 All right, so it's a pleasure to have CeCe Shen from Northwestern and she will talk about metrics of constant churn scalar curvature and the associated Calabi flow. Go ahead. Perfect. Thank you, Tamash. Okay. So today we talk about metrics of constant churn scalar curvature and a churn's Calabi flow. So let's begin. So firstly, I'm going to outline my talk when I start by discussing some background and setup. And then from there, we'll move into problems in the scalar setting where we will all give some examples of constant scalar curvature, scalar metrics, that's what CSEK stands for. And then we'll discuss existence problem per CSEK metrics. And from there, we're going to discuss how all the theory in the scalar setting can be generalized to the non-calar setting. And so here's where we'll talk about estimates for metrics of constant churn scalar curvature. We'll discuss a churned Calabi flow for Hermitian metrics and then we'll outline the proof of the estimates. Okay. So here's just some setup. So we're going to let x omega be a compact complex manifold of complex dimension n and omega a Hermitian metric on x. And so we can define our local coordinates, z1 through zn on x. And what I will refer to as a metric omega is in fact this 1, 1 form given in this way where gij is a positive definite Hermitian matrix. And we see that this 1, 1 form omega is scalar if and only if it's closed. That is d of omega vanishes. And so throughout this talk, I'll start off with a case where everything is scalar and the latter half the talk, everything will be in the context of non-calar. Okay. So let's discuss notions of curvature. So the churned Ricci form is a 1, 1 form given by negative dd bar log determinant of g. And so similarly, the first churned class is the double or dd bar class of this Ricci form. And the churned scalar curvature, which I'll denote by just capital R of a given metric omega is just the trace of the churned Ricci curvature for the churned Ricci form. I'll keep referring to these things nonfully. And so in the Kaler setting, the churned and the lovey-tuvita connections agree. And so the churned Ricci and churned scalar curvature are equal to the standard Ricci and scalar curvature. So in the Kaler setting, you have nice symmetries in the indices of the curvature tensor. And so no matter how you trace that up, you always end up with the standard scalar curvature. But in the non-Kalar setting, you don't have these symmetries. And so in fact, there exists four different possible Ricci curvatures and two possible scalar curvatures. And so I'm specifically going to focus on the case where we trace it a certain way. Okay. So let's discuss the clavifunctional. So the clavifunctional is introduced by clavie. And it's defined as the square of the L2 norm of the scalar curvature. And so you can rewrite this integral here as the square of the L2 norm of the difference of the scalar curvature and the average scalar curvature. So underline R denotes the average scalar curvature with addition of this L2 norm of this average scalar curvature. And so this is in the Kaler setting. So in the Kaler setting, this average scalar curvature quantity is in fact invariant of the Kaler class of omega. So as you can see, it's an invariant. It's a homological invariant. So just giving you some definitions here, extremometrics were introduced by clavie and they're defined as critical points of the clavifunctional. And constant scalar curvature metrics are a subset of extremometrics where they satisfy that the scalar curvature is equal to the average scalar curvature everywhere. And from the above equation, you can see that these in fact minimize the clavifunctional. Since this part is an invariant of your class, you cannot change. This is always me a square, thus positive quantity. It's minimized when this thing vanishes, which is exactly when the metric is CSEK. Okay, so let's discuss the claviflow. So given x omega not a compact Kaler manifold, the claviflow, starting at this metric omega not is defined by this evolution, where you evolve omega, this evolving metric omega of t in the time direction by dd bar of this scalar curvature of this evolving metric also dependent on t, where you start from this initial metric. And you can see that this doesn't change the Kaler class or dd bar class of the metric. And so you can in fact write the evolution in terms of this Kaler potential. So you can write the evolving metric omega t starting at omega not and then evolving by dd bar of this Kaler potential for some smooth Kaler potential, which we will call phi. And you can normalize phi so that it's unique. So let's give it, this is the standard way of choosing it. And in normalizing it this way, then the claviflow can be written in terms of evolving phi of t by the scalar curvature of the evolving metric minus r bar. So this is just some constant, which is to determine by the fact that we normalize in this way. And you again can just set the initial function to be zero. And so this flow here, the claviflow, is in fact a gradient flow for both the clavifunctional as well as the Buczyk energy functional. And fixed points are constant scalar curvature or Kaler metrics. As you can see here, if this vanishes, then you have a CSUK metric. So convergence of the claviflow. So there are known results about the claviflow. So Chen, he proved short-term existence of the claviflow and global existence of a flow under the assumption of a uniform Ricci bound. And Zekli Hidi proved that uniformly bounded curvature along the flow and proper Mabuchi energy gives that the flow converges to a CSUK metric. And Chen Sun proved that if the initial metric is essentially close to a CSUK metric in some sense that I'm not diving into, then the flow exists and converges uniformly to the CSUK metric. Okay, so now let's discuss examples of CSUK metrics. So in general in complex geometry or it's the existence of these canonical metrics within a given killer class are of large interest. And so CSUK metrics, six-year-old metrics and killer Einstein metrics can, people can consider them as potential candidates for a canonical metric within a given class. So firstly, killer Einstein metrics are a subset or are themselves constant scalar curvature metrics. And they're those defined to satisfy that the Ricci of the metric omega is equal to a scalar multiple of the metric itself. So Ricci omega equals lambda omega. And then up to rescaling, since the left-hand side here doesn't change up to rescaling, lambda you can assume is either negative one, zero or positive one. And so examples of these, in the case where lambda is positive one, we have the classical example of the Fubini study metric on complex projected space. For lambda equals zero, this is the whole class of clubby metrics. Example, the flat metric on Cn or on the torus. And then hyperbolic metrics on Riemann surfaces of units greater than equal to are examples in the case where lambda is negative. And so, loosely speaking, if you have a killer Einstein metric, then that is also going to