give me a moment Okay, and so now what about the bending angle? I hope I'll answer your question by in the next minus three minutes So Again, we might have two limiting configurations Fine infinity a zero infinity if you like think of this as fuchsian or just think of this as a base point and another one Which was alpha away from it in terms of these limiting configurations in terms of these one forms of values in this real line bundle okay Now one of these these are one forms and I can check that that if we're on a Riemann surface and I have a zero p and q so p and q that's bad notation p and And r Are in the zero set of the quadratic differential q I can check that if I integrate along the path between p and r that Remember each one of these points is a zero. So each one is Going to an ideal triangle in this sort of bent lamination so this path is developing to some path on this pleated surface And I can check that that that the integral formally defines a bending co-cycle That's it. This is now technical, but it formally defines a a way of Describing the bending of a pleated surface It measures the total bending of an arc. So with that So therefore alpha determines two pleated surfaces from two different perspectives the first one is I can take my original pleated surface associated to that limiting configuration and I can use alpha and this bending data to bend it so now I have a Have a pleated surface that way just by taking on one pleated surface and bending it more The other point is that a zero infinity plus alpha is a limiting configuration So it is it is a limit of representations. So it also corresponds to a shearing class of pleated surfaces So that's another pleated surface, which I wrote sigma sub alpha here to be cute So that arises from the first proposition. So we have two pleated surfaces associated to this alpha Okay, and so the theorem is of the same thing. So So so alpha real is so alpha here is representing the bending of the pleated surfaces that are associated to To these limiting configurations now the point is that When I'm talking about shearing and bending I'm no longer referring to the background remon surface x the the Domain of the harmonic maps the fixed object on which we built the Higgs bundles. So Forget that Nope, maybe I should say. Oh one way of reading this theorem is that These alphas these harmonic forms are it's a sort of hodge theory result There's in some sense harmonic representatives of bending co-cycles through these topological objects You give me a remon surface and now I find a harmonic version of those bending co-cycles Okay, let me not tell you the proof Say that in a minute. And so now if we slightly change x to x prime what happens to this description to limiting configurations Well, Phi infinity will change to Phi infinity prime, but those will share the same vertical measured foliation That'll be the by the construction that comes out of it and alpha will change to some other alpha prime but But the bending is already fixed geometrically So these alpha and alpha prime have to have the same periods Oh, there's a missing technical hypothesis here on the quadratic differentials that I'm not going to talk about So the point is that this gives us some topological description of some of the generic limiting configuration independent of the choice of background remon surface and I'll stop Well, there's a we were talking about limiting configurations which are Defined for genetic in the generic stratum so Certain parts of this theory work fine in other words There is right This sort of description of you go far out and you get Things that look like geodesics that works fine. And so you have some You find some surface which has boundaries which are geodesics As Scott points out if it's an ideal try if it's as a simple zero then you there's only one Possibility for those complementary domains those are ideal triangles But in general you might have You can still sort of talk about You know have some control on the harmonic maps and in particular, you know, there's one of the questions here I many people is You know what happens at the higher order zero case and one can sort of imagine that what happens is in the My super joy brings up the in the case where the if you start with an SL to our Representation so you're mapping to H2 and so your pleated surface is not bent at all It just all lives inside a hyperbolic two space which can't be bent Let's see and then you start to to bend You might leave a a hyperbolic ideal quadrilateral Which would correspond to a second order zero and then the sort of ways that the Two different frontiers would fit together Maybe possibly I don't know would be you know by insert a diagonal in one of two ways neither bend this way or bend that way, but that's the Hieracodimension strata and have to spill those and fill it in and glue things together Then the answer I have to I'm not I'm not sure what an acceptable answer is well, oh, okay, so there's a description of Okay, so there's a number of ways of saying this Here's one way so there's a Famous compactification of the representation variety by real trees due to Morgan and Shalen, okay, so what is going on there? So what is happening there is that? They're looking at lengths at you take a collection of curves a gent nice Generating set for pi one on the on the Riemann surface and Under the representation if you're leaving all compact sets in a representation variety Some of those curves are getting very very long. So one can look at like the projective limit so what is happen and So they the harmonic map we know is respecting this representation It's the harmonic map is becoming like a pleated surface. So what is happening to these curves? Well, this is I don't know how well I can draw this but If I'm looking at a curve whose Whose which lifts Or which you know starts here and here's another point in the orbit and it's you know makes its way from here to here Well, then what happens is you can represent it by a curve Which does something like that now remember I'm shearing a lot So from here to here Along the foot of the geodesic along the foot of the triangle that's always bounded ideal triangles There's only one ideal triangle. This has length 1.3 some number then from here to here It's like t to the half times the vertical measure and So here's t the half times the vertical measure There so one can predict the lengths of these curves and From this one learns that the Morgan-Shellid Compactification by real trees is exactly what you get by projecting this Fullyation down to its leaf space Right, so you sort of see the leaves here So this projects to I don't know if I can do it right, you know some some tree the issue is that when you do that you have Collapsed all these tori these compact things to nothing because we're only looking at the quadratic differentials So what the pleated surfaces do is somehow refine the Morgan-Shellid Compactification by inserting these sort of bendings Another feature which I'll and I we have to stop I know I'll release the prisoners in a moment the So Francis Bonahan is a very sophisticated theory of pleated surfaces and he has a description of Some portion of the frontier that is reminiscent of this Where he fixes a maximal lamination so lamination where all the complementary domains are ideal triangles and he Parametrizes the end or the space in terms of bendings and shearing along that lamination so this is some version of And that will work for any lamination any Generic lamination, so this is somehow like a diagonal version of that Maybe I'll just shut up now