 Right. Yeah, so I want to thank Shemiao and the other organizers for inviting me. I guess I should be able to share my screen somehow. Or maybe I don't have the… Yes, yes, one second. Permission. Good. Okay. Let me spotlight you. Okay. Actually, I'm still not seeing the option to share screen. Am I? You, I think, are allowed to share screen. Huh. That's strange. Yeah, I don't have the normal option here. Maybe I can also try. Oh, wait. Let's try this. Here. Sorry about that. There we go. Can you all see this? Okay. Sorry for the slow start there. Okay. Yeah. So once again, thanks for the organizers, for the invitation to talk. I actually have just gotten my first COVID vaccine. So I have… I'm very hopeful for maybe someday being able to come to Maryland in person. But for the time being, I'm excited to give this talk today. Okay. Right. So I'll be talking about killer Richie solitons and Toric geometry. So we'll start with some introductory things here. Just a check. Can you guys see my cursor when I move? Yes. Okay. Good. Just as I highlight things. So we'll start with a complex manifold. And the complex dimension will be n. This j is… The standard notation I'll be using is an endomorphism of the tangent bundle, which squares to the negative, the identity. And we'll say that Romani metric is scalar if this j is an orthogonal transformation with respect to this metric. And if this metric makes this complex structure that was given, parallel. So, you know, the… One of the standard things you will do in scalar geometry is instead of really worrying about the Romani and metric so much, you get all the information you need from a corresponding symplectic form called the scalar form, which you get in this way. And you can… If you introduce some complex coordinates, then you can see that the… This gij bar here is a Hermitian positive definite matrix that everything you need to know about the metric as well. So that's the setting we'll be working in. So, a Kailer-Ritchie soliton is a pair of things. It's some information. This Kailer form and the holomorphic vector field. We asked that to satisfy an equation. So here I have the Ritchie form and the term was involving the lead derivative. And then this is a multiple just of the metric itself. And so if you are familiar with these things, if you've seen these things before… Oh, what happened here? Sorry, technical difficulties just don't seem to stop here. Is it something… Yeah, sorry. Can you see again now? We were always seeing you. We were seeing you. Sorry, something might have just gone more wrong on my end. So you can see this now and everything's back to normal. Okay, apologies again. Yeah, so just if you've seen these things before, there's a lot of different conventions about the signs and the coefficients in front of all three of these terms. So just for simplicity in my setting, this is the most convenient way to express this. So in particular, you might see a negative sign on this term. And it's the same thing, just switch the sign of the vector field. The metric is called Kaler-Einstein. If there's no vector field term here, it's an old-standing thing to study. And we can rescale things so we can just as in the Kaler-Einstein setting, if you're familiar with that, just to make sure that this lambda is plus one, zero, minus one. When it's positive, we'll say that this is a shrinking soliton. When it's zero, we'll say steady and expanding in the negative case. And this has some correlation because the Regi form represents the first-turn class of your manifold. This has some condition on what the homology class of this closed-to form can possibly be. So in the shrinking case, it's going to be inside the first-turn class. Okay, so we say that it's a gradient if this holomorphic vector field, which is part of our original data, is the gradient of a smooth function with respect to our given metric. Remember, right? So if given a soliton, this G is the soliton metric, I say that the vector field happens to also be the gradient with respect to G of some function. Well, when you work things out, it turns out that this lead derivative term becomes exceptionally simple in the Kaler case. And I get a simplification of the equation that looks like this. So I just have an ideal bar of this function f, which we're going to call the soliton potential. If you hear me say that, that's always referring to this function f. And so there's a lot of adjectives, shrinking gradient, Kaler, Regi, soliton. These things are, I think, of interest originally because they are models of what can happen along the Regi flow when you reach a singularity where the curvature is not getting too large as you approach the singular time. So it's kind of the nicer singularities of the Regi flow are, in some sense, modeled off of what these metrics, at least usually on a non-compact manifold. But for reasons which are not really so much that, for the rest of the time, we'll be mostly considering these. So shrinking gradient, Kaler, Regi, solitons. So just some examples, pretty fast here. So on Cn, you take the standard Euclidean metric, whose simplecate form looks like this. If I just take my vector field to be the Euler vector field here, well, obviously the Regi curvature of this metric is zero. And if I look at this lead derivative term, it's a fast computation to check that this is equal to the original metric itself. So this actually just says that the metric is a cone, if you're familiar with that. So this is a shrinking gradient, Kaler, Regi, soliton. With respect to this vector field, it's called the Gaussian shrinking soliton. The reason that it's gradient is, I mean, you can see if you like that this, if I let this function just be the norm of the radius squared, then the gradient with respect to the Euclidean metric is just the Euler vector field. So Gaussian has something to do with the associated measure space. And we'll talk about really. Okay, so another key example that we'll need to see again is the Fubini study metric on Cpn. You choose the appropriate normalization in front of this thing so that it