Pluripotential Kähler-Ricci flows
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Guedj, Vincent
We develop a parabolic pluripotential theory on compact Kähler manifolds, defining and studying weak solutions to degenerate parabolic complex Monge-Ampere equations. We provide a parabolic analogue of the celebrated Bedford-Taylor theory and apply it to the study of the Kähler-Ricci flow on varieties with log terminal singularities.
 Thank you. I'd like to thank you for attending the picture. I'm very excited for giving me this opportunity to speak here. So I will report on a recent joint work with Wang Qinlu from Hasei and with Amit Zariaï from Toulouse, who is in the audience. And we are trying to develop the first steps of a parabolic, very potential theory. So we have posted two papers on the archive last October. Both are quite long and a bit technical. There are close to 60 pages each. So I'm not going to explain all of them by far. But I will try to focus on one of them that deals with the compact setting, which is a bit easier to explain. And so what we try to develop is to understand complex mojampere flows, especially the case of the Kele-Arichi flow, by means of very potential techniques. So we try to define and understand the weak notions of solutions to these flows. And there are basically two natural approaches. So one is by approximation by smooth flows and developing a priori estimates. This is the path I'm going to try to explain today. And another one, which is quite useful as well, but maybe a bit more technical to explain, is by means of Péran technique of envelope of weak sub-solutions. So this one I will skip today, but the inspiration nevertheless lies behind what I'm going to explain. So the main motivation is the study of the Kele-Arichi flow, which you can write like this. And we try to understand this evolution equation, not only at first on smooth manifolds, but you try to run it for as long as you can and then try to extend the definition of the flow as soon as you reach finite time similarities. And very soon you need to understand this object on a mildly singular varieties. And actually it's quite common to study the so-called normalized Kele-Arichi flow, which can be written that way. Where lambda is a real parameter with sign with this way of writing things is the same as the sign say of the first-gen class of the canonical bundle of the manifold. So if you pick, maybe I should write here that at the level of homology classes, so if you write alpha t, which is the homology classes of those scalar forms theta t, then alpha t dot solves a very simple ODE, namely that maybe I should write that way. Alpha t dot is given by this. So you are from there and from the initial Kele-class that you give yourself, understand that alpha t is a definite and simple form. So we can pick, maybe I should write it here, a smooth representant of this form in alpha t, a fixed class, and then solving this nonlinear equation turns out to be equivalent to solving the following complex monjamper flows. So n will be the complex dimension of the manifold. Where dv is a fixed volume form on the manifold, h is a smooth and fixed function, and phi t is the unknown function, which is assumed to satisfy the following positivity property. So it's a well-known fact that which is a particular to Kele geometry that solving this Kele version of the Ricci flow or the normalized version is actually equivalent to solving a complex monjamper equation at the scalar level. So rather than having an unknown tensor, you actually end up with trying to find and solve an evolution equation for some unknown function, scalar function. So that's the kind of equation I'm going to focus on. But as I mentioned, it's also extremely important to understand this type of equation on mind-leasing larvalities. So the precise setting will be the following. So let's assume that the underlying variety is actually might be singular. It still makes sense of considering such an evolution equation. And in order to understand the corresponding complex monjamper flow, we fix a resolution of singularities and lift the problem to the smooth manifold x, and the corresponding flow transforms into, let me write this notation for complex monjamper flow, transforms into the following type of equation. So f looks similar to what we had before. So let's take some function h of t of x plus lambda phi of t. So this hasn't much changed. So the new, or maybe h tilde, there is a new guy here, g, which has the following form. And the zeros, so these are homomorphic sections of some line bundles, and the zeros of those guys actually corresponds to the nature of the singularities you started with. So we have defined both poles and zeros for this density, but the important point from an analytic point of view is that you nevertheless end up with a density g, which is LP integrable, and this is one possible meaning for the fact that the singularities you are dealing with are might. So the next one, the difference is that you have lifted here, omega t is actually the lift of some eta t. So eta was the caliform on the manifold v, but so omega t is no longer strictly positive, it has some degeneracies above the singular locus of the manifold you started with. So you have a lack of positivity on the reference form, and you have also some analytic difficulties coming from this density g. So in the sequel, I will focus on this general form of the complex mojampere flows, and I would like to make sense of weak solution to this flow in the sense of pre-potential theory. So rather than working on the manifold x, the first hint that we follow is that we work on the manifold xt, which is a real manifold of dimension 2n plus 1. So this is the maximal existence time for this flow, and this is a real manifold of dimension 2n plus 1. And we want to understand this equation on xt. So we are going to interpret both sides of the equation as positive measures on xt, and trying to make sense of this both left hand side and right hand side in the sense of measures. So the left hand side is the easiest one to deal with, namely the only part you need to understand, or somehow the delicate