be a CSUK metric. And since those are critical points or minimizers of the clubby functionals are also going to be critical points. And so they are also themselves extremal metrics. And so explicit constructions of CSUK and extubal metrics has been done by clubby, Huang, Guan, Tonneson, Friedman, and Huang Singer. Okay, so now we'll dive into the problem of the existence of killer Einstein metrics on a complex manifold, complex compact killer manifold. So if you have a killer Einstein manifold that is one that permits a killer Einstein metric, then that means that the first term class of the manifold, which is defined as this dd bar class of the Ricci curvature, is going to be equal to lambda omega in these three cases of the different signs of lambda. And so it was shown that the existence of a killer Einstein metric is equivalent to finding a solution phi to the complex Monjampere equation given by omega plus dd bar phi to the n, equaling e to the lambda phi plus f omega to the n, where f is the reach of potential of omega. And so we have three different cases for the lambda. So in the case where lambda is negative, or rather when the first term class is negative, the existence of a killer Einstein metric, omega, belonging to negative of the first term class. So omega is always a positive form. In this case, the first term class is negative. So now we're requiring that the metric belong to negative of this negative first term class was proved independently by Yao and Ovan in 78. And for the Khabi Yao case, that is the case where the first term class vanishes, the existence of a Ricci flat killer Einstein metric omega, it was shown that you can find one in any given killer class. And this was the result that was proved by Yao. So in the negative first-term class case, you can always, the statement is finding one within the negative of the first-term class. For the vanishing case, you can find one in any given killer class. So it's a bit broader of a statement. And then, now we'll talk about the Yao-Tian-Donaldson conjecture for killer Einstein metrics. So the most difficult of the three cases is when lambda is positive. These are the phono-manifolds. And so for phono-manifolds, the Yao-Tian-Donaldson conjecture states that the existence of killer Einstein metrics is equivalent to the algebraic notion of case stability. And the forward direction was established by Chen-Donaldson Sun, building on the work of Tian Yao and Tian in the case of phono-surfaces. And the result, sorry, the reverse direction was shown by Tian, Donaldson, Stopa, and in the most general form by Berman. Okay, so that was the Yao-Tian-Donaldson conjecture for killer Einstein metrics in the phono case. There's also a conjecture now for the case of constant scalar curvature metrics. And so more generally, the Yao-Tian-Donaldson conjecture is that the existence of a CSEK metric in a given killer class is equivalent to a relative notion of case stability. So in the killer Einstein case, it was relative to the first-term class, but here it's relative to any given killer class. So it's a slightly adjusted notion of case stability. And it is known that the existence of CSEK implies case stability, as shown by Stopa and Berman, and the converse remains open. And the difficulty, the main difficulty with the CSEK case is that, from a partial differential equations point of view, it's a fourth-order PDE versus in the killer Einstein case, it's a second-order PDE. So those two additional orders implies that the problem is a little bit more difficult to work with. And then Chen Chang in 2017 made progress towards showing existence of CSEK metric within a given killer class. Okay, so before we go more into existence, I'm going to discuss an obstruction to existence of a constant scalar curvature killer metric. So again, underline R denotes the average scalar curvature of omega on the manifold X. And so it follows that by definition that if you integrate R minus the average scalar curvature, it's going to go to zero. And so by the existence of a solution to the Poisson equation, you can always find this potential H, such that the Laplacian of this H is equal to R minus underline R. And so in a Chen by Tutaki that he introduces an invariant for the killer class where given a holomorphic vector field V that takes on this form. So V super I can be written as partial J bar of F for this potential function F, then the for talking variant for this vector field is defined as negative of the integral of this vector field acting on this potential function of pure H integrated against omega to the N. And so via an integration by parts, which works out straightforwardly in the Kayla case, this in this integral indeed equals the integral of R minus R underline R multiplied against F times omega to the N. And so it can be shown that this is invariant of the killer class. So up to the addition of DD bar V of a state function, this was won't change. And so you can see that if there exists a CSDK metric in a class, then this right hand side will vanish and since it's invariant, it will vanish for any F because this R minus underline R part is vanishing. And so this gives us an instruction, an obstruction if you do happen to find some V such that F of V does not vanish because that would imply that there cannot exist a CSDK metric inside that killer class. And let me know if I'm going too fast. Okay, so now I'll move into talking about a continuity path for CSDK. So a continuity path was also the approach that all Ben and now used in the killer Einstein case. Here we consider a different path, of course, because this is a different problem that Chen proposed in 2015. So for some real number T between in the closed interval 01, we define this equation here, which we refer to as star sub T. And so just looking at this, you can see that when T equals 0, the equation at hand is simply equating this trace sub omega phi of omega to a constant term. And at T equals 1, this part vanishes. So your equation at T equals 1 is simply that R of this resultant omega phi is equal to this C. So this resultant thing at T equals 1 is constant scalar curvature. And so our goal is to show this is a constant scalar curvature. So what this path is doing is like trying to continually extend a solution from T equals 0 to a solution at T equals 1. And so in order to do that, we define I to be the interval for all T in this closed interval 01, where equation star sub T has a solution. And so to show existence of such a metric, we need to show one way of doing it would be to show that I is firstly non-empty. And this is true because at T equals 0, you just have this trace term. And so simply choosing phi to be 0 or any constant rather will give you a solution at T equals 0. And then you just show that this interval I is open. And so Chen in the same paper where he proposed this path proved openness away from 0. So for all for the open interval 0 to the closed interval 1. And this was because at T equals 0, the situation at hand is a little bit trickier because this is a second