 Leonardo Sennatore z Stanford University. Vse je zelo izgleda o kosmologi in pozitivnih kosmologi. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. Čekaj. kot t heard or so, we didn't know how to make progress, in the last few years, thanks to some mathematical techniques that I will explain, we were able to make progress, and so I think it's interesting for you to know about this. So, first of all, let me explain the physical motivation. So, what we want to do? So, before I start, if something is unclear or my handwriting is not beautiful as it is, početno načine si predtiv način, sem da ne predtiv si vse zradič, okej? Zato v svojj universičnih je zabezet, ali v過去i odrečno je bilo zabezet, in vese z različenje universičnih je zabezet za veliko različenje, različenje cosimo, kakva gradične, zazvala pribenik, ima najbolj problem, ker u universičnih tako, načine je stabil naredil in homogenitizem. Zato, aso tukaj homogenitizem se naredil in vzelo, da je tudi držav, in vzelo, da je tudi zelniko. Zelo, da je tudi tudi zelniko, zelniko je stavil, ki je tudi zelniko. In po delu izviti, je to bilo sreč, če je tudi zelniko. Vzelo, da je tudi zelniko, da je v rupo, pošla, da je tudi zelniko. In zelo, da je tudi zelniko, Some people came out with the theory of inflation. Inflation is a period that is postulated to happen before the ebook of the consideration of the universal before B. benuclosynthesis, when the atoms are formed, the nucleus are formed. And it is supposed to be an epoch where the universe takes as the following form. Zato je, ta metara, ta metara v univiru, je domenjena z skalarstvom, z grandža d5 squared plus v of phi, minus v of phi, in je potenčnja, ki je začala, tako je skalarstv, ki je potenčnja energija, in je potenčnja, je več, več vse. Zato je vse, kaj je potenčnja vse, in več, več vse. Mlajte, da se činica kgtenje tukaj z vredanje in pejte to z deli phi z vredanje tukaj ko je ak dishesna tukaj. Ni ta kalderama s r Vikingskeh pejci, listedte to monksne xi tubi. If the scalar field in this region is as large as the inverse length of this region, h inverse, we call h inverse i, such that this is a choice given by the potential energy at this point over n plus square, or g Newton. If the universe, there is a chunk of space where the field is homogeneous on this region, on this size, and the potential is flat enough, then one can solve the equation and find that the universe expands, the universe is homogeneous, then the matrix is the following form. The square is minus dt square plus a square of t dx square. So this homogeneous universe is called the Friedmann-Roberts-Wolcker metric, and the universe expands exponentially. It is the ht where h is a constant given by this inflationary h. It is constant because the potential energy is constant. What will happen is that the field rolls down very, very slowly down the potential because there is a slope, but the slope is very small, so it is blown down very slowly, and therefore changes very slowly in time because the potential is quite constant, and therefore the universe pretty much expands exponentially during this time. This is an acceleration accelerating universe. A double dot over a is h square, it is positive, and the universe expands exponentially, and locally it is the same as the city space. For some time it is the same as the city space, and then at some point the flat region stops, it drops to the bottom and inflation ends, and the normal epoch of Big Bang cosmology, radiation domination, Big Bang neklosintesis starts. So this is a very important phase that we think it happened before what is called the Big Bang. It is the first epoch of the universe that we normally know of, and it is called inflation, and there are many experimental evidence supporting this, observation evidence supporting this, what happened. Ok, now the problem was the following. There is a problem however for inflation, since a few years after the initial formulation of inflation. The point is that as you saw, I repostulated inflation to solve why the universe is so large and homogeneous. On scales, the scales that we probe now, which is the Hubble scale that we probe now, which is the Hubble scale now, that is the scale that we probe. But in order to the, what is this scale so far, I assumed that initially there was a scalar field homogeneous in a big region of space, how big as the Hubble patch, Hubble's Hubble rate during inflation. So now the Hubble rate during inflation is very small, we are talking about maybe 10 to the 13 gv, so it is a very, very small length, but in terms of Hubble is still an Hubble length. So for a very few years after the formulation of inflation, there was the question of how likely is to be in these conditions. How likely is to find yourself in this top, in this smooth region so large on top of the potential. And what in general I like is for inflation to start. And this is called the initial patch problem. And this was born maybe 85, this problem, and it was very hard to make problem, but it is highly debatable. As typical, we promise that one cannot solve, there is a lot of discussions and angry people. If you open a scientific American every five years, there is the cover saying, blah, blah, blah, very people upset. And the reason why it was hotly debated is because if you want to understand how the universe becomes homogeneous, you have to start from a very non-homogeneous universe. And therefore, very non-homogeneous universe, hard, are hard to make any progress for two reasons. One could try to solve numerically the evolution. One can take a non-homogeneous universe and solve numerically. But the non-homogeneous universe, as you just heard in the former lecture, tends to have singularities. So numerically, you need a code. Numerically, one needs to handle singularities. And this is hard. In fact, it wasn't possible until the codes that were developed to study gravitational waves were developed. And analytically, as you heard in general, it's hard whenever there is, we cannot do, I mean, analytically, it's obviously hard. We can make a statement about non-homogeneous universe. Like, for example, the Perron singularity or incompleteness theorems. I was at the lecture, so I called them incomplete theorems. And so, but it's hard. In particular, it probably was not made. Now, in the last few years, we were able to make progress in these two camps. First, when you