part you need to understand is this one, which is a complex mojampere operator, which has been studied in the late 17th by Beethoven Teller. And they propose a very convenient definition of the complex mojampere operator. And as soon as, say, for all t, the map x goes to phi t of x is omega t paracelonic and bounded. So I assume that my object here, I will impose that they are locally bounded, and I want to impose that they have some pruricic bombacity property, which means that, so definition, phi t is omega t paracelonic. And if and only if, if you write locally omega t is i dd bar of some guy, say ut, then phi t plus ut is a pruricicobamonic function. So the first condition we impose is that this function phi of t and x is, you can think of it as a bounded family of omega t paracelonic function. And if you do so, then by Beethoven Teller theory, this is well defined on x for each fixed t as a positive measure, and then you average this with respect to the big measure in time, and you have a nice well defined measure, positive measure on x t. So that's for the left-hand side. And maybe I should emphasize that we have good, very good convergence properties of this operator. So phi goes to dt, which omega t plus i dd bar phi t clear. For example, if you, so I forgot to mention that, of course, in practice, in both in theory and in practice, you can almost never compute this object in reality. What you do is you approximate them by smooth guys, and you want to be able to pass to the limit in the approximation process. So it's crucial in this week's theory that you have reasonably good convergence properties. And here in the Beethoven Teller setting, there are tons of such continuity properties. So if the phi t, for instance, converges, if you have an approximate that converges uniformly, then you can pass to the limit inside this mojant operator. So there are many other properties, but let's stick to this one for the time being. So now I would like, and this is perhaps a bit newer, to make sense of the right-hand side. So what do we have on the right-hand side? Well, something reasonably nice, something that is independent of time, and then there is this time derivative that shows up. So something that is quite natural is that we assume that t goes to phi t of x is, say, locally Lipschitz. So it's uniformly in the space variable. And then dt phi is well-defined almost everywhere, and therefore you can make sense of the right-hand side in the Lebesgue sets. So this is well-defined almost everywhere. So you're plugging the rest of the picture, and you end up with almost everywhere well-defined right-hand side, the density of the right-hand side, and so well-defined right-hand side. So that's first naive observation that you might hope that you will be able to produce estimates so that your objects are indeed locally Lipschitz in time, and we will do so. But it's also a problem that it's not sufficient to pass to the limit in approximation processes. So we actually add for more, namely we wish that our solution have some convexity property in time. So we want that t goes to phi t of x is almost concave, and again locally uniformly in 0t times x. So now if you have, so forget about the semi-concave problem, imagine you have a sequence of concave functions. If they converge pointwise, then you can indeed take the time derivative and you have convergence of the time derivative. So the convergence properties are guaranteed if you can make sure that your object have concavity properties in time. One last remark is that we don't, or we shouldn't, ask for Lipschitz property or concavity property all the way down to time 0, because as you may know, for the heat equation and the same property holds for the Kelel-Ritzchitz flow or complex monjampere flows, those flows have very strong smoothing properties, so you can start from an initial data that is poorly regular, and the flow will have the tendency to smooth out the initial data. So you can, it's natural to expect that you have good regularity in time for positive time, but you shouldn't expect good time, regularity in time up to time 0. So the local here is, you should stop the uniform estimate before time 0. Okay, so that's the definition of the left-hand side, the definition of the right-hand side, and what can we do with this? So the zero end I would like to explain is the following. Assume that capital F is continuous in the three variable. Assume moreover that it is quasi increasing in the last variable, which I will denote by R, so this is F of txR, and locally semi-convex in the last two variable, sorry, in t in R, so these are the assumptions on F. Assume g is in LP, like for instance this example, p bigger than 1, and the set where g vanishes has a volume 0. And finally assume that you pick an initial data, which is omega prosomonic with respect to omega 0, and say bounded to start with. Then there exists a unique phi, which solves the problem with the appropriate regularity, namely, so for all t, x goes to phi t of x is omega prosomic on x, 0 maybe phi is bounded, let me write it like this maybe, you cannot get bounded up to a time t, but it's bounded up just a bit, if you start just a bit before t. Okay, so it's locally bounded, but it's essentially bounded. So second property, t goes to phi t of x is locally uniformly semi-convex. Phi solves the complex-Majan-Père flow equation in this sense I just explained. And finally, phi t goes to phi 0 as t goes to 0, say in L1. So with this type of regularity, and with this definition of weak solution, there is a unique weak solution in this sense of such complex-Majan-Père flows. So capital t is the mean time where omega t is a correct class? Well, actually you give yourself data like this, so it could also be the, you need for f to be well defined as well, so if your data is well defined on this manifold, then you can solve the problem like this. So for the, of course you have in mind that for the Kele-Ritchie flow, capital t will have the tendency to be the maximum homological existing type. Okay, so I would like to explain the proof of this result, but before that I would like to stress, to mention a few earlier works