lies in the first term class. So this is actually a Kaler-Einstein metric. So like I mentioned before, this is just a Kaler, Regi, soliton with no vector field. Any such thing is clearly gradient. You just take the zero function. So that's another key example we'll run into again. Some less trivial examples. There's these classic ones, Feldman, Neumann, and Kanoff in 2003. They looked at negative line bundles over Cpn minus one and used the natural UN action and sort of reduced that problem to solving an ODE. Found some shrinking gradient Kaler-Regi solitons on the total space of this line bundle, these line bundles. And much more recently is a generalization of that. We have examples of Futaki. So if I replace Cpn minus one by some sort of any suitably nice compact Kaler manifold and I replace this O-negative K by any negative line bundle with this again, sufficiently nice here. So these are going to be roots of the canonical bundle over Torre quantum manifolds and we'll encounter what that means a bit more later on. There are also shrinking gradient Kaler-Regi solitons on these spaces as well. Okay. So what I've been concerned with is the question of uniqueness. So just a few words on like the history of solving this problem. So for the moment, we'll focus on compact manifolds. That's the historical starting point here. And the original question is about Kaler-Einstein metrics. This is due to Kalabi in the 50s. He showed that these are, you know, if one of these exists, at least in its unique, assuming that this lambda is zero or negative one. So that would be zero first-turn class or negative case. Lambda being positive is a sort of substantially more difficult problem. And one of the key reasons is that you'll have holomorphic automorphisms in that case, and you won't in the others. So the issue is that if I have a, you know, a diffeomorphism that happens to also pull back the complex structure to itself, then this pullback metric will still be Kaler-Einstein. And that's going to be a similar problem in the soliton case. But it is unique up to this action of the automorphisms. That's proved by Bandua-Mabuchi in the 80s. So far later. So even later, talking about more general Kaler-Einstein solitons. So actually the zero and negative constant case turns out to be simpler. There's kind of a quick trick you can do to show that in fact, the vector field has to vanish if you're going to have on a compact manifold, a shrinking or a steady Kaler-Einstein soliton. So that kind of, you know, you rule it out immediately. It reduces to the Kaler-Einstein case. For them to equals one, this is known now, proved by Tian and Zhu in the early 2000s. Again, this type of result of Bandua-Mabuchi unique up to that action of automorphisms. Okay, so I'll be mostly interested in the non-compact case, because the compact case is solved. And in general, non-compact manifolds, it's very hard, or it's a subtle issue, the issue of uniqueness. And it's because of maybe the obvious reason that there's a lot of possibilities of what can happen as you go off to infinity. So some results that I want to highlight that are things that are something you can obtain that your techniques could be maybe more similar to the compact case. These results of Coxwar and Wong show that on some special manifolds, if you look at metrics, which are already asymptotic to some fixed nice metric, then there's only one such shrinking gradient Ricci soliton, which is asymptotic to that fixed thing that you start with. So if you're interested in some cone metrics or a cylinder, you can get some results like this in there. And there are more results, more people I probably should mention that I don't know about. The result that I want to highlight as something that, you know, I guess I would like to aspire to is this recent result of conlanderable and sun, which says that if I have a complete shrinking gradient, Kailor Ricci soliton, and if I just assume that the Ricci curvature is bounded, and I'm interested in a few key underlying manifolds, like the Euclidean space, or the total space of these line bundles that were considered by Feldman, Ilmenin and Knopf, then you also have uniqueness up to automorphisms essentially. So here this says, if the manifold is Cn, then this soliton has to be the one that I mentioned earlier, the Euclidean case, the Gaussian shrinking soliton. And if your underlying manifold is the total space of these negative line bundles, then again up to automorphism, this, your thing is isometric say to those ones I mentioned earlier, constructed by Feldman, Ilmenin and Knopf. So this doesn't mention much about the behavior of the metric at infinity, just maybe bounded Ricci curvature. So that's a bit more flexible than saying it's already asymptotic to those, has to be equal to them. This is just saying pure uniqueness as long as the Ricci curvature is bounded. And there's some good indications that Ricci bounded hypothesis could be removed. So to state my results I want to, I need to introduce a bit more. So to the setting I'm working in is that of Toric manifold, the title of the talk at least advertises that. So again, we're going to be considering a complex manifold of complex dimension N. And we're going to suppose it meets an action of the complex torus which is just C star to the end. So notice that this N here is the same as the complex dimension. And with some nice properties. So first of all, it has to be an effective action. So that means that there's nothing in C star to the end that does nothing. So if there's a, every element of C star to the end moves at least one point. It's holomorphic, that's probably a natural assumption. And I'm going to also put a slightly more technical assumption that the fixed points that consists of finitely many points. Of course, if the manifold is compact that's going to come from free. That's the data that I'm going to call a Toric manifold. So this complex manifold together with this action. And it turns out that in this setting there's always going to be at least one point and of course once there's one point will be