order and this is a fourth order. So this jump in regularity or order makes it a little bit tricky given what he was trying to do. But later that year and earlier next the next year Hashimoto and Zang used approximation methods to prove openness specifically for the case T equals 0. Open is at T equals 0. And then finally, as recently Chen Chang in 2017 were able to show that this interval I is in fact closed but only under the condition where a bounded entropy along the path where the entropy quantity is defined as the integral of the log of this ratio of volume forms integrated against omega p to the n. Yeah, so it just means that there can't always exist a CSEK metric or yeah. And so they require this bounded entropy to actually show existence. OK, so how did they show existence? So the CSEK equation for omega phi, you're looking for a CSEK metric omega phi differing from omega by addition of dd bar of a smith function phi. And so like I mentioned earlier, it's going to be a fourth order equation. But the trick that Chen Chang used was that they broke it down into a pair of second order equations. So the first equation is just defining this variable f to be the log of this ratio of volume forms. And then the second equation is taking the Laplacian of this f. And then in here you technically will get a trace omega phi of reach omega phi plus trace omega phi of reach omega. And they're imposing the constant scalar curvature condition by equating this part to underline R, or rather, the average scalar curvature of omega phi. And so Chen Chang proved closeness under the assumption of bounded entropy, like I mentioned earlier. And the key ingredient was showing the following a priori estimates. And so the exact statement of their theorem is that if you have x omega phi, a CSEK metric where omega phi equals omega plus dd bar of the smith function phi, then all derivatives and the supernorm of the killer potential phi can be estimated in terms of an upper bound of the entropy. So the bound will depend on x, the background metric omega, and the entropy. And so Chen Chang used their estimates to prove in 2018 in a subsequent series of papers. There were three in total. The propellance of k energy, or Babuchi's k energy in terms of the L1 geodesic distance in the space of killer potentials, modular the automorphism group, implies the existence of a CSEK metric. And this builds on theory developed by Darvash and Rubenstein, where the problem reduces to regularity of minimizers of the k energy. And the reverse direction was established by Berman, Darvash, and Liu in 2016. OK, so that's the theory for the killer stuff. Now we get to a point where the question at hand is from this, how can we extend this theory of existence to the non-killer permission setting? So I'm still maintaining that we're still working with permission manifolds. We're just loosing the fact that omega no longer has to be closed. OK, so let's first just talk about metrics of constant Chen scalar curvature. So like I mentioned earlier, there are two different ways you can trace the curvature tensor to get a scalar curvature. And the way that we'll consider is the Chen scalar curvature. And that's the one where you're using the Chen Ricci form. And we choose this one because using this one, the Chen Ricci form is given by an equation that looks like the complex monochrome pair equation. Versus if you chose the other one, then it differs from the killer case in a lot more ways. And so firstly, the Chen-Yamabi problem with the existence of a constant Chen scalar curvature metric within a given permission conformal class was investigated by Engelis, Kalamai, and Spade in 2017. And they show existence in the case where the expected constant Chen scalar curvature is non-positive. So they prove existence within a given permission conformal class. And then Koka and Lejmi in 2019 explicitly constructed solutions or constructed CCSC metrics on rolled surfaces. So for example, they worked on here's a book surfaces that are known to not permit, admit any CSCK metrics. So no constant scalar curvature killer metrics, but they're able to construct still constant churn scalar curvature metrics that are non-killer. OK, so now we'll talk about how the results of Chen Chang can be extended to the non-killer setting. So firstly, in the killer case, you have the double-colmology class to work with. And so the question of existence is asking when there exists this omega-phis CSCK within the killer class of omega. And in our case, in the non-killer setting, we don't have the metric as closed, so we cannot define this co-mology class. So we'll simply ask, when does there exist a constant Chen scalar curvature metric differing from some given permission metric up to addition of dd bar of a smooth function fee so that the theory still feels pretty similar without requiring closeness? And then like I mentioned earlier, going from the standard scalar curvature, specifically now choosing honing in on the churn scalar curvature, because that one most closely resembles the one in the killer case, at least in the set of computations. And so the question you would like to ask is if we have x omega, which is a compact, complex manifold, under what conditions does there exist a constant churn scalar curvature metric of the form omega-phi equaling omega plus dd bar phi for a smooth function fee? And so we prove the estimates in the same way as Chen Chang under the assumption of bounded entropy. So the entropy is defined exactly the same way as in the killer case. And since we've loosened the killer assumption, we still have to supplement by making this slightly weaker assumption that dd bar omega to the k vanishes for k equals 1 and 2, which we'll later see. In fact, it means it vanishes for all k. And this three notable things that this assumption provides us with is that it ensures that the volume is preserved in this dd bar class of omega in the sense that if you add dd bar phi to this omega, its integral of its volume form stays the same. And that just makes many parts of the computation more simplified. Another part is that in a similar fashion, it means that the average scalar curvature is an invariant of the dd bar class. So this average scalar curvature of metric omega is still going to equal the average scalar curvature of metric omega phi under this assumption. Because otherwise, and we'd have another situation where there'd be time dependence on this, what would have been a constant that was independent of time. And then finally, this condition, we're specifically just the Go to Sean condition, which is that dd bar omega to the n minus 1 vanishes, gives us that integrals of all plushions of smooth functions vanish. And that's also just a component of the proof that arises several times and needs to be dealt with. Okay, so let's dive into the assumption a little bit further. As I said earlier, I write it as dd bar omega to the k vanishing for k equals 1 and 2, but in fact, this is equivalent to it vanishing for all k. Most interestingly, k from 1 