and stress some differences. Maybe one comment, but maybe more for the experts, but the convergence that we've got at time 0 is actually quite strong. So I put the weakest one, it is sufficient to get uniqueness of the solution, but as we will see at some point in the proof, the convergence is in a reasonably strong sense. So in capacity for those who know what it is, are almost by above. So the, or if you start with some phi 0, which is continuous, the convergence is actually uniform. So there are strong information on what happens at time 0. But for the, this statement, especially for the uniqueness, which is quite delicate, L1 convergence at time 0 is sufficient. There's one comment on previous works. And that's actually motivated this one. So first there has been an important work by Song and Tian. It has been published only recently, but the paper goes back to 10 years ago. They developed weak solutions in similar contexts, but asking for the manifold V to be algebraic. There is a projectivity assumption in the process that is actually unnecessary. And perhaps more importantly, they prove strong reliability on the regular part of V. But since this is very demanding, the method is not very flexible. So besides the fact that it's natural to expect that this kind of techniques will be applicable in the Kehler context, more importantly, the techniques that they propose, which are quite interesting, produce stronger results, can be obtained, but brings techniques that are not very flexible. So it's important to develop software tools. And this is something we have been doing with Philippe Essidieu some time ago. And with Amedzariah. And I should also add that part of the work was completed by our PhD student, Thau, the last few years. And in these works we have developed parabolic viscosity theory. So rather different approach producing weak solutions as well. And here the drawback is that we need to assume, maybe with these notations, that there is a G missing, no? It was here, maybe. So we need to assume that the density G that I was mentioning, for the viscosity technology to work, we need data that are continuous. And in particular, the density that causes some trouble has to be continuous, not LP, a bit more demanding than that. So for instance, for the application to the Kela-Rachie flow, or the study of the Kela-Rachie flow, it implies that the singularities that you can deal with are canonical. And there is an important class of singularities that show up in the minimal model program, the so-called terminal singularities, that are not covered by this approach. Why the one that we propose today applies to the most general singularities encountered in the classification of a G-Rachie meter. And finally, a last comment on previous works, so they have been worked by many authors. Let me maybe mention a work by Peter Topping in the few years ago, and definitive work by Dinezza and Lou, which essentially shows that you can start from a very arbitrary initial data. So the initial data, which I denoted by phi 0, can be almost arbitrary. The presentation here I asked for the initial data to be bounded, but you can extend this hypothesis by far, and this is a work that Eleonora and Chinlu have done. So this could be also done here, but it brings some extra difficulties that we hide under the carpet. Okay, so the goal now is to give you a few hints on the proof of this result, and the technique, it's not a surprise if you've been following carefully the lectures by Gabor. You approximate by smooth flows, and you try to prove a prior estimate. So that was the main hint in Gabor's first lecture. And this is exactly what I propose to do. So let me write it that way. Assume you have this following question. Let us assume just for now that omega t is scalar, a whole new scalar form, that f is a smooth function, and that g is a smooth and positive function. So in practice, it means that you will have indices, approximations, and that fj will be a family of smooth function uniformly convergent towards the f, which is merely continuous, that omega t will be omega tj, omega t plus some extra positivity factor, etc. etc. But the notation will be awful, so I skip that, and I focus on proving a prior estimate for this approximate, without too many indices. So the first claim, so the claim will be that first in this context with the initial data phi 0, then phi of tx is uniformly bounded, maybe on 0t minus epsilon cross x. Second, that t times dt of phi will be bounded by yet another constant. And lastly, that the second time derivative is also under control. So in the following way, t square times d double dot t is less or equal than the third constant on 0t minus epsilon dot x. So that's the goal. And once again, you should imagine that there is an index, an extra index j that you are approximating, and of course in the statement, all the constants that you end up with are actually independent of j. So how to prove that? So we try to give some hints. So maybe let's have a look at the first, which is the easiest estimate. And it relies on Yao's solution to the calabic injector plus some extension due to colloj and a paper that has already appeared in previous lectures, and then you pick a function, or maybe I should write some notation. So you fix some uniform bound. So I haven't been precise on what are the assumptions on the family of forms. And the precise assumption that we actually need will show up for each app you estimate. So for the first one, we need to have a uniform lower bound, uniform in time. It just depends on x. It just depends both on time and space variable. This depends on x. So you can bound from above this family of semi-positive form by a fixed caliform that's given for free, that's cheap. But more demanding is that you need a positive lower bound that is uniform in time. And by Yao's solution to the calabic injector, you can fix some potential, rho plus or rho minus, with some respect to this form and bound it such that theta plus minus plus i dd bar of rho plus minus to the n. So the equation, mon-jamper, equals gdv. So this you cannot exactly do because of mass constraints, but you can adjust by a