many such that if I look at the orbit of C star to the end of this point. It is open and dense inside of the, the manifold I'm considering so you can you can view your manifold as just C star to the end with some extra things thrown in at infinity. And again, if it's compact you to completely compactify as a partial compactification because you might throw something in on some spaces and then it may still stand off to infinity in other directions. So an important feature of this is that there is the inclusion of the real tourists maybe obviously inside of C star to the end which just looks like the unit circle on each of the C star factors. And so that gives rise to an action of this real tourists and we're going to be interested in Taylor metrics on these manifolds which are which have this torus as their symmetry or inside of the symmetry group. And some silly pictures I drew just to get a to get a sense of what these things look like. Right so on C, you have a C star to the C star action just given by multiplication right. And on CP one you have essentially just the stereographic inverse stereographic projection of the one I drew C star action, which you know in the right homogeneous coordinates looks the same as this. I think that's fairly straightforward I just wanted an excuse to draw some pictures. So just to note that the Euclidean metric is preserved under the S one action here, and the Fabini study metric is preserved of course under rotations along an axis. So you know a few, a few less trivial examples but still fairly trivial I guess CC to the end I can I can look at the same action that I just drew just in more dimensions. So just given by multiplication. And on Cp and I do exactly the same thing right so I just take that action on CN and I look at that I add a CP and minus one and infinity. So if you look back on this slide here, you'll notice that I have a copy of see without the origin inside of C. And if I delete these two points I drew purple here I have that that's my copy of C star inside of CP one. So that's this compactification picture. I'm also looking at the total space of O minus K. This is also Toric so I just explained to you why CP and minus one is Toric. Well, then I also have this line bundle, but the fibers of a line bundle come with the natural C star action you just multiply those fibers right. That's what you did on C. And so that will turns out is it is nice enough to to qualify with my extra conditions I gave gives us the structure of a Toric manifold. And indeed, the Fik examples are invariant under this real corresponding real action. It this, I told you these were UN invariant and it turns out that this real tourists it's as the natural sort of diagonal subspace inside of UN sort of standard maximal tourists in UN. Even better, you can do the same sort of thing on any on a negative line bundle over a Toric manifold by just taking the C star action which rotates the fibers and futakis recent examples of shrinking gradient killer cheese all tons are on spaces like this so they're over a Toric manifold, and they do turn out to be invariant under this natural real tourist action. So if you're not familiar with any of this they're constructed by this club beyond that's just a general way of producing some kind of metrics on line bundles and total spaces of line bundles. There's tons more examples of torque manifolds coming from algebraic geometry that I won't have much time to get into how to get those. So what is the my main result says that up to automorphisms. There's a most one complete TN invariant shrinking gradient killer Richie solid on a torque manifold so if I start with a torque manifold, and I have a killer Richie solid Tom. There's only one TN invariant one. So a few notes about this so first of all. Again, I'm not making any actually assumptions on what the behavior at infinity is at all. I think it's kind of cool it turns out to just work without without anything there so I don't say anything about boundary curvature even. I definitely am not requiring its asymptotics be prescribed. But this TN invariance hypothesis is fairly strong. So I replace that with extra symmetry. Another thing is that completeness is really key there. If you're familiar with any of these things you probably will have thought of 1000 counter examples already if I don't have the word complete here. So if I just look at the any inclusion of CN inside of CPN and I restrict the Fabini study metric. That'll give me something which is not the Euclidean metric and is also a shrinking gradient killer which is all a ton on CN is just not complete. And to generalize that example, there's this classic result of one too which says that on any compact torque follow manifold, there's a shrinking gradient killer which is all a ton. So I can just take the complement of any torus invariance of variety inside of this restrict that thing on my. So that's some non compact thing. And I'll have, you know, potentially many different metrics that I could have taken this way, none of them will be complete. Okay. But so that's just a note about completeness will come back to that in a bit. Another note here is that a lot of the time this T again invariance is not an ideal assumption so maybe we could replace this with some more natural geometric condition. So again, same setting here we're going to start off with a complex manifolds possibly non compact and just I'm always going to highlight the complex dimension is n. So real dimension to n. And now we'll work in a slightly different setting so now I won't assume that it has an action of this full big algebraic or complex torus, but just some an action which is still effective and holomorphic but of the real torus to answer some product of s ones. And again this and here is going to match the complex dimension, just sort of the maximal dimension you can have an action like this. So if you differentiate this action, then you'll get the, the lead algebra of this torus inside of the space of holomorphic vector fields on your manifold right so this action itself is holomorphic. So that