 So I'm going to talk about the existence of twisted big Caleb Einstein metric This is a joint work with Kewei Zhang who's at the Banging Normal University Okay, and I'm in the fortunate situation I will complement some things In his presentation so starting off about the set thing right so we are Working on a compact Taylor manifold it the setting will also force it to be projective But we won't use too much of that and then I will want to first explain to you. What's the Ritchie curvature? So This this perhaps should complement something that was already in Remy said, you know Metrics our solutions to this PDE and Then lost last. Oh, but where's the Ritchie curvature? What not? So maybe this could this could help in Giving a more rounder picture. So how do we so so we have a? right, so according to the Swedish school, right one most always fix a reference reference for right Okay, okay. Well, maybe not, but I'm doing it anyway, right? So the data is going to be fixed for the whole talk representing a big homology class and I am taking a potential Corresponding to data and I'm trying to understand how to define the Ritchie curvature of that. Are you okay? so one thinks about this a little bit and Maybe the definition is elusive, but one that one thing that we perhaps agree upon is that this discrepancy Formula for Ritchie curvatures should hold whatever whatever definition I take so in case of smooth metrics, we have this nice formula between the difference of two Ritchie curvatures Omega here is smooth So I know what this is and then if I could understand what this Thing is maybe I should I can take this as as the definition. So so that's what I'm doing here so I Will define the Ritchie curvature of positive currents only if I can make sense of this the DDC of log of the density of non-pluripolar Complex monjampere measure of that to you to the end over omega to the end so if I happen to be in the fortunate case that The non-pluripolar complex monjampere is Absolutely continuous with respect to omega then this quotient makes sense I take a log and I only ask of it that it's integrable. Okay, so I'm kicking out a bunch of Elements from PSHX, but there's plenty here. Why well because of the BEGZ right so in BEGZ they prove yaw's theorem meaning that This theta u to the end can be for example any F times omega to the end So there's always going to be plenty elements here and for those I know what the Ritchie curvature is Okay, so I'm looking at those and I want to see if I can find a canonical metrics among them right so then we get to this Understanding of what Taylor Einstein metrics are and I'm going to talk about twisted Taylor Einstein metrics in the big sense, so I'm going to assume for this talk that the first churn class splits as my Big class theta plus a pseudo-effective class So etta is a smooth form that will represent the pseudo-effective class This only means that this the plurisopharmonic function with respect to etta is non-empty and Automatically big plus pseudo-effective is always big So I get the condition for the rest of the talk that negative Kx has to be big Okay, so I'm picking a psi here and I want to find u with minimal singularity such that Ritchie theta u is equal to theta u Plus this twisting etta psi So when when there's nothing here, so it's possible that I take the trivial class Which is to the effective then I'm back to the case of Taylor Einstein metrics. Otherwise, I'm twisted Okay, and if I manage to find Such a metric so such metrics are called of course Metrics now you can nibble around with Coefficient here you can put here a lambda. We know how that works You can suck that into the cohomology class, so Here I'm strictly focusing on lambda equals one and okay, so then you look at this equation You also look at the defining formula for Ritchie theta u you play around with that exactly as in the smooth case And then you arrive at Remy's equation down here So up to a constant this equation is equivalent with this complex one jump their equation where on the right-hand side You have e to the f minus u minus psi psi was again the potential of the twisting form here and F is the corresponding Ritchie potential Which comes about you solving this? elliptic PD right so it's a smooth Function Okay, so we're staring at this second-order PD now we want to find the solutions with minimal singularity and You know the story right from the ample case Remy sort of expanded on it, so I will hobble over it unfortunately solutions don't exist in general and the goal often is to find algebraic criteria to guarantee existence that in a very sort of economical manner, so you're hoping that these Conditions that guarantee existence are also be necessary. So you want to really characterize Situations algebraically where these canonical metrics exist and there's a bunch of such conditions Usually they fall under the umbrella of case stability. So our choice for this work is to work with delta invariance. So These have been developed in the last Danish 15 years started