to n minus 1, because by straightforward computation, dd bar omega k can be expanded into this term where you have dd bar omega, wedge some power of omega, and then this other term where you have d omega wedge d bar omega. But these purple terms still vanish using our assumptions, because this term vanishes by the assumption where k equals 2. And then when you expand that out, this part vanishes by the assumption where k equals 1. And so that gives us this thing, in fact, also vanishes. So this always vanishes 2 for all k. So these are actually equivalent statements. And so this might seem like a strong condition. So you might ask when such a metric exists or do they ever even exist outside of the Kailer setting? Because of course Kailer metrics satisfy this property. And so they do in fact exist in some elementary examples. So for example, if you'd let x be a product metric, sorry, product manifold, where you cross n a complex surface and m a Kailer manifold of any dimension you'd like, then on the complex surface n, you can always find a Godishon metric. And so a Godishon metric will satisfy this condition because you're working in the case where it's just n equals 2, where it's just a complex surface. And so omega n satisfies this condition. And then when you cross it against this Kailer metric, you can obviously choose the Kailer, sorry, when you cross with a Kailer manifold, you can always choose this Kailer metric omega m, which will also satisfy this. And thus the resultant product metric where maybe I should have put like projection, basically this product metric satisfies the condition we are talking about. And so this manifold x is certainly non-Kailer because you're crossing it with this non-Kailer component. So it is definitely extending it beyond just the realm of Kailer manifolds. This is my elementary example I could think of. Okay. So now let's discuss estimates for constant turn-scaling curvature metrics. So we prove the following a priori estimates similar to the Chen-Cheng result. So we're gonna let x omega be a compact complex manifold with omega permission metric satisfying our assumption. And so if omega phi equaling omega plus duty bar phi is a metric of constant turn-scaling curvature, then for all K we have that the CK norm of phi is bounded by a constant depending on K x omega and the entropy. So ENT denotes the entropy of omega and omega phi. This is an alias to the Kailer setting where we just have to make this additional assumption or rather we've swapped out the Kailer assumption to the slightly weakened form. And so this is the CSC, so the constant scale curvature case, but more recently these estimates in fact, very straightforwardly extend to the non-constant scalar curvature case where you can still get CK bounds on phi for all K just that now the right-hand side constant will also depend on the C0 norm of our phi. So it doesn't have to be constant anymore. If it were constant, you obviously don't have this dependence which is what we have above. But if you wanted to keep it around and not assume it was constant, then the estimates still work out, you just have this additional assumption. And so this latter set of estimates was useful recently for showing results about this analog of a clabby flow in the Hermitian setting. So before we dive into that, I'm gonna talk about the Mibu GK energy on Hermitian manifolds. So we're gonna work with x omega naught, a compact Hermitian manifold with vanishing first spot Turing class. So it's a slightly more refined version of the standard Turing class, first Turing class. And so in this case, it coincides with where, when the average scalar curvature R is zero. And so Tassati-Winecove observed that the Mibu GK energy can be defined in this way. So this is, normally you see the Mibu GK energy defined in a variational point of view, but if you integrate it out by parts, you can see that you can write it in this form. Where this form I've written here explicitly uses the fact that we're working within the vanishing first spot Turing class case. And this F is the Turing-Reachy potential of omega naught, which is just the one, because you're working in the vanishing first Turing class case, you know that the Reachy form of this is gonna be DD bar of some function F. So we refer to this F as the Turing-Reachy potential. And V here is just the integral of your initial metric volume form, which indeed still stays preserved under the flow, since the flow, yeah, moves along the DD bar class of the metric. And so if omega naught is scalar, then this agrees with the standard Mibu GK energy in the scalar setting. And critical points are Turing-Reachy flat metrics. So typically the Mibu GK energy's critical points are constant scalar curvature metrics. But in the case where you restrict to vanishing first Turing class, there could a whole point, like the only constant scalar curvature metrics on manifold of vanishing first Turing class are gonna be Turing-Reachy flat by a straightforward computation. Just perhaps one quick question if you can go back. So what happens or what goes wrong if something indeed does go wrong in case you don't have that C1 BCX equals zero? So you can see here that, so the original, if you compute the non-variational equation for the Muge energy, it gives you the entropy quantity integrated over T, and then it gives, or sorry, I forgot the T because you're taking it then, but you still have to add to it, a sum of like Reachy wedge omega to some K wedge omega phi to like N minus K minus one or something, a bunch of times over. And so in the case where you don't have vanishing first Turing class, then that sum doesn't reduce nicely because when you take the Reachy potential, so here the Reachy potential itself is simply DD bar F. But in the case of if you assume, so firstly, if you assume positive first bar Turing class or negative first bar Turing class, then you immediately are again in the scalar setting, right, because imposing that condition gives you Kalerness. So if you don't impose that condition, then you just have to, then you can write in terms of the Reachy without expanding it further. And so you can't jump into this sort of formulation. I see. Right, because you can't break down the Reachy anymore. Yeah, so that's the main problem there. Okay, so now we'll discuss a Turing claviflow on Hermitian manifolds. So here in the same way as the claviflow, we can evolve, the flow can evolve this function phi of T. And so you can write the flow starting at omega naught by the evolution in terms of this potential function phi of T. And so this analog is given by the evolution that phi of T evolves in T by the scalar curvature of T plus this giant term involving torsion. So torsion is something that measures the inability of an electric to be Taylor. So in the Taylor setting, torsion always vanishes. So, and then since you know that the average scalar curvature is zero, this agrees with the standard claviflow. If the initial metric is Taylor. But since it's not, then we have this non-zero term that's