when you differentiate that'll mean the vector field you get this holomorphic. Okay, so that's just to understand the statement here. So, the, the, another version of this of this uniqueness. And I approve says that up to automorphisms by start with a non compact torque manifold. But maybe I shouldn't say torque I should say supposedly I'm in this setting so MJ admits the section of this real torus, and I have a shrink ingredient complete Caleb Richie soliton here, and the and the Richie curvature is bounded. And I assume again that the vector field is in this lead algebra. So I put some assumption on the vector field here and I put some assumption on the Richie curvature but the reach curve assumptions fairly mild. And this is, this is somewhat more restrictive. But the conclusion is that if it does indeed admit so, so there's only one such metric, which has this property so this uniqueness with among those which have found a Richie curvature and whose vectors will satisfy this condition that if I apply the complex structure to them. It's one of these special holomorphic vector fields obtained from my action. And if it does admit some, then it turns out this action can be complexified and I will say in just one second what that means. But this thing is by holomorphic to my original space is by holomorphic to a quasi projected torque variety. So what do I mean when I say this thing can be complexified it means that the remember I was starting off in this case with only an action of this real torus. And that means there exists an action of this far bigger torus. And corresponding, you know, that extends this action right so I look at the underlying real torus action of that thing it's my original one. So, if I start off with just this real torus action I don't know a priori that I have this big open dense set which looks just like she started the end, but it turns out that you do. If you admit killer Richie saw Tom, which is complete. I can summarize what I'm saying here this is similar to the previous theorem just instead of assuming a priori that I'm T and invariance. I only now assume this condition on the vector field and I assume that the Richie curvature is bounded. But the conclusion is basically the same with this extra nice thing which says that this thing, which a priori didn't have as nice restructure turns out to. Okay. So, this business about complexification if you're used to working on compact manifold might seem silly. The TN action can always be complexified on a compact manifold, given the my previous setups. And the reason is essentially that all vector fields are complete right so if I have a TN action, I get this inclusion of the Lie algebra inside of the space of holomorphic vector fields, and I can take J of all those things, and then flow out in that action and try to get a C star action from that. And if all of those vector fields turn out to be complete then you can do that. Right so you can you can obtain a C star action from each of those vector fields, put that all together to a C star to the N action. But if your manifold is not compact then it may well be. And we'll see an example in a second that if I do that, even if my original vector fields complete if I take J that may not. I apologize for all the sirens on my street here. We can hear that. Right so that's like I just said if them is not compact this isn't true anymore so I could a nice example is this unit disk inside of see right so by look at the S one action on that that certainly preserves the unit disk. That's effective didn't know elements of this one does nothing. But there's no way that I can if I if I was able to take J of that corresponding rotational vector field. Well what I get is the Euler vector field and the C corresponding C star action just would would give me a few morphism between the unit disk and see which we know can't exist right so certainly this this doesn't work in the non compact setting in general. And what the theorem says is that there's a that you can fix that problem. Right so lastly, here. Remember that we have these solitons on CN and on CPN. And I'm going to choose the appropriate vector field here. And the salt on equation is really good about products. If I have a one on this space one on this space I take the product I can just take the product metric it will also satisfy this equation with the product vector field. So what I'm going to just for in terms of names I've already kind of said this I think once if I, the product of the Fubini study metric on Euclidean space or sorry on the on projective space and the Euclidean metric on Euclidean space that product metric I'm going to call the standard cylinder. So it's maybe a bit silly to say that since none of them are actually that the normal cylinder but in any case. The standard cylinder on CP one cross C is the unique shrinking gradient killer which is all time with bounded scalar curvature. Now again just up to these automorphisms. So here, I only assume bounded scalar curvature and I don't need to make any assumptions on the vector field so this is a corollary of this previous thing, and the proof basically says, you know, if I have such a thing. I can find in one automorphism which makes the vector field whatever it happens to be in this nice set of vector fields right so if you go back real quick. This condition that Jx is in the Lie algebra is going to be the key thing there. Okay, so that's the last result that I want to mention. And for the rest of the talk I'm going to sort of get into some of the key steps of how you prove things like this. So introduce a little bit more background to do that. And so, so we're just in the setting of having a killer manifold now so I'm suppressing the, the complex structure in this notation here. And we're going to be assuming that there's an effective TN action, which preserves our original killer metric, right. So again there's a complex structure that this preserves as well too. As I mentioned, this is in particular a symplectic form and we're going to actually make use of the symplectic properties explicitly. So