out by Fujita Odaqa if I'm not mistaken and we will be specifically looking at the analytically Interpretation by blue Johnson, right? So we are looking at this delta invariant That's attached to my class data and the singularity the twisting singularity psi So this is infimum e of a psi e over s e. Okay, let me explain what all these symbols are So e e is going to be a divisor over x So e is a divisor that doesn't live on x, but on a birational model of x and then here I take a Bimeromorphic max from y to x and then he lives in y now You can do this also in the projective role So so so you you can squeeze in the word projective in there. We have a note somewhere in the paper that Okay, so you know what he is you're taking the infimum over all such e What are what is a psi e so a psi e is the twisted log discrepancy? So it's a x e the log discrepancy minus the long number of psi along e Right, so what's the twisted log discrepancy? Well, it's one plus The vanishing order of the Jacobian of pi Along e right so that's essentially what's encoded here minus the long number of psi along e What's that? Well, what you need to do is you need to pull back psi to y and then along e Check all the long numbers The smallest Will be this the long number of psi along e and then by sue Semicontinuity you get that actually this There's a hundred percent chance that you get the minimum minimum number Which is which is this one here? Okay, sorry? All right, so that explains sort of what's here in the top at the bottom This has many names. I know it by the sort of average The long number of the class with respect to e so there's a lot to sort of process here It's an integral going from zero to Tau theta e's the tau theta e is essentially the Sashadri constant of the big class That up with respect to e Right, so you it's the biggest number tau Such that the class of tau minus Sorry, it's a class of theta minus tau e is still big Okay, so there's a various ways to say what this is Another way of saying is that it's tau is the biggest the long number a Potential in psh x tau can have So it's some sort of extremizer Okay, so then you integrate from zero to tau here the volume of this class Which is big so it makes sense what you're doing here and you're averaging up over the total volume So you get some quantity here and you know if you think about it a little bit This can be interpreted as the sort of the long number along e of an average element of Randomly picked tether psh function. Okay. Good. So anyway, we have divisorial data and we're testing stability of our Geometry using this divisorial data. Okay, so I think that in a nutshell sort of explains the so called delta invariant so now our main result can be Stated that way. So if this delta invariant is strictly greater than one to the twisted delta invariant the psi twisted delta invariant greater than one Then there exists a twisted That the twisted Taylor-Einstein metric Satisfying this equation which again is equivalent to the scalar equation All right, so a few words Need to be said. So this is an extension of a result that is already found in Berman books to be on so in the ample case there It's a slightly different terms. Perhaps this result is written down again in that their case negative kx is ample and Teta and Eta or maybe not integral q the u-line models Exactly Now conversely conversely if the unique Taylor-Einstein metric exists, then one can also show that Delta invariant is greater than one This is more or less in line with what Remy talked about and the proof of this is is also very similar To what they do so from you having a unique The emphasis on unique a conical metric Existing techniques Showing that Dating energy is coercive and then you would go from Caursivity of the energy is basically descent to a Algebraic stability conditions in their case it was by test configuration in this case. It is via delta invariant Okay, but again, I said in the beginning we want the condition that characterizes So what's missing to have a two-sided result is the word unique from here So notice a twisted Taylor-Einstein metric not a unique one So we definitely expect that there's a word is missing from here, but we can't get that yet Because we don't have a bundle my butchie type uniqueness theorem in our transcendental setting here Since unfortunately or fortunately it is tied up in a conjecture of burnson I'm going to remind you so Burnson in his inventionist paper where he obtains all sorts of different families of Theorem mentions the possibility of a sort of very general Version that will probably have all or most of the ones stated there on the umbrella Whereas you take a sub-geodesic in the class of negative kx. So that's Teta plus etta in my case right, so you take a sub-geodesic segment here and then by I'm going to And burn some it's known that this this correspondence so t to negative log this expression is actually convex Now if it happens to be linear then it's expected that this Sub-geodesic is actually a geodesic induced