also here with us. But the integral of this whole thing will still vanish. And again, F is a Turing Reachy potential. And this specific term is the trace of the torsion. So if you're interested, you know, the torsion of the evolving metric where it's Ti sub i against the thing that I'm inter-producting together with this latter term. And so like I mentioned, if omega naught is Taylor, then this agrees with the standard claviflow since this vanishes and you have an implicit negative R under line R here. And this flow was derived basically as the gradient flow of the Mbouche energy. So the Mbouche energy will vanish along this flow in the case where we have a definition from Mbouche energy which is why we work within the realm of permission manifold with vanishing for a Scott Turing class. It would be nice to know how to extend this more generally. Well, like I said earlier, the definition of the Mbouche energy cannot be written in as nice a fashion when you don't assume signness of the first Turing class. And you can also see from this that, well, it's not super immediate because you have this giant trailing torsion term but in fact, six points of this flow are constant Turing scalar curvature metrics, which again, because we're working the context of vanishing for a Scott Turing class coincide with those metrics that are Turing Ricci flat. So fixed points of this flow are Turing Ricci flat metrics. Okay. And so in this setting, we can use the estimates we showed previously to conclude convergence properties of the flow. So again, if we let X omega naught be a compact permission manifold of vanishing for a Scott Turing class, you impose the assumption, this assumption that I've been talking about this whole time on the initial metric omega naught, then we can conclude that a solution to the Turing claviflow starting at omega naught exists as long as a Turing scalar curvature remains bounded along the flow. So those estimates we have on fee have a dependence that constant depends on RFE like I showed earlier, if you guys remember. And so we can only conclude existence of this flow for as long as that constant stays bounded. So we need to assume that the Turing scalar curvature still remains bounded along the flow, which is similar to the results in the Kehler setting. And in addition, if the Turing scalar curvature remains uniformly bounded for all time, then we can show that we have smooth convergence of the flow to the unique Turing Ricci flat metric in the DD bar class of omega naught. So it's gonna be able to form omega infinity equaling omega naught plus DD bar, this potential function limited to infinity. So this is an application of the estimates that I discussed previously. And it's kind of like, it introduces a method of approaching the existence problem of constant Turing scalar curvature metrics using a parabolic flow. Obviously, the conclusion here is not particularly unique or interesting, but it's the method that shows that the fact that it still works is kind of interesting. And hopefully if this can be extended, then it might provide more meaningful results about the existence of more general constant scalar curvature metrics. Okay, so the estimates in the constant scalar curvature case and in the non-constant scalar curvature case follow in a very similar fashion. So I was gonna quickly go through the method used to compute the estimates in the constant scalar curvature case. So again, we use the pair of second order coupled equations, which looks exactly the same as in the scalar setting, where in this case, the Laplacian and Ricci form denote the Turing Laplacian and the Turing Ricci curvature. And so the approach to showing these estimates follows from that of Chen Chang, where they first prove a C0 estimate for F and phi, then they improve a C1 estimate on phi followed by an LP estimate on the trace of omega phi, which with right-hand side constants still depending on P. And then from there, they compute further and use a Moser-Duration argument to remove dependence and in fact get this L infinity bound on this trace where the constants on the right-hand side, the constant on the right-hand side here will depend on the entropy and so that gets passed along throughout. So the final constants will depend on x omega and the entropy. And so why is that sufficient? Well, once you show an L infinity bound on this trace, then it's very straightforward if you have a bound on their volume forms, which you do because this is basically E to the F or E to the negative F. In this case, that you can obtain the trace bound the other way, so the other trace. And then once you have those two trace bounds, there's a straightforward bootstrapping argument that you can use involving the fact that you have this coupled second-order pair of equations, which gives you the desired estimates. Okay, so notable differences in the computation from the Kailer setting. So like I mentioned, torsion terms will arise now whenever you're commuting commuter derivatives or trying to commute indices of the curvature tensor, et cetera in many instances. And then of course also when you do, there are many steps in the proof that require integration by parts. And so if you do integration by parts, this red term here would ordinarily vanish in the Kailer setting. Interaction by parts is very nice in the Kailer setting, but now since we don't have closeness of the metric, the single D of omega to the minus one is not assumed to vanish, even with our assumption that there's only a single D here. And so we have to handle these extra torsion terms that arise, okay? So do I have eight minutes left or how much of this proof do I... One second, you have it, I'd say at least 10 minutes. Yeah, we started a little bit late, so. Okay, sounds good. At least 10. Cool, so I can dive into the proof methods for these estimates. So step one of showing the estimates is, like I mentioned, the C zero estimates on F and phi where F is log of this volume form and here we have the CSEK equation. And so this first component is proving the C zero estimate. So to do that, firstly, we can normalize phi so that it's super zero just to set it to be unique. And then we normalize omega such that its volume is one for simplicity. And from there, so we're working in a non-Kehler setting. So the non-Kehler setting, there's a analog of the Yavis theorem for complex manifolds, which is proved by DeSalle and Wine Cove, which states that for every smooth real valued function G on X, there exists a unique B and a unique smooth real valued function, psi on X, solving that this complex Monchampere equation so that omega plus D D bar psi to the N is equal to E to the G plus B for this constant B times omega to the N, where you ensure that this resultant thing is still Kehler, or sorry, not Kehler, it's still a metric, it's still positive. And then you assume that the C of psi is zero just for normalization purposes. And so ordinarily, you can't deduce much about B, but in the case where you impose the assumption that we've been using this whole time, the