fixed constant here, which guarantees that the mass of the left hand side and the mass of the right hand side are the same. And once you do that, there is a unique such solution up to normalization. So for the plus, this is really Yao's solution for the minus because here the form has the tendency to degenerate. It's not really a killer. This is essentially the work that was mentioned by Koalje and IgZ. So if you do that, then you observe, for instance, that u plus of t and x, which is rho plus of x plus ct plus 1, is such that it's actually greater or equal than phi t of x. So that's what you can do, basically. And let me say one word to how to justify this. So what we can observe is that due to this inequality, phi t is a sub-solution to the flow, to the following flow, plus plus ddc v to the m dt is e to the dt v plus f tx v gdv. So actually, the point is that theta plus plus ddc phi t to the m because theta plus is a bit more positive than omega t. This is greater or equal than omega t plus, or sorry, I tried to avoid ddc since the beginning, but they catched quickly up. So you have this inequality. This is the equation satisfied by phi t. And now, okay, so that's the first observation. So phi t is a sub-solution for this monjamper flow. And now you compare to the fact that u plus, the claim is that u plus is a super solution to this flow. If c is large enough. So I leave that as a computation. And then you apply the so-called comparison principle. Once again, we are working with smooth approximations. So it's a classical maximum principle that you plug in here, which tells you that as the names suggest, the sub-solution is smaller or equal than the super solution. So I'm running out of time. So I will skip the Leap Sheets estimate, which is a bit tricky, but I would rather say one word about the last one, which is probably the newest of the three estimates. And maybe let me give a proof of this when under simplifying assumption, namely, I will assume that f is zero. It simplifies a bit, but it's still a bit technical. So I will assume moreover that omega t is given like this. So it's a fine. And in this case, the claim can be made, the estimate is even better, namely that t times the second derivative of phi is less than n. So it's a very precise estimate on the second time derivative. And why that? So set h of tx, sorry. So in this case, if I under this simplifying assumption, you can get rid of the square. So consider t times, let me use dot notation to simplify my life. Consider this and t0 x0, where h reaches its maximum. Either t0 is zero and then you're done. Or let me assume, maybe here, that t0 is positive. Then I claim that if you apply the following heat type operator to the function h. So s t is omega t plus i dd bar of phi t. Then this is less or equal than phi t double dot minus t divided by n phi t double dot square. So that's the important and slightly tricky computation. But let's assume that for a while. And then look at what happened at this time t0 positive and x0. Then at that point, this operator applied to i h should be a negative, which tells you that if you multiply by t0, this quantity, t0 phi t double dot minus t0 phi t double dot square divided by n, is less or equal than, sorry, is positive. Therefore, this quantity, no, that's okay. What is missing? No, so this is non-negative, I multiply by t0 which is positive and I end up with that. And this, you can send this guy on the other side divided by t0 phi t double dot and you end up with what you wanted. Is it n or no, no, no, no, no, no, yeah, okay. So I was planning to give you the full tricky computation, but pull you, I'm running out of time, so you will miss that. That was the funniest part of the story. Okay, how much time do I have, Daniel? Five minutes, yeah. So I will skip that because I need more than five minutes. So where are we? Here are the estimates. So let's pretend we have a full detailed proof of this. And now remember that we are actually arguing by approximation and we have this kind of information. So for each j, phi j is a solution of an approximation flow and we have this kind of estimate on phi j. Then first observation, families of quasi-prosthumatic functions, if you understand well the negative part of the curvature, are weakly compact in the L1 topology. So this first bond tells you that you can extract convergence subsequence, say in L1 sense, in a poor sense. Then, as I said, if you go now down to the third one, the second and the third one tells you that those families of approximants are locally leapshipped in time and furthermore, essentially, concave in time. So you can pass to the limit in this weak sense but also pass to the limit in the time derivative, thanks to the concavity. So where is my flow? It has disappeared from the board? Yeah. Well, did you say? It is hiding, no? No. Okay, so let me just rewrite that. Okay. Oh, thanks. Here it is. So what I just said is that here, on this right-hand side, we can pass to the limit, thanks to this information. And I cheated a bit at the beginning saying that we have tons of continuity results for the Mongeampere operator. There is one that we haven't got, is that it's not continuous along L1 convergence. So we need some extra information on how, if we extract here in the L1 sense, how this converges toward the limit. So, if you look at the equation that these approximants are solving, what you get here is the Mongeampere of the approximants are given by a density times the bake. And the densities are essentially uniformly bounded from above, thanks to the Lipschitz estimate. In that setting, we have strong stability results for solutions of a Mongeampere equation, and we can prove that actually it's not L1, but actually converges in the L infinity sense. So the approximants uniformly converge towards the limit. And therefore we can pass to the limit and produce a solution. With some extra minutes that I haven't got, you can apply this for instance to the long term behavior of the Kedavitchi flow on singular Kala-Biaux varieties, and get in a few lines that the flow deforms any initial data towards a singular Kala-Biaux metric that Gabor is trying to understand better for us. Thank you.