by automorphism. So if one can get that then One one should be able to show existence So uniqueness in our theorem and then we would have a two-sided characterization now regarding this conjecture I mean there have been some progress in our brain Optimistic that maybe maybe in the near future will be able to get this and then this whole picture will be Okay, but this is where we are right all right good. So some some some further notes Some remarks that need to be said here about the perceived novelties of our results and also limitations Yes So the whole the whole segment has to be a very simple Okay, so some notes here right so transcendental Right so our techniques are transcendental I wouldn't dare to call it sort of out of out of the blue new approach to Case stability If you you know spend a little time with our work you will see all these sort of divisorial criteria, which is also the Cornerstone of the non-archimedean approach. So if I think one way of putting it Co-op really I think our techniques are sort of the analytic cousin of the of the non-archimedean approach So due to the fact that you know, we we sort of fixate on transcendental Ingredients we obtain a transcendental ytd type theorem so this Is seen by the fact that teta and Etta are really Transcendental Big classes in the big code now before I do a victory lap with this Immediately I have to mention the limitation because there is one so there's the setting Forces something that needs to be mentioned. So if a Kieler Einstein metric does exist, which we definitely hope to exist then Negative Kx has to contain a positive KLT car namely the kill the current coming from the Kieler Einstein Plus et up see the twist, right? So I mean this is easily seen a few if I roll back to the equation right look at the right-hand side of the equation I have to be able to integrate this Look, what do you see here in the exponent u plus psi, right? So then this current here needs to be Okay, this current needs to be KLT now if there's a KLT current in negative Kx Right the second homology class of holomorphic sheaf is zero implying that every class is Even though we are in the big cone, right? Both teta and etta are essentially our line one this okay, so The question is like really how transcendental is this you can definitely? Approach our line bundles like you line so question is can you recover the result? In in the current form for our line bundles coming from you line bundles right so playing a bit of devil's advocate Well, the issue that you run into immediately is The neither the twisted delta invariant or its own twisted And unless this is argued Which which I believe to be true that the delta Invariant or in our case the twisted version is continuous on the big cone this type of approximation by Q line bundles will not work. So perhaps you allow me to insist still that we do have a Transcendental like Theorem perhaps the first the first one no even though right The second could be even more transcendental right now again. Is it possible to avoid? This condition all together well There is a way Potential way we can't do it yet So the limitation would not persist so now though would not be applicable If you would be able to twist by a not necessarily pseudo-effective current at upside here and this in the ample case is doable right so my Collaborator is exactly just that we can't do it in the big case. He did it using quantization Which is in the realm of algebra? We purposefully want to avoid that in our case, but even if we wanted to use His techniques unfortunately as you know in the big case things as basic as Sava takigushi type theorems, you know Go out the window because potentials with minimal singularity type do not contain cards, sorry Kailer cards, right? So this that problem persists, but We do expect and you know either us or maybe somebody else will be able to obtain a version of our result without at upside the positive and that will be truly truly Transcendental in the sense that you could go deep deep into the Kailer corp away from the Neuron 70 class Okay. Yes In There is a spot that we got stuck by not being able to use a Safa Takegoshi. And then, again, the sticking point is that among potentials of minimal singularity type in a big class, there is no scalar current. If there were, you would immediately be a scalar class. Okay, good. Let me tell you some more remarks. So, again, another perceived novelty, our method gives a very general framework to prove why did the existence theorem of the following nature. So, delta invariant greater than 1 implies existence of certain type of scalar Einstein metrics. There is really one ingredient that's non-negotiable, convexity of the appropriate take function. So, in our case, that is handled by... And then, many of you could think of other situations where this result is known or expected to be known. As soon as that result is established, I believe that, you know, following our steps, you will be able to prove a YTD type existence theorem of the