the most important thing for us is going to be the notion of a moment map for the TN action. And remember, so I'm going to try and highlight here. The difference between this real torus action and the complex torus action so here I'm only considering having a real torus action on the killer manifold, possibly not compact. And as I say a moment map what I'm referring to is smooth map so it takes you from your space into the dual the algebra of your torus. And it satisfies some equation. I've got it here. What does this mean. So here, this le algebra like I said is we identify that with the vector fields on M. So that's what it means to take the interior product of your killer form with something in the le algebra. And the pairing here is just denoting the dual pairing between T, the le algebra and its dual. So if you like to choose this natural metric on the le algebra. This is just the dot product on our end. Right, so that some some equation like this so I mean maybe that looks like nothing if you haven't seen this before. But if you've seen any basic symplectic geometry you might know what a Hamiltonian function for a vector field is. What a moment map is is just a collection of Hamiltonian functions for all the possible vector fields to be considering. Right so just kind of stitch it, you get one for each one you stitch them all together in a nice way. So I think the action is Hamiltonian if we have such a thing right so with the way to think about this is I have a killer manifold I have an action that thing may or may not be Hamiltonian if it is that means there exists one of these moment maps. And one of the features that that should keep in mind in terms of thinking about the geometry of this thing is that essentially these are constant on you can make a choice such that these are constant on the TN orbits right so if I have this TN action on my manifold. The moment map doesn't change value here so we'll think about this as a sort of vibration. Okay, so we'll have to touch on a little bit of the algebraic geometry of this I'll try to keep this to a bare minimum here. So, even I guess farther down and I have to talk a little bit about combinatorics. So, part of the what I need to introduce here is polytopes and polyhedron so what does that mean. Just to make sure you guys have seen these things before but in terms of the terminology when I say polyhedron, that's going to be anything which is a finite intersection of half spaces. And a polytope is going to be inside of our end it's a finite intersection of hacks half spaces which happens to be bounded right so when I say polyhedron should imagine like something possibly extending off to infinity but polytopes like a square. So that's hopefully not too bad. And then there's some sort of magic algebraic geometry that you can do which if you're given a suitably nice polyhedron, you can cook up a quasi projective torque variety. Well the only thing I want to note about this process is that if I have a polyhedron inside of our end, then the dimension of this variety you cook up is the complex dimension is also in. So I guess not the only thing I want to mention I want to say this is turns out to be always a torque manifold. And the compactness of this thing is directly related to the compactness of the corresponding polyhedron so if I start off with a non compact polyhedron so just a regular old polyhedron, then the thing that I'll produce is going to be non compact algebraic torque manifold. And if the thing I start up with it is compact, then I'll end up with a compact one. So I don't want to say anything about how this process works just there's some sort of you can wave your magic wand and it makes one for you. So some key examples. Again, you'll see this over and over again. If I start off with will certainly just the half space inside of our consisting of all positive numbers. And if I do this magic I'll end up with C. And if I choose a closed interval inside of our and I sort of wave my magic wand would I'll get a CP one. So I don't want to say too much about that just that there exists such a thing you could do. And there's a reason why we need to care about that. So going back in time a little bit. So, in terms of what we want out of these moment maps. How does this relate to geometric analysis. We'll see the kind of this a few steps and then we'll see what what we end up getting. So, this classic result of a Tia and simultaneously Guileman and Sternberg says, but have a two and real dimensional compact symplectic manifold with an effective and Hamiltonian TN action. So what that means is that there's a moment map and then the image of this moment map is always a polytope. Okay, so what that means is you can have this nice picture of my manifold, I got my moment map maps down onto some nice on easily easy to understand subset of Euclidean space, and it's kind of it's like a vibration over most of these points you just have TN orbits. So you'll see we'll see this will help us understand the geometry. The connection with complex geometry is that in that same setting, if, if my manifold was cooked up by this magic algebraic procedure, then the polytope that you get is the image of the moment map is the same one that you started with. So if I start off, for example, with the open interval zero to infinity and R the map that I'll get C. And if I have some, if I pick some TN invariant scalar metric on this thing and I have a moment map, then the image of that moment maps going to be that interval. I guess I should have chosen a compact example, but we'll see that with the suitable choices that's going to be true in a non-compact setting as well. So the conclusion, like I said, is that you have a correspondence, this polytope gives you a correspondence between compact Hamiltonian manifolds with the TN action and compact complex manifold to the C start TN action. And what do you want to do with this? Well, what you would like to do, if you're interested in studying geometric analysis, is to understand equations better. Using this. So how can I do that? Well, essentially this moment