 Thank you.  I'd like to thank you for attending the picture.  I'm very excited for giving me this opportunity to speak here.  So I will report on a recent joint work with Wang Qinlu from Hasei and with Amit Zariaï from Toulouse,  who is in the audience.  And we are trying to develop the first steps of a parabolic, very potential theory.  So we have posted two papers on the archive last October.  Both are quite long and a bit technical. There are close to 60 pages each.  So I'm not going to explain all of them by far.  But I will try to focus on one of them that deals with the compact setting, which is a bit easier to explain.  And so what we try to develop is to understand complex mojampere flows, especially the case of the Kele-Arichi flow,  by means of very potential techniques.  So we try to define and understand the weak notions of solutions to these flows.  And there are basically two natural approaches.  So one is by approximation by smooth flows and developing a priori estimates.  This is the path I'm going to try to explain today.  And another one, which is quite useful as well, but maybe a bit more technical to explain,  is by means of Péran technique of envelope of weak sub-solutions.  So this one I will skip today, but the inspiration nevertheless lies behind what I'm going to explain.  So the main motivation is the study of the Kele-Arichi flow, which you can write like this.  And we try to understand this evolution equation, not only at first on smooth manifolds,  but you try to run it for as long as you can and then try to extend the definition of the flow as soon as you reach finite time similarities.  And very soon you need to understand this object on a mildly singular varieties.  And actually it's quite common to study the so-called normalized Kele-Arichi flow, which can be written that way.  Where lambda is a real parameter with sign with this way of writing things is the same as the sign say of the first-gen class of the canonical bundle of the manifold.  So if you pick, maybe I should write here that at the level of homology classes, so if you write alpha t, which is the homology classes of those scalar forms theta t,  then alpha t dot solves a very simple ODE, namely that maybe I should write that way.  Alpha t dot is given by this.  So you are from there and from the initial Kele-class that you give yourself,  understand that alpha t is a definite and simple form.  So we can pick, maybe I should write it here, a smooth representant of this form in alpha t, a fixed class,  and then solving this nonlinear equation turns out to be equivalent to solving the following complex monjamper flows.  So n will be the complex dimension of the manifold.  Where dv is a fixed volume form on the manifold,  h is a smooth and fixed function, and phi t is the unknown function,  which is assumed to satisfy the following positivity property.  So it's a well-known fact that which is a particular to Kele geometry that solving this Kele version of the Ricci flow or the normalized version is actually equivalent to solving a complex monjamper equation at the scalar level.  So rather than having an unknown tensor, you actually end up with trying to find and solve an evolution equation for some unknown function, scalar function.  So that's the kind of equation I'm going to focus on.  But as I mentioned, it's also extremely important to understand this type of equation on mind-leasing larvalities.  So the precise setting will be the following.  So let's assume that the underlying variety is actually might be singular.  It still makes sense of considering such an evolution equation.  And in order to understand the corresponding complex monjamper flow, we fix a resolution of singularities and lift the problem to the smooth manifold x,  and the corresponding flow transforms into, let me write this notation for complex monjamper flow, transforms into the following type of equation.  So f looks similar to what we had before.  So let's take some function h of t of x plus lambda phi of t. So this hasn't much changed.  So the new, or maybe h tilde, there is a new guy here, g, which has the following form.  And the zeros, so these are homomorphic sections of some line bundles, and the zeros of those guys actually corresponds to the nature of the singularities you started with.  So we have defined both poles and zeros for this density, but the important point from an analytic point of view is that you nevertheless end up with a density g, which is LP integrable,  and this is one possible meaning for the fact that the singularities you are dealing with are might.  So the next one, the difference is that you have lifted here, omega t is actually the lift of some eta t.  So eta was the caliform on the manifold v, but so omega t is no longer strictly positive, it has some degeneracies above the singular locus of the manifold you started with.  So you have a lack of positivity on the reference form, and you have also some analytic difficulties coming from this density g.  So in the sequel, I will focus on this general form of the complex mojampere flows,  and I would like to make sense of weak solution to this flow in the sense of pre-potential theory.  So rather than working on the manifold x, the first hint that we follow is that we work on the manifold xt,  which is a real manifold of dimension 2n plus 1.  So this is the maximal existence time for this flow,  and this is a real manifold of dimension 2n plus 1.  And we want to understand this equation on xt.  So we are going to interpret both sides of the equation as positive measures on xt,  and trying to make sense of this both left hand side and right hand side in the sense of measures.  So the left hand side is the easiest one to deal with,  namely the only part you need to understand, or somehow the delicate part you need to understand is this one,  which is a complex mojampere operator,  which has been studied in the late 17th by Beethoven Teller.  And they propose a very convenient definition of the complex mojampere operator.  And as soon as, say, for all t, the map x goes to phi t of x is omega t paracelonic and bounded.  