sort, again, delta plus greater than 1 implies existence of your canonical metric. Now, throw in uniqueness, throw in uniqueness, and then you get a two-sided characterization. That's the part that we're missing here. Okay, now, again, I'm somewhat biased. I've been following this study of K-Romax Einstein metrics. We've described singularities by Truciani. I think this is a prime candidate to sort of carry out this type of analysis. So, I'm sorry, but just the convexity of the data actually isn't... I mean, isn't that really where the problem with the negative form enters? I mean, that would not be true. Yes, yes, yes, yes, yes. So, with the negative, with the f-upside being more positive, that would go. Yeah, yeah. So, but that's a good way to deal with these, right? So, that is, one doesn't even expect any full characterization, right? We just expect that stability applies to this graph. Okay. But we need, that's the question, this method. So, this is the interview. Yes. Yes, yes, yes, yes. So, it's just good, there's bad, and then doggily. So, the good is that I think that this is a generic method, but the bad is that transcendental classes are really very close to the nine-on-seven o'clock code. Yeah. You have to decide what you look at. Everybody has to do that, so do we. Okay. Yeah. So, along these sort of very general framework to provide the D theorems, let's look at the algebraic case when there's no twisting. So, we give a very short proof of this result of Li-Tian Wang, where you take a log-fano pair with positive delta invariant, and then you get, in that case, a unique Keiler-Einstein metric, and until a few days ago, I mean, this was sort of the end of the story here, right? But there is a plot twist by Chen Yan-Chu. I mean, we knew a couple slightly earlier, but those, the work appeared now in archive. So, it turns out that, yes, our methods imply Li-Tian Wang, but in this algebraic case, it's actually true vice versa as well. So, what's going on? So, Chen Yan-Chu noticed that in case negative Kx is big, and you're stable, X is actually log-fano. So, log-fano by BCHM means that it's a mori-dream space. So, every big line bundle has finitely generated section ring. Okay? So, in particular, the section ring of negative Kx is finitely generated. So, you know, then you're extremely tempted to look at the proge of this section ring. You do that, let's call that Y. You immediately get a birational map into Y. So, this is not the map, actually, written by that. You resolve this map and then some sort of very, very attractive analysis arises without going into details, right? So, what you can show is that this Y is singular, but a Q-fano that's stable. So, it admits a Kaler-Reinstein metric. So, again, slightly stretching precision. What you can do is you can pull back this Kaler-Reinstein metric from here onto X, and that will be essentially the Kaler-Reinstein metric on Y. Not quite. In reality, it will be the Kaler-Reinstein metric will be the pullback Kaler-Reinstein metric plus a divisorial singularity, but that you can always subtract, right? So, all in all, right, we have here in the very important algebraic case sort of two possible ways of doing things, either using pretty potential theory, as we develop it, or you use this landmark result in minimal model program, and in that case, our result can be reduced to the result for log-fano pairs by lithium-fan, right? So, geometric problem having two almost... I mean, nonintersecting approach is one from algebra, one from analysis. I think this is essentially the definition of our field, right? Now, before I get very emotional, I should mention that we've seen instances of this. So, this in our field, right? So, in BeGz, there is a very, very similar situation, right? So, if you read BeGz as carefully as I did, then there's a passage there that points out something eerily similar, right? So, if Kx is big, in that case, the authors put together finite energy pre-potential theory to construct a Kaler-Reinstein metric on Kx. And even there they point out, you can do this, or you can use there also BCHM, more or less the same way as here, to reduce the problem of Kx being big to Kx being ample and X singular, and this is a case handled by BeGz. So, again, over there, you also have this alternative, either completely analytical or minimal model program. Same thing here, the only extra ingredient is the stability. This was very surprising to me. I did not expect this. So, yeah, definitely for me, this was a plot twist. Right, good. So, with regards to the general case, obviously this BCHM approach, it's not going to help, but this is definitely a very important particular case where lots of different options are available. Okay, so in the remainder of my talk, let me expand a little bit on the analysis and then how we approach this. Okay, so, ingredients. So, our main ingredients are this sort of Rottwitz-Rosswitt-Nichtrom correspondence between geodesic rays and maximum test curves, aided by the relatively repotential theory