constant B in the solution of this complex Monchampere equation in fact has to equal this specific ratio. So in the case where you have this assumption, we know explicitly that B is gonna be log of this ratio and that's an important part of the proof, since now we can tell what the dependence on G is gonna be. So our G will be roughly speaking RF. Okay, so using the previous theorem, we're gonna let G be F times log of this thing that approximates the absolute value of F. And we'll let psi be the unique function, solving this thing. So we're gonna use that theorem to show existence of the psi solving that complex Monchampere equation where we've chosen our G very specifically, yep. Now we're gonna use Tans-Alphen variant. So Tans-Alphen variant is a well-known thing that Tans proved in the case of Kehler metrics and Kehler manifolds, but in fact very straightforwardly extends to the non-Kehler setting. And it states that if you have X omega, a complex manifold, then there exists a constant alpha positive and C positive depending only on X omega, such that for all PSH functions on X omega, the integral over X of E to the negative alpha of psi minus sup integrating X omega to the N is bounded. So this alpha and this C are fixed across all PSH functions psi that's satisfied that it's, yeah, that our PSH. So this is a pretty useful and strong statement. And so by the above theorem, we can choose that Tans-Alphen variant alpha so that for phi and for psi, these integrals are both bounded. So we've normalized them so their supes are zero. So this part vanishes. So the statement basically just says that these two integrals are zero as long as we're working the Tans-Alphen variant alpha. Okay. And now a bit more complex is just like you will choose a very specific quantity to apply the maximum principle to and then you will choose a very specific cutoff function. And then from there, you will, you have a bunch of constants TB determined, which we'll choose strategically. And I guess, yeah, here you just compute and this part is not super technical. It's very computational, but it's not super difficult. And so the crux of it is that you apply the ABP maximum principle to this quantity Q multiplied against this cutoff function eta. And so the ABP maximum principle tells you that the soup of Q eta over the any given ball is gonna be bounded by its value on the boundary plus this large, this large integral where the integral you're integrating only on the part where this parentheses thing is negative. That's just the statement of the ABP maximum principle. And since we know that this, the square root of F squared plus one is always positive. This is why we didn't just choose F. We chose this approximation of absolute value of F. And this is basically the entropy quantity. We find that this right-hand side large integral over the negative part, over all of B is in fact can be bounded above by B intersecting space where F is less than equal to C for some constant C just by the fact that this is only positive when F is bounded above by some fixed constant C. And then working from there, you can unravel these definitions out using the fact that F is bounded and then use the fact that you chose your constants nicely giving you a resultant thing that looks like the 10 alpha invariant bound. And from that, you get that this right-hand side is bounded. And so we know that, yeah, so in one step we know that side's always negative, so that goes away. So that's why we are left with this right-hand side. Okay. And so this tells us that we assume that P was a maximum point for Q, a point at which Q attains a maximum. And so Q of P, which is equal to the soup of Q and the whole manifold X can then be bounded in this way where this arose from the fact that we chose our cutoff function specifically. And basically given what we showed earlier, this implies that we have an upper bound on the quantity which was in the exponent previously, which was F plus epsilon psi minus lambda phi. And so to prove an upper bound for F we simply have to bound phi and psi. And phi and psi can be bounded using uniform estimates of the complex non-jumper equation for permission metrics proved by Dnevin-Kolijic as well as Blotsky, generalizing from the killer case, which immediately gives us that phi and psi are uniformly bounded. So this then, once we have bounds on phi and psi, then we know that F is gonna be bounded above. And then a lower bound for F follows straightforwardly from an elementary maximum principle argument. And the dependence on the entropy arose from this large integral that we dealt with to get the, in the process of using the ABP maximum principle. And so it gets passed along. Okay, so the next part is the C1 estimate on phi. So here we consider the quantity Q, which is this large kind of disgusting thing where I want to note that this differs from the quantity used by Chen Chang. So in the previous proof, effectively had to use non-killer analogs of several key ingredients that Chen Chang used in the killer setting. Here, there's not so much existing theory needed. It's more just straight up computation. And so we have to actually adjust our quantity Q here from the killer setting by a simple addition of this plus one, which gives us, you know, one more of these exponents. So in the killer setting, there isn't a plus one here and it works out beautifully. But because in our case, we have the presence of torsion terms, adding this extra exponential of all these little things is sufficient to then have the rest of the computation carry out. So that's the only notable difference. So of course, when you compute, like I mentioned, we have presence of torsion terms, which I will highlight here as these red terms. And so you can simplify this, getting these other two additional torsion terms and you complete a square. And then it turns out in this case of the C1 estimate that these torsion terms are pretty harmless and you can, Young's inequality slash Cauchy Schwarz in the way into terms that start to look the same as the other terms arising in the killer setting. So you can bundle them all together, just like in the killer setting. Obviously where the constants are gonna be a little different, but the essential parts are the same. And so if you choose lambda sufficiently large, then we can get that this little portion of Q for the giant quantity Q, then below by this large right-hand side, which simplifies using an elementary consequence. So the fact that F was defined this way to give us this part. And so you know that at a maximum of Q, the little portion of Q is less than equal to zero. And so what we computed here was this inequality. And so from here, we know that at a maximum, if you move this to the left-hand side, the power of the quantity that we're interested in is higher on the side with this extra one over N versus on the other side. And so from that, we can deduce or conclude an upper bound on d phi squared with the vector omega. So that gives us the C1 bound. Okay, so now we'll talk about the LP