map lets you get some nice kind of coordinates on your manifold. So more precisely, if I look at the interior of this polytope, then it turns out that this moment map is actually a diffeomorphism. So remember, I told you there was always, so how should I say, there's a big open set where this is going to be a diffeomorphism and between the interior of this polytope cross your torus and this open set. What it says is that even in that just the purely non-complex, symplectic setting, I can write my torus manifold as a sort of compactification of something big. All right, so this is something easy to understand. This is a product of some nice subset of Euclidean space and this torus. And moreover, if I have a TN invariant scalar metric on this thing, then I have a nice explicit representation for what it is in the quote coordinates given by this product here. So it turns out there's a sort of potential function, which I'm calling you. And this is just a function on the polytope, right? So this doesn't depend on the real torus and that makes sense because I'm asking for the metric to be invariant under the torus, right? And your metric can always be given in such a form. So this is when I say Uij here I'm referring to the Hessian matrix with respect to the coordinates on P. And this is the inverse of that. So the entire metric structure is just determined by one potential function. Again, even in the non-complex setting. And that sometimes referred to this as a symplectic potential. Like it's similar to a scalar potential. So why would you go through all this song and dance? Well, it turns out that when you do all of this, you can get a sort of massive simplification in a lot of cases. So the famous example of how this was used was to study scalar curvature. This was done originally by a bro and then really, really made use of by Donaldson in 2002. It was sort of a big inspiration for a lot of this conversation around case stability. That's going on now. So you get this nice expression for the scalar curvature which looks like, so I take the Hessian of U and I take the inverse. And then I look at all the second partial derivatives of those terms and I take the trace of that thing. So maybe that sounds complicated, but it's a fourth order equation. So this turns out fairly simple, relatively speaking. What we're going to be more interested in is that the Kepler-Ritchie soliton equation, which turns out to look, in my opinion, even nicer. It's only a second order equation. What it turns out to be, so if this metric G happens to satisfy the Kepler-Ritchie soliton equation, then this potential function U satisfies this equation. So determinant of the, remember, this is just a real valued function. This is a real Hessian is equal to some term involving the first derivatives of U. And the second piece of information here is this constant, which is uniquely determined by the soliton vector field. And so the upshot is that I can study complex monjeon pair equations on my original manifold. The moment map lets me sort of push that down to just studying a real monjeon pair equation inside of, on this domain in Rn for my potential function. It turns out to be a lot simpler to do in a lot of cases. So this is all in the compact setting. And there's a lot more subtleties that arise in the non-compact case, I guess, as you might expect. So, first thing is that if I just, if I state this in as big of a generality as I possibly could imagine, the image of a moment map on a non-compact, the amount of all doesn't even need to be convex. So certainly not going to be a nice polytope always. And there's even worse things that can happen. We'll see some of them in a minute. But the real crux of how the previous thing worked is that how can we reduce to a monjeon pair equation to a real monjeon pair equation, right? I mean, we needed to be able to work inside of Rn using the moment map. And to do that, to actually study sort of metrics which are varying, we need to make sure that that image is always the same of the moment map. So we can study all the possible metrics by just studying the one nice open subset of Rn. So that's going to be the goal. And remember this was achieved for us with Delzant's theorem, right? Delzant's theorem said, this is worth. If I have any on a compact manifold, Hamiltonian Tn action, then the image of the moment map is this one fixed thing, this one polytope which I gave, which gave me my original manifold, right? So if I'm working in the setting of manifolds with the C star to the n action, then I always have a fixed image of the moment map given to me by Delzant's theorem. And so we need some analog of that in a non-compact case in order to make this picture work. And so actually the first main step and the proof of all of the theorems I mentioned at the beginning is to prove a Delzant type theorem for certain non-compact manifolds. So for us, it's going to be, if they admit a complete shrinking gradient in Kepler-Ritchey's soliton, then the conclusion is that you know what the image of the moment map is going to be at priority. So then I can study only things on that. It turns out, unbounded polyhedron. So that's the first step is to see if you can reduce searching for shrinking gradient Kepler-Ritchey's solitons, which is a complex monjon pair equation on the manifold. See, it turns out you can reduce that to studying real monjon pair equations on the polyhedron set of Rn. You have to identify what that is. And then once you do that, then there's a little bit more standard machinery that can kick in. So you need to prove a theorem which establishes the uniqueness of those real monjon pair equations itself, right? So I'd like to prove uniqueness for Kepler-Ritchey's solitons. I reduce that using the moment map to studying a real monjon pair equation inside of Rn. And then I prove a uniqueness theorem for those real monjon pair equations. And the way you do that is you study the ding functional, and this is something that is kind of classic in Kepler geometry now. It has a really nice representation in the TORRIC picture. So