developed by my collaborators, Dinez Alou and myself. The valueative criteria for integrability, as I've learned it from, BBJ, and books go far, but I also understand that in the early days, multiple times, strong guans who opened this theorem, again, we use the versions that I learned in the appendix of BBJ. And last but not least, the convexity of the ding energy that was again pointed out as the non-negotiable ingredient in all this story, due to burdensome power. Okay, good. So, being even more specific, this sort of twisted ding energy, lambda-twisted ding energy is going to play an important role. So, you might recognize this expression from many different works in case lambda is equal to 1, right? So, in the case of lambda is equal to 1, we're talking about the actual ding energy, but here we allow lambda to be positive. Hopefully I'll be able to say why this extra lambda parameter plays a role. I think this lambda trick already probably goes back to BBJ, right? Did it? Maybe not. I've learned it from papers of my co-author, Kewei. But again, given what I'm about to say, I wouldn't be surprised that this is coming from other places. Okay, so our first result that I think is novel, and it's extremely helpful, it's actually very simple. So it says, okay, so you have a sub-geodesic ray. Okay, I won't define that. It will take me too much time. Again, so think of a sub-geodesic ray as a gloria-subharmonic function on, right? So you can put such sub-geodesic ray into these lambda ding functionals and study the slope as t goes to infinity. Now, one thing that I have to note here, for lambda equals 1, we know that this is convex, right? So that, in that case, the limit exists here, but typically it doesn't, right? So the best thing I can put here is either lim-inf or lim-soup. I put lim-inf. So there's a formula for this in the following manner. So this surprises nobody, right? So you have the Mont-Jean-Père energy here. So then this is nice in convex, so that having a nice slope at infinity, check mark. The interesting thing here is perhaps here. So the supremum of all tau's, such that you had tau with negative lambda, is integrable. Now, the thing that I have to explain to you is what you had tau is. You had tau is the time-legeant transform of my sub-geodesic ray. So this is where we start sort of deep diving into the Ross-Witt-Nistrom correspondence, right? So in the Ross-Witt-Nistrom correspondence, I know that I'm actually better off studying the sub-geodesic or geodesic. By studying, it's time-legeant transform, right? So here, this is what we get, right? Now, there's an integrability exponent in the interpretation of this formula as well, right? So what this says is that the integrability exponent of you had tau is at least lambda, right? At least lambda for tau in R. And I'm looking for the biggest tau in which the integrability exponent is at least lambda. Again, I'll expand this a little bit more if I will have some time. Similar formulas I've seen in BBJA, also various papers of books of Jonsson, not quite in this form. So this is again tailored to a very, very analytic treatment. And then one thing I should say is I definitely inspired from a not paper not too long ago by Ming Chen and myself, itself inspired from a paper, a lemma of Boo from a conference volume 2. I wouldn't be surprised if he wouldn't recognize it though, but yes, thank you, Boo, again. And that's not the first time that was said during this conference either, right? Yes, so let's try to absorb this as much as we can and I'll get back to it. It's sort of a critical observation. Okay, so the other sort of critical observation is the next theorem. This gives you an interpretation of the delta invariant as geodesic semi-stability, or uniform geodesic stability threshold, right? So we prove that the delta invariant, in this case the twisted delta invariant, is the threshold of semi-stability using sub-geodesic rays of the family D lambda, right? So you start... So for lambda very, very small, D lambda is actually... For small, positive lambdas, this class is not empty. And the question is how big of a lambda will this condition still hold while the supremum is actually the twisted delta invariant? And there's sort of this uniform version. So if you look at unit speed geodesic rays, then you can ask the same question. So for what... What's the biggest lambda for which the slope of these ding energies is bigger than a fixed constant? And that will also give you... Let me just... This is some of you might know. So D lambda squeaky bracket UT is the slope of D lambda along this geodesic ray, right? So as I said, this limit might not exist as T goes to infinity, so it takes a limit. So every time you see a squeaky bracket, it means basically the slope of that energy going that way. Okay, good. Now, again, I've seen versions of this in the ample case in papers by Buxom Jonsson, sometimes in a monarchy minion context, sometimes off. Also saw it in a paper by Kewei Zhang. And this, again, I wish my memory