So I assume that my object here, I will impose that they are locally bounded,  and I want to impose that they have some pruricic bombacity property,  which means that, so definition, phi t is omega t paracelonic.  And if and only if, if you write locally omega t is i dd bar of some guy, say ut,  then phi t plus ut is a pruricicobamonic function.  So the first condition we impose is that this function phi of t and x is, you can think of it as a bounded family of omega t paracelonic function.  And if you do so, then by Beethoven Teller theory, this is well defined on x for each fixed t as a positive measure,  and then you average this with respect to the big measure in time, and you have a nice well defined measure, positive measure on x t.  So that's for the left-hand side.  And maybe I should emphasize that we have good, very good convergence properties of this operator.  So phi goes to dt, which omega t plus i dd bar phi t clear.  For example, if you, so I forgot to mention that, of course, in practice,  in both in theory and in practice, you can almost never compute this object in reality.  What you do is you approximate them by smooth guys, and you want to be able to pass to the limit in the approximation process.  So it's crucial in this week's theory that you have reasonably good convergence properties.  And here in the Beethoven Teller setting, there are tons of such continuity properties.  So if the phi t, for instance, converges, if you have an approximate that converges uniformly, then you can pass to the limit inside this mojant operator.  So there are many other properties, but let's stick to this one for the time being.  So now I would like, and this is perhaps a bit newer, to make sense of the right-hand side.  So what do we have on the right-hand side? Well, something reasonably nice, something that is independent of time,  and then there is this time derivative that shows up.  So something that is quite natural is that we assume that t goes to phi t of x is, say, locally Lipschitz.  So it's uniformly in the space variable.  And then dt phi is well-defined almost everywhere, and therefore you can make sense of the right-hand side in the Lebesgue sets.  So this is well-defined almost everywhere.  So you're plugging the rest of the picture, and you end up with almost everywhere well-defined right-hand side,  the density of the right-hand side, and so well-defined right-hand side.  So that's first naive observation that you might hope that you will be able to produce estimates so that your objects are indeed locally Lipschitz in time,  and we will do so.  But it's also a problem that it's not sufficient to pass to the limit in approximation processes.  So we actually add for more, namely we wish that our solution have some convexity property in time.  So we want that t goes to phi t of x is almost concave, and again locally uniformly in 0t times x.  So now if you have, so forget about the semi-concave problem, imagine you have a sequence of concave functions.  If they converge pointwise, then you can indeed take the time derivative and you have convergence of the time derivative.  So the convergence properties are guaranteed if you can make sure that your object have concavity properties in time.  One last remark is that we don't, or we shouldn't, ask for Lipschitz property or concavity property all the way down to time 0,  because as you may know, for the heat equation and the same property holds for the Kelel-Ritzchitz flow or complex monjampere flows,  those flows have very strong smoothing properties, so you can start from an initial data that is poorly regular,  and the flow will have the tendency to smooth out the initial data.  So you can, it's natural to expect that you have good regularity in time for positive time,  but you shouldn't expect good time, regularity in time up to time 0.  So the local here is, you should stop the uniform estimate before time 0.  Okay, so that's the definition of the left-hand side, the definition of the right-hand side, and what can we do with this?  So the zero end I would like to explain is the following.  Assume that capital F is continuous in the three variable.  Assume moreover that it is quasi increasing in the last variable,  which I will denote by R, so this is F of txR,  and locally semi-convex in the last two variable,  sorry, in t in R, so these are the assumptions on F.  Assume g is in LP, like for instance this example, p bigger than 1,  and the set where g vanishes has a volume 0.  And finally assume that you pick an initial data, which is omega prosomonic with respect to omega 0,  and say bounded to start with.  Then there exists a unique phi, which solves the problem with the appropriate regularity, namely,  so for all t, x goes to phi t of x is omega prosomic on x, 0 maybe phi is bounded,  let me write it like this maybe, you cannot get bounded up to a time t,  but it's bounded up just a bit, if you start just a bit before t.  Okay, so it's locally bounded, but it's essentially bounded.  So second property, t goes to phi t of x is locally uniformly semi-convex.  Phi solves the complex-Majan-Père flow equation in this sense I just explained.  And finally, phi t goes to phi 0 as t goes to 0, say in L1.  So with this type of regularity, and with this definition of weak solution,  there is a unique weak solution in this sense of such complex-Majan-Père flows.  So capital t is the mean time where omega t is a correct class?  Well, actually you give yourself data like this, so it could also be the,  you need for f to be well defined as well, so if your data is well defined on this manifold,  then you can solve the problem like this.  So for the, of course you have in mind that for the Kele-Ritchie flow, capital t will have the tendency to be the  maximum homological existing type.  Okay, so I would like to explain the proof of this result,  but before that I would like to stress, to mention a few earlier works and stress some differences.  Maybe one comment, but maybe more for the experts,  but the convergence that we've got at time 0 is actually quite strong.  So I put the weakest one, it is sufficient to get uniqueness of the solution,  but as we will see at some point in the proof, the convergence is in a reasonably strong sense.  