estimate. So again, we consider this quantity Q, which differs from the scalar quantity by addition of this plus one. So it seems like in these two cases, having a little bit of this extra exponential with these special coefficients chosen a certain way is sufficient for dealing with torsion terms that arise. And so you compute in very similar manner. And voila, again, we have some red terms, i.e. the torsion terms arising. Also a good trick that we use frequently is just completing the square. So you can make these torsion terms look like a different set of torsion terms. And then the new set of torsion terms that we get, again, using Young's inequality slash Cauchy shorts be banded below by something of this form, which looks more similar to existing terms in the scalar computation. And so if you choose lambda, so they're always constant TBD, then following a very similar computation is the one used by Chen Chang, we can compute that the fact that the laplacian of this quantity Q that we're interested in to the power of two P plus one is banded below by these other things, which after an integration, this gives us something slightly more interesting. Okay, so firstly, it's not yet interesting, but you'll see it soon. Along the way, you collect those torsion term, but that is actually not so harmful and you can band it by this. And then after some cleaning up and bounding things and applying the arithmetic geometric mean and equality, you get to this place where you find that the L, the two P plus one plus one over N minus one norm of this trace is banded above by simply the two P plus one norm of the trace. So this is great because the power here is higher than the power here. And so we can iterate to get a LP bound as long as we can show a base case for the iteration. And so the base case we choose is simply one P equals zero. So we wanna make sure that it's bounded for P equals zero so that we can conclude a bound for all higher P. And so if P equals zero, this right-hand side still holds true, so it's banded above by simply the trace. And then when you simply compute this trace, it's just gonna be equal to N plus Laplacian of V. And an integral Laplacian always vanishes by our assumption. So all we're left with is basically some volume term. So there you have an upper bound on the base case which then gives us the LP bound for all higher P where the constant will increase as you increase P. Okay, and then finally, we wanna show the infinity bound on this trace. This is probably the most involved computation of the four steps. So here we consider this quantity where A of F is now a not yet chosen real value function and B and N are our natural numbers to be determined. So in the Kehler setting, they just had B equals zero and N equals one and A of F was just, I think, something very simple. Like, yeah, which I'll get into in the next slide. So when you compute the Laplacian of the first part of that term, you get a bunch of torsion terms. But you also get some potentially good terms. So this term, this nabla tilde, nabla tilde bar F squared term is actually a very good term because it's positive as long as this coefficient is positive. And you can see that in here, this is gonna be a second order F term. And so you need to control that using this term. Additionally, over here, this is a potentially good term as long as this coefficient is positive. And this term is a first order F term, which will be used to kill, or to account for these two things arising. And so to control these non-kehler terms, we have to choose our real value function A of F so that the coefficient here is something positive. So I have the flexibility of requiring that's at least a half. It doesn't, it can be anything positive and that's fine because you can young inequality. It'll just be some infinitesimally small positive amount. And then you also wanna ensure that this coefficient is simultaneously positive. So here I just wanna require that it's equal to some positive epsilon. And so now we have to solve this series of, ordinary differential inequalities where you wanna just choose your function A to satisfy these two things. And so it can be seen that if you just choose A prime to be kappa e to the F plus half minus epsilon, then you have that it's derivative A prime prime where it's just a function of F is just gonna be kappa e to the F. And so if you choose kappa and epsilon sufficiently small, then in fact, you can have that these two inequalities be simultaneously satisfied. And so that's kind of the one limiting place that was a little bit tricky to deal with. Yeah, so in the killer case, the function A was just F over two. So A prime was just a half. And because they didn't have these red terms, they didn't have to worry about positivity of these two guys because they had nothing that they had to, there's nothing nasty that they had to deal with in addition to just the standard terms. Okay, so now you notice that Q is a sum of two different components. It has this first order F component and it also has a power of the trace component. And that's because each term, one compute when you compute its Laplacian will give you a good term and a bad term. And the other will have like a complimentary good term slash bad term. So then it's a balancing act between the right ratios of these two, such that when you add them together the resultant Laplacian will have nice cancellations in the terms that you don't know how to deal with. So firstly, if you take the Laplacian of the trace, you get this purple term, which is offset, this is a bad term is offset by this good term arising from the Laplacian of that first order F term. But the first order F term has this, it's also just a third order term basically in fee that will then be accounted for by a term that arises in the Laplacian of the trace. So this is a very standard trick used when you generalize things from non-killer geometry to killer, sorry, from killer geometry to non-killer geometry where you just balance you choose your quantities very specifically and you wanna balance it so that the terms that aren't easily controlled can be canceled out in such a way that you're only left with the terms that you do know how to control. And so we also use a nice observation that Laplacian of the high power of a function of the trace can be banned below by the high power of the function of the trace, cancel Laplacian of just the trace. And that's nice because that means that in some cases we knew the high power, but fundamentally the good term, bad term ratio stayed the same in how it comes out in the trace, sorry, in the Laplacian. Okay, and so from this, we get that if you compute out Laplacian of Q, which was computed as where Q is defined as this thing inside the parentheses, then the resultant thing after much computation gives you that span below by negative of this power of trace times the first order F term and then subtract like this other trace term to higher power. We have a lower bound on the trace and so you can play with this and just rewrite it in terms