would serve me better, but it's definitely implicit in BBJ already. Robert disagrees, okay? I'm giving you too much credit. Who knew? Exactly, right? Exactly. Okay, good. So, yeah. I think it's implicitly there. Let's move on. Now, again, so what I'm trying to say here is that I wouldn't be surprised if something like this, you know, people already saw, hence it's more believable, right? Now, the thing that I want to emphasize on, depending on how much time I actually have left, how much time do I have left? Lots of time, very little time, no time. That's not some time. Okay, good. Okay, so Robert asked. So there's one thing I didn't mention and it's also one of the, as I said, perceived novelties here. We don't use the K energy in our discussion. So that was missing. I just, as I said, convexity of the ding energy is non-negotiable, but K energy, we don't bother. Now, even in the Kaler case, this could, so even if you don't care about the big case at all, this could change the perspective a little bit on some things. So since this was already asked and we'll discuss it also with other people, let me try to emphasize in like two, like three slides to get away with not having convexity of the K energy, which in the big case, by the way, we don't have at all, we don't even know. So there's a result by Eleonora and Shin that treats the big NF case and they point out there that in the big case it's not even clear what you should write down as your K energy in general. There's a twisted term and it's a good question. So one month to get away with it. So let's see what we propose. So again, we're in the Kaler case, so the minimal singularity type is just zero, so the minimal potential with minimal singularity type is zero, and I assume that I'm stable. So this twisted delta is positive. Okay, so if the ding energy were coercive, then there would exist automatically a Kaler ish type method. So I'm assuming it's not coercive. So the story pretty much starts in the same manner. If only I wouldn't introduce this sort of auxiliary ding energy. So this auxiliary ding energy with the running parameter beta that I'm fixing between zero and one is the ding energy minus a sort of J term. So this is like a J energy. So immediately you might see that this is a bit strange. So this is convex along geodesics. This is also convex because this is linear and this is convex. So convex minus convex typically is nothing. So it's almost as if I'm doing something really, really crazy and bad. If not, if you don't recognize that I am working with potentials later on whose supremum is always going to be zero. So I will be constructing geodesic rays starting from the zero potential. In that case, you know that the soup is going to be linear. So there's this observation. So this term here which is typically convex along geodesics is going to be linear. So linear minus linear. So this d beta that's sort of adjusted by a J term is for the data that I will put in will be convex. So, you know, maybe not so crazy. But again, I haven't told you what the point is yet. Okay. D beta is still convex. Good. Let's see how that's going to be helpful. Okay. Okay. So now comes what I... Okay. Sorry, sorry, sorry. So since D1 is not proper, then I play the same game that many people that read BBJ will recognize. That means that for every J, I will find a potential whose supremum is zero, giving you this bad inequality. Right? So if you were proper, such a thing you could not do for some J eventually. And also, not too difficult to argue that if something like this holds, then these bad potentials, 5J, has to be sort of further and further out from the scalar potential, theta. Okay. Good. So again, you know what I'm about to do, right? Maybe, maybe not. So again, now we're involving this twisted energy D beta. So look what's going on. So look at this condition. Okay. I have this inequality. If I were to place D beta here, then I would get this extra term here. That's beta times a J term, which is, again, the J energy is proportional to D1. Right? Beta is fixed. So this same sequence will give me this estimate that D beta over 5J is actually less than minus epsilon over the length of, sorry, the distance between zero and 5J. And again, so this is the same sequence. So I'm just repeating here that the distance between zero and 5J goes to infinity. Okay. So again, I'm just carrying this information with me, and it's still not clear how this is going to help me. And now I'm doing the thing that is done in BBJ, which is I'm forming these segments from, so it's a unit speed geodesic segment joining zero and 5J. Okay. So zero has supremum zero, and 5J has supremum zero, so all the potentials along all these segments have supremum equal to zero. So the D beta is convex along all these geodesic segments. So then I will get these estimates for free for any time in between, so in between time on these geodesic segments. Right? So, so far, everybody knows that nothing crazy should happen. Okay. Now, I noticed that the supremum of... I