I mean, this is a bunch of stuff, maybe hard to read all at once. I think the important thing to sort of recognize about it is that if I remember what my Kepler-Ritchey's soliton equation looked like, if I differentiate this thing, then you might imagine sort of this part, or sorry, this part here coming from this term and then the rest coming from this term. And so it turns out that that intuition is correct and that the critical points of this functional are solutions to this real monjon pair equation. Okay, so in the remaining time, I just want to sort of break down a little bit about how you prove the second part. Since I guess this is a geometric analysis seminar, I'll focus more on the analytic perspective. So, right, like I mentioned, there's always this dense orbit. Inside of M, sorry, that's just a little bit of the setup of how you like, how do you do this in practice? There's this dense orbit inside of your manifold. So we'll be working on that. And the nice thing is it's like a big coordinate chart. And it's sufficiently big that we can kind of ignore everything else in a sense, right? So it's open-ended stents. So that's what I mean by when I say by continuity, it often suffices just to work exclusively on this open set. If you know that things extend properly, then you just basically have everything that you need. So we'll be studying invariant scalar metrics, which you can write as I double bar of a function on you. It turns out that's not a very restrictive thing to ask for. So we're on C star to the end. Just imagine we're on C star to the end now. And we're looking at scalar metrics, which are just I double bar of a potential function, which do exist since we're on C star to the end, right? So we'll take these natural coordinates on C star to the end, and it's going to be super convenient to work in the logarithmic coordinates. So I'm going to call C is going to always mean the real part of the log of Z, and then theta is the translation version of the universal cover of the real torus, maybe. Okay, so this is still some big open set in here. And so the condition that this metric is T and invariant is equivalent, right? If it's I double I double bar of a function for this potential scalar potential function phi here to only depend on C, the real part of the log. And that's just saying it only depends on the radius, not the what have you rotating part. So what that says, if I now go back and say what does it mean that Omega is I double bar of this function, then if I write this in sort of these real coordinates in the logarithmic, the real coordinates corresponding to the logarithmic whole morphic coordinates, then my Omega is the hashing with respect to C of phi in front of this standard symplectic form, which says that that has to be positive definite matrix, which in turn says that this function here is convex right so you can think that you have studied any like very potential theory here used to your your killer metrics being related to I double bar of place of harmonic functions. So we kind of the T and invariant was to cut out some of the complex geometry there, and the corresponding analog is just convex functions. Okay. So a little bit of the, what's the this link between this complex picture and the symplectic what I was describing earlier. Well, again, we're going to be working on this dense orbit here. And we have that our Omega is I think it's useful to just see this here or Omega is basically even by the hashing of this real function, real value function phi convex. So we have a moment map earlier right so suppose that we were in the in the setting where it says Hamiltonian. So we have a moment map. Then it turns out, if I'm working on on this side. So here I'm sort of working on C star to the end. Then I can find what my moment map is explicitly. So if I look at the Euclidean gradient of this function by right so just the vector valued function whose components are the partial derivatives of fire with respect to see so nothing to do with the metric. Then that that's what turns up to be my moment map and so if you roll back, if we're working in a setting where I know what the image of the moment map is a priori that tells me that the gradient of phi is this is this one fixed. It's compact, it's a polytope. So the connection between the complex and the symplectic geometry, this I mentioned the symplectic potential function earlier. That was something which was defined on the image of the moment map right and this function phi is something which is defined on our end. So the connection between these two things is determined actually one to one by each other which you might expect, since they both determine the metric. It's this thing called the Legendre transform, which is something you can do to convex functions. And it's given like this so this U turns out to be the unique function which satisfies an equation of this form. So this is perhaps a bit strange if you haven't seen it before but in any case, there is some way to determine the one from the other. Okay, so I think I may skip a bit talking about uniqueness of the vector field just in terms of time we might be able to come back to this at the end if there are questions. Instead, I will mention just uniqueness of the equation so the first step would be to prove that the vector field is unique. So there's only one vector field, which could be the vector field of a shrinking gradient killer, which is all it's on, you know, in such a way that it works in this torque setting. So then that basically just fixes what these constants in the real modern pair equation can be. And remember step one of a sketch was to fix the domain that these functions you depend on. So first we fix the domain, then we fix these constants see so the equation is determined. And now we want for the fixed equation, we need a uniqueness theorem for that. And so this is where the ding functional comes in. And the real issue here I mean the real thing that needs to be worked out is what is the right space of fun convex functions on this thing, once you're at this stage to be working