of Q for some possibly much bigger C. And so then you can prove the L infinity bound using motor iteration and applying several instances of the holder and solve of inequalities with again, we have to specify the base case. The base case here for Q is that you wanna make sure that the integral of Q vanishes, sorry, not vanishes is bounded. So firstly, we integrate this L one, sorry, this first order F term, you can use integration by parts because F, you know, you're doing it to two same functions F. So in fact, this term is still equal to this term as you see in the killer case. And then this can be expanded. So F is itself bounded and then Laplacian of F can be expanded into these two terms, which can then be bounded. And of course, the trace term. So this was the first part of the Q quantity. The second part of the Q quantity space case is bounded by our LP bound that we showed in the previous section. So this is bounded by some C that depends on B plus one, which is fine. And then, and of course, the entropy dependence is passed along from the first step. Okay, and so that concludes the exhausting list of computations needed for showing the estimates. From here, open problems that are natural extensions of things I've discussed are the most important to be defining and specifying a continuity path to show openness for the existence of this concentrated scalar curvature metric. So far, these estimates give us closeness, but I haven't specified any path and this would be an interesting problem to work on since our overarching goal is proving existence of the constant scalar curvature metric. Another would be understanding a geometric interpretation of the spouted entropy quantity. So like in the killer case, they've made sense of it and from a geometric point of view, here it'd be interesting to work with what we know in the non-killer setting or extending theory to the non-killer setting to better understand what this entry of quantity means. Another thing would be to maybe see if something along the lines of like a futaki invariant type of quantity or slash invariant exists that would provide us with an obstruction to existence of a constant scalar curvature metric. And then finally, it would also be nice to see if this parabolic approach to showing existence works out in which case we would need to figure out ways of generalizing this term clobby flow or really some related flow to a broader class of complex manifolds. Okay, and that's everything. Thank you very much. Thank you very much for the nice summary making these very, very difficult computations friendlier for us. Are there any questions from the audience? I already asked some before the talk. Go ahead, if there's anything it's difficult for me to point people out. So go ahead and unmute your microphone and ask. There's anything. If nothing, I did have one additional question, perhaps. So when I was tracing the steps in the also important C0 estimates, I could not find anywhere where you use this vanishing del del bar conditions. So I used it when I use the society wine cove result for the complex monochrome per equation and Hermitian manifolds case. I used it to know what the constant B is. So the constant B basically gives you something that looks like the entropy. So you know that the dependence on the resultant constants will depend on this entry quantity. If we didn't assume the DD bar omega vanishes for those two, those powers, then the constant B could be some exotic thing involving F, but in a way that we don't fully understand. And since we're trying to bound the C0 norm of F, we wanna know what we're bounding it by as a function of F, you know what I mean? Yeah, so in this case we know that it's gonna be bounded by this entropy quantity. But yeah, in other cases, it might be bounded by, I don't know, other exotic integrals involving F. That's something that I think would be interesting to see like if the dependence on the constant dependence on F can be more clearly stated in the society wine cove results that we can still conclude some sort of estimate there. Yeah, yeah, thank you. Thank you for clearing that up. And there was one more thing on your very last slide if you go back, I had sort of a flash. So towards the second line, so understanding geometric interpretation of bounded entropy. So at least a very biased point of view in the Kehler world, the first thing that one thinks of when you say bounded entropy, you think of compactness with respect to this L1 Mabuchi type metric. So the question is, so perhaps for that you need to have some geometry on this space of your admissible functions. And perhaps as we discussed, that's a little bit out of reach, but even beyond that, so historically, this D1 topology was preceded by a more potential theoretic point of view in this, or you might know of these BB, EGZ, BB, GZ papers. So first of all, if there's a, it looks like to me that you could have this kind of pluripotential theoretic approach. If you could set up this E1 space, this kind of finite energy integrability, perhaps it would be possible to discuss compactness there even without these infinite dimensional geometries. Historically, that's what happened. So somehow this L1 Mabuchi came much later. I see. Yeah, that would be interesting. I guess I haven't dug deeply to the starting points of what happened there in the Kehler setting. And I'm most recently. Somehow geometric interpretation, at least in my mind, it's sort of synonymous with compactness and finite energy. That's true. That would be interesting since my work currently on trying to understand the geodesics in the space of permission potentials hasn't really led to anything yet. So I'm planning a different view approach. This could potentially sort of circumvent that. And historically, sort of this geometric and infinite dimensional approach came much later. Mm-hmm. I see. Okay. That's a good note. I'll look into it. And also, you mentioned this phototype invariant. Mm-hmm. So instead of looking into if you have something like that, if such a thing existed, that really resemble futaki invariant, my impression is that that would indicate that you have this kind of invariance by holomorphic self maps of your manifolds, of your CSK metrics. So that might be an indicator. If you have that, then you might, it might be worth looking into this futaki situation. If not, it's a little, I would be a little bit pessimistic, but of course I could do it. Okay. Yeah, that makes sense. Yeah, so far, the whole thing with trying to find something analogous of the type invariant is challenging for the same reason that most of this extension is challenging because you want invariance under addition of dd bar of smooth function, but then all of your integration by parts becomes doesn't quite work out nicely anymore. Yeah. Well, if no other questions came up, did somebody think of a question? If not, then we'll thank CeCe again for again the very nice talk. Thanks a lot. We invite you to a virtual dinner, whatever that may be. I'll definitely stop the recording.