So in capacity for those who know what it is, are almost by above.  So the, or if you start with some phi 0, which is continuous, the convergence is actually uniform.  So there are strong information on what happens at time 0.  But for the, this statement, especially for the uniqueness, which is quite delicate,  L1 convergence at time 0 is sufficient.  There's one comment on previous works.  And that's actually motivated this one. So first there has been an important work by Song and Tian.  It has been published only recently, but the paper goes back to 10 years ago.  They developed weak solutions in similar contexts,  but asking for the manifold V to be algebraic.  There is a projectivity assumption in the process that is actually unnecessary.  And perhaps more importantly, they prove strong reliability on the regular part of V.  But since this is very demanding, the method is not very flexible.  So besides the fact that it's natural to expect that this kind of techniques will be applicable in the Kehler context,  more importantly, the techniques that they propose, which are quite interesting, produce stronger results,  can be obtained, but brings techniques that are not very flexible.  So it's important to develop software tools. And this is something we have been doing with Philippe Essidieu some time ago.  And with Amedzariah.  And I should also add that part of the work was completed by our PhD student, Thau, the last few years.  And in these works we have developed parabolic viscosity theory.  So rather different approach producing weak solutions as well.  And here the drawback is that we need to assume, maybe with these notations, that there is a G missing, no?  It was here, maybe. So we need to assume that the density G that I was mentioning,  for the viscosity technology to work, we need data that are continuous.  And in particular, the density that causes some trouble has to be continuous, not LP, a bit more demanding than that.  So for instance, for the application to the Kela-Rachie flow, or the study of the Kela-Rachie flow,  it implies that the singularities that you can deal with are canonical.  And there is an important class of singularities that show up in the minimal model program, the so-called terminal singularities,  that are not covered by this approach.  Why the one that we propose today applies to the most general singularities encountered in the classification of a G-Rachie meter.  And finally, a last comment on previous works, so they have been worked by many authors.  Let me maybe mention a work by Peter Topping in the few years ago, and definitive work by Dinezza and Lou,  which essentially shows that you can start from a very arbitrary initial data.  So the initial data, which I denoted by phi 0, can be almost arbitrary.  The presentation here I asked for the initial data to be bounded, but you can extend this hypothesis by far,  and this is a work that Eleonora and Chinlu have done.  So this could be also done here, but it brings some extra difficulties that we hide under the carpet.  Okay, so the goal now is to give you a few hints on the proof of this result,  and the technique, it's not a surprise if you've been following carefully the lectures by Gabor.  You approximate by smooth flows, and you try to prove a prior estimate.  So that was the main hint in Gabor's first lecture.  And this is exactly what I propose to do.  So let me write it that way.  Assume you have this following question.  Let us assume just for now that omega t is scalar, a whole new scalar form,  that f is a smooth function, and that g is a smooth and positive function.  So in practice, it means that you will have indices, approximations,  and that fj will be a family of smooth function uniformly convergent towards the f,  which is merely continuous, that omega t will be omega tj, omega t plus some extra positivity factor, etc. etc.  But the notation will be awful, so I skip that, and I focus on proving a prior estimate for this approximate,  without too many indices.  So the first claim, so the claim will be that first in this context with the initial data phi 0,  then phi of tx is uniformly bounded,  maybe on 0t minus epsilon cross x.  Second, that t times dt of phi will be bounded by yet another constant.  And lastly, that the second time derivative is also under control.  So in the following way, t square times d double dot t is less or equal than the third constant on 0t minus epsilon dot x.  So that's the goal.  And once again, you should imagine that there is an index, an extra index j that you are approximating,  and of course in the statement, all the constants that you end up with are actually independent of j.  So how to prove that?  So we try to give some hints.  So maybe let's have a look at the first, which is the easiest estimate.  And it relies on Yao's solution to the calabic injector plus some extension due to colloj and a paper that has already appeared in previous lectures,  and then you pick a function, or maybe I should write some notation.  So you fix some uniform bound.  So I haven't been precise on what are the assumptions on the family of forms.  And the precise assumption that we actually need will show up for each app you estimate.  So for the first one, we need to have a uniform lower bound, uniform in time.  It just depends on x.  It just depends both on time and space variable.  This depends on x.  So you can bound from above this family of semi-positive form by a fixed caliform that's given for free, that's cheap.  But more demanding is that you need a positive lower bound that is uniform in time.  And by Yao's solution to the calabic injector, you can fix some potential, rho plus or rho minus,  with some respect to this form and bound it such that theta plus minus plus i dd bar of rho plus minus to the n.  So the equation, mon-jamper, equals gdv.  So this you cannot exactly do because of mass constraints, but you can adjust by a fixed constant here,  which guarantees that the mass of the left hand side and the mass of the right hand side are the same.  And once you do that, there is a unique such solution up to normalization.  