told you, I'm sort of reminding myself that the supremum of all these potentials is zero. So I can run a diagonal argument in L1. So weak L1. In weak L1, I run a diagonal argument, and I get a sub-geodesic, ut, such that, well, perhaps after taking a subsequence, of course, which I can always do, and each ujt will convention the ut with respect to L1 in this weak topology. Okay. Now, again, it's a very small line here that I want to gloss over, right? You can only do this for t, not for all-time t, but for almost every-time t, right? I mean, it's sort of a Fubini theorem that gives you this, but you let me get away with this because you're definitely sure that the titanic will sink in a minute, right? So we're getting to the critical situation. The critical situation is here, right? So the limit will satisfy some things and not others, right? So this slope formula I still have because d beta is lower semi-continuous with respect to L1, right? So the exponential part is actually continuous with respect to L1, but i bar is upper semi-continuous, so minus bar is lower semi-continuous. So this gets the check mark. The supremum is preserved because sup x is continuous with respect to weak L1, but here's the problem, right? So this is where the titanic is sinking. I would prefer to have this equal to negative t, right? So now, collapsing might have occurred. I cannot rule out that this ut in the worst possible scenario is actually the zero geodesic, right? That's no good, right? And then previously, this is where the wisdom stopped. Okay, this is where d beta comes in. So, okay, I don't know if this ut is trivial or not, but I can put it into my d beta and look at the slope formula. It's theorem 3, so it was this slope formula. And I get the following thing. So, okay, maybe. Oh, the thank you. Okay, that's coming, that's coming. Don't worry, one more slide. Yes, we're getting close to the punch line. So now, right, so the d beta, the slope of ut along the d beta, it can be expressed in this formula. And I know that this has to be less than epsilon. Okay, but I'm very concerned that ut is collapsing. So what does that mean? So there's this notion of maximal geodesic rays. There's a geodesic ray that's sitting below it, but it's sitting on top of it, but it's sort of minimal in nature. Now, you have collapsing in this situation if the slope of the monjampere energy is identically zero, right? I'm soup normalizing. So if this guy were to vanish completely, then it's bad, it's bad. That's what I want to avoid. Okay, so I take this maximization. If my sub-geodesic collapses, the maximization will be the zero geodesic ray. Now, in the Ross-Witt-Nichtrom correspondence, the maximization corresponds to taking the Legendre transforms, p maximization in every count, right? So again, I will have this nice formula. So v is my maximization, v hat is p, u hat, down. And this is the formula for that, for those that don't know. Now, using... Now comes the value of the criteria for integrability. This v operator does nothing to the long numbers, not just on x, but everywhere actually. So all the integrability exponents of v tau hat and u tau hat will be the same. Moreover, also the slope of the i energy will be the same. So the slope of ut and its maximization will be the same. So I have the same inequality here with v's, as I had with u's. Okay, so let's assume the v was trivial, meaning this is zero. Okay, if that is zero, then the only way this is negative, if some of the integrability exponents here, right, will have to go become, what? So yes, so if this is negative, then the integrability exponents have to be less... Sorry, what do they have to be? Yes. Yes, so the supremum... What's going on? So the integrability exponent of all the v tetas, yes, has to be equal to the integrability exponent of zero, greater than one. But that's absurd, because that means that this will be zero as well. So I will get zero is strictly less than minus epsilon. No good contradiction. So, you know, I get... Basically, even though in this condition there might be collapsing, I will not get that the slope of the i's, as t goes to infinity, is zero, right? So this sub-genetic sort of survives the collapsing. It might collapse a little bit, but not too much. So then what you do is essentially renormalize this geodesic ray vt, because it might have lost some speed. Since d beta had negative slope, it will still have negative slope. And okay, so now comparing the formula for d beta and d1, what, that this renormalized ray will have ding slope less than beta. So I have a family of unit speed rays, whose ding slope is as small as t. So then this says what? Well, this says that the uniform geodesic stability threshold cannot be bigger than one, right? And then this is a contradiction with the other theorem that I mentioned, with this line, right? So we started out with asking this guy to be positive. And I just cooked up a family of rays that for lambda equals to one, will give you no such epsilon-lambda uniform constant.