on. The rest is kind of something which has been done before. Like I mentioned earlier the critical points of this functional solutions to the real modern pair equation. And so the the theorem is that this is a strictly convex function. If I have a function you, which is coming from a shrinking gradient killer which is all it's on on some torque manifold. It is the unique critical point right so it's strictly convex it has a unique critical point. And this is like the potential corresponding to my original problem that I care about is the unique such thing, if it exists. So how do you do that. Then my probably last slide I'll say here. Well, the, like the key is the moment map like I mentioned. And so you reduce this to a couple of elementary facts about convex functions. So basically we'll consider the, the, if I'm supposed I'm given two such things as this is a notation I'll let you sub t be that convex combination there. And then remember I told you there was this thing called the Legendre transform which lets you take any, any one thing and on. So there's this, it's a duality. So if I have a function on the convex function of the polytope, I can produce something on our end using that and vice versa. So if I have something on our end, they can produce something on the polytope, you make the right assumptions. So let phi sub t be that that process applied to this path. And then there's this kind of old school sort of, I don't even know what to call this it's a well it's an inequality. I think it was first observed by Berman and burns and this would be really useful in this setting, because what it turns out to mean is, let me just skip to that here. It turns out to mean, pretty much directly from this that you get this convexity of the ding functional from this. So the way that you do that is by observing that. If you think about the Legendre transform art enough, then this term here which appears in this ricopa inequality is exactly this term here, which appears in the ding functional. So you kind of have this kind of classical result which tells you automatically, if you can nail down the space of functions well enough, then you'll get convexity for free. That's what I was mentioning about how the real trick is figuring out the space of convex functions well enough. So I think it's just just because of time I'll stop here. So, thanks again. Thank you, thank you. Virtual applause all around. Are there any questions in the audience. I have a novice one so I'm, I'm not really familiar with this non compact complete world but I have to ask. So, in this direction you mentioned that there's already some uniqueness results about solid tones when there's some specific fall off conditions at infinity. Then you obtained some assuming is kind of global symmetry. It's either on the manifold or on your metrics or a combination. Is there sort of an overarching conjecture that is, you know, supposedly going to contain many or all of the results of this time in this area. Yeah, that's a good question. So, I think there are some such things floating around. I think that at least so what I know mostly about so I think that there's kind of two. There are worlds but they're kind of two worlds here and one of them is the, the, the Kailer world and then there's a sort of more general sort of shrinking solid tones world and in the Ramonian. I personally would be specific more more than the Kailer. In the Kailer setting I think that we mostly think these things are unique. So, essentially what you say is complete non-compact plus Kailer plus shrinking solid tone. Shrinking specifically, yeah. Yeah, that's I think the basic idea I think is, is that but it doesn't seem so easy to get your hands on that. I mean, that's so far hard without at least a curvature condition. So, the curvature condition that I use comes from this work of Colin Durwell and some, they need, it's kind of related to this completeness of vector fields issue. So that they use the boundary curvature there but there's kind of some indications that it shouldn't be necessary but kind of no idea how to remove it. So, so that's that. So, like, for example, how do I prove this one about CP1 times C. So the statement there was that there's only one with bounded scalar curvature. So no assumptions on anything except bounded scalar curvature. So this already kind of takes a lot of work. I basically have to show that you can cook up an automorphism of CP1 times C that takes your solid tone that you have and makes it invariant under the natural T2 action. So I think in terms of achieving that it's not it's not so clear, but I do think that's the thought. Okay, thanks a lot. Did any other questions arise. Yes, actually about this theorem, if you take this to the logic of a bundle of CP1, can you say anything that possible. Yeah, it might be the, but but I don't know where so I thought about a few possible extensions of this but it turns out to actually be quite a bit of work even just to get this from what I already mentioned. So like the way that the way that I do this. And there might be a better way this is just the only way I knew was to like. I mean it involved like a basically doing some Morse theory using the the what do you call this thing solid on potential. So you kind of, you know some facts about what the shrinking gradient killer Ritchie solid tons that potential function f turns out to be a kind of almost a Morse function and so you got to understand some things about the critic like the critical function of that that Morse function is basically I need to nail down where the fixed points of the action are, which can be kind of a priority. Crazy. So I think in this case I use this some parts of it that kind of are a bit special I think for two complex dimensions. But it doesn't seem like those are really necessary I just happened to need to use them for my argument. So I thought a little bit about it but it's not super clear to me how it would go just yet. Thank you. Any other questions. All right, so if not then we'll thank Charlie once once more accept our apologies on behalf.