So for the plus, this is really Yao's solution for the minus because here the form has the tendency to degenerate.  It's not really a killer.  This is essentially the work that was mentioned by Koalje and IgZ.  So if you do that, then you observe, for instance, that u plus of t and x, which is rho plus of x plus ct plus 1,  is such that it's actually greater or equal than phi t of x.  So that's what you can do, basically.  And let me say one word to how to justify this.  So what we can observe is that due to this inequality,  phi t is a sub-solution to the flow, to the following flow,  plus plus ddc v to the m dt is e to the dt v plus f tx v gdv.  So actually, the point is that theta plus plus ddc phi t to the m because theta plus is a bit more positive than omega t.  This is greater or equal than omega t plus, or sorry, I tried to avoid ddc since the beginning,  but they catched quickly up.  So you have this inequality.  This is the equation satisfied by phi t.  And now, okay, so that's the first observation.  So phi t is a sub-solution for this monjamper flow.  And now you compare to the fact that u plus, the claim is that u plus is a super solution to this flow.  If c is large enough.  So I leave that as a computation.  And then you apply the so-called comparison principle.  Once again, we are working with smooth approximations.  So it's a classical maximum principle that you plug in here,  which tells you that as the names suggest,  the sub-solution is smaller or equal than the super solution.  So I'm running out of time.  So I will skip the Leap Sheets estimate, which is a bit tricky,  but I would rather say one word about the last one,  which is probably the newest of the three estimates.  And maybe let me give a proof of this when under simplifying assumption,  namely, I will assume that f is zero.  It simplifies a bit, but it's still a bit technical.  So I will assume moreover that omega t is given like this.  So it's a fine.  And in this case, the claim can be made,  the estimate is even better, namely that t times the second derivative of phi  is less than n.  So it's a very precise estimate on the second time derivative.  And why that?  So set h of tx, sorry.  So in this case, if I under this simplifying assumption,  you can get rid of the square.  So consider t times, let me use dot notation to simplify my life.  Consider this and t0 x0, where h reaches its maximum.  Either t0 is zero and then you're done.  Or let me assume, maybe here, that t0 is positive.  Then I claim that if you apply the following heat type operator to the function h.  So s t is omega t plus i dd bar of phi t.  Then this is less or equal than phi t double dot minus t divided by n phi t double dot square.  So that's the important and slightly tricky computation.  But let's assume that for a while.  And then look at what happened at this time t0 positive and x0.  Then at that point, this operator applied to i h should be a negative,  which tells you that if you multiply by t0, this quantity, t0 phi t double dot minus t0 phi t double dot square divided by n,  is less or equal than, sorry, is positive.  Therefore, this quantity,  no, that's okay.  What is missing?  No, so this is non-negative, I multiply by t0 which is positive and I end up with that.  And this, you can send this guy on the other side divided by t0 phi t double dot and you end up with what you wanted.  Is it n or no, no, no, no, no, no, yeah, okay.  So I was planning to give you the full tricky computation, but pull you, I'm running out of time, so you will miss that.  That was the funniest part of the story.  Okay, how much time do I have, Daniel?  Five minutes, yeah.  So I will skip that because I need more than five minutes.  So where are we?  Here are the estimates.  So let's pretend we have a full detailed proof of this.  And now remember that we are actually arguing by approximation and we have this kind of information.  So for each j, phi j is a solution of an approximation flow and we have this kind of estimate on phi j.  Then first observation, families of quasi-prosthumatic functions, if you understand well the negative part of the curvature, are weakly compact in the L1 topology.  So this first bond tells you that you can extract convergence subsequence, say in L1 sense, in a poor sense.  Then, as I said, if you go now down to the third one, the second and the third one tells you that those families of approximants are locally leapshipped in time and furthermore, essentially, concave in time.  So you can pass to the limit in this weak sense but also pass to the limit in the time derivative, thanks to the concavity.  So where is my flow?  It has disappeared from the board?  Yeah.  Well, did you say?  It is hiding, no?  No.  Okay, so let me just rewrite that.  Okay.  Oh, thanks.  Here it is.  So what I just said is that here, on this right-hand side, we can pass to the limit, thanks to this information.  And I cheated a bit at the beginning saying that we have tons of continuity results for the Mongeampere operator.  There is one that we haven't got, is that it's not continuous along L1 convergence.  So we need some extra information on how, if we extract here in the L1 sense, how this converges toward the limit.  So, if you look at the equation that these approximants are solving, what you get here is the Mongeampere of the approximants are given by a density times the bake.  And the densities are essentially uniformly bounded from above, thanks to the Lipschitz estimate.  In that setting, we have strong stability results for solutions of a Mongeampere equation, and we can prove that actually it's not L1, but actually converges in the L infinity sense.  So the approximants uniformly converge towards the limit.  And therefore we can pass to the limit and produce a solution.  With some extra minutes that I haven't got, you can apply this for instance to the long term behavior of the Kedavitchi flow on singular Kala-Biaux varieties,  and get in a few lines that the flow deforms any initial data towards a singular Kala-Biaux metric that Gabor is trying to understand better for us.  Thank you.
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