heard from the lecture from Nico Nunes, we have now codes that can handle singularities, the black hole singularities that describe mergers of binary black holes. Well, there is a numerical code, the Solger equation, solving with singularities. And we can do the same for early universe cosmologies. So, and also, yes. So, there are numerical codes that are available. Codes are now available. And if you ask yourself, let me simulate a big universe, a torus. Full of enom... We discovered field completing homogeneous. This is a numerical experiment we did a few years ago. You take a total other universe. You let it go, completing homogeneous. You will find that after some time, here there is a big black hole. But in most of the... So, the numerical code finds some black hole here, here. But in the space between, the universe expands exponentially and becomes actually infinitely large. OK, so numerically, we first experimented evidence that inflation actually was starting out of the homogeneous universe. There was singularities were forming, but somewhere in the manifold inflation was always starting. And in particular, in the numerical experiment we were finding that the universe always starts inflating. And so, this suggested the existence of a theorem that we will formulate. We suggest that the universe... Depending on the topology of the manifold, if you have an homogeneous universe but initially spanning everywhere, inflation always starts somewhere. And there are analytic techniques that are not completely new, but at least were new to us, or to some physicists that allows us to handle this problem. OK. Of physical interest, so there will be an analytic theorem that we are going to prove in this lecture, saying basically that inflation always starts in some topologies. And of physical interest is the case of 3 plus 1 dimension, which is our universe. We live in 3 plus 1 dimension. Here there are only partial results. Work is in progress and I think it's a very interesting topic. But in this lecture I will show you the proof of the full theorem in 2 plus 1 dimension. The theorem that basically says that if you take a 2 plus 1 dimensional cosmology, initially spanning, for all topology except one, inflation always starts. Even though now we stand in the size of the initial perturbations and now we stand in the fact that one forms many black holes and singularities here and there. OK. In this lecture we will prove the following theorem, which I will formulate next, but he has the following assumptions. So the first assumption is that we are dealing with cosmology. The manifold is a cosmology. What is a cosmology? The cosmology is what we also in physics we call a spacetime manifold, which means it's m. OK. Sometimes for simplicity I will keep general n dimensions and then at some point we will specify to 2 plus 1 dimension. So n plus 1 to 2 plus 1. n plus 1 dimension is a n plus 1 dimensional manifold. It's a cosmology if the topology is r cross m, m, n, where r is time and m is the spatial manifold. And if the m... OK, let me call it mn. mn is a compact. OK, for simplicity we take the compact n manifold and the property has been proven by Gerov that this statement is equivalent to the following fact, that the manifold mn plus 1 can be foliated by a family mt of manifolds where t is the parameter called time. OK, it's the time of the spacetime and mt are all... with all the syntopology. Syntopology and mt are Cauchy surfaces. For physics this is the manifold we are interested in because they tell you that each slice, there is a Cauchy surface and the topology doesn't change. It's what we really mean by spacetime. OK, so roughly speaking there are many... the manifold can be foliated by maybe complicated mt slices all with... with some... some time directions orthogonal to them. OK, then another assumption we'll do. So this is the manifold we deal with. Then in particular... OK, you can see that we check in this t where t is time here goes from t0 to plus infinity. So there is a beginning of time t0 and then the universe goes and we want to see what happens to the end. Then we have to say something about the the matter. The matter satisfies... the matter satisfies the dominant energy condition that you already saw in several lectures and the strong energy condition. But there is also... there is also a cosmological constant bigger than zero that is separated. The cosmological constant does not satisfy the strong energy condition but it is there and the rest of the matter satisfies this. So the dominant energy condition is the following quantity. It means that... OK, this is called DEC and this is SEC. So DEC means that minus T mu nu k nu is a future directed time like or NIL vector for any time like k nu vector k nu. A consequent of DEC notice that DEC implies the weak energy condition which tells that T mu nu k nu k nu is bigger than zero bigger equal to zero for any time like vector k nu and this simply tells you that the energy density is positive. OK, the energy density is positive. And usually I always find these energy conditions very reasonable apart from the cosmological constant that tends to evaluate many of them but indeed here we are talking about the stuff that is not the cosmological constant. And the strong energy condition is the statement that T mu nu minus one over N minus one g mu nu T contracted with the same time like vector k nu is bigger equal to zero. OK and this is for any future directed maybe I forgot future time like vector future directed time like vector k nu and this for the perfect fluid plus p is bigger equal to zero and rho plus 3p is bigger equal to zero for normal fluid physically means that gravity is attractive. OK, if a matter is dominating is the matter the energy content of the universe is dominated by this term by this kind of matter gravity is attractive. So this is the method that we will deal with. And then as I mentioned in numerical experiment but also as it is intuitive the universe we are dealing with an initial expanding universe it starts going and it is an homogeneous so we will form singularities so we have to talk about singularities there will be singularities in the manifold and but we don't know where they are we are interested in very complicated initial conditions so we are going to make an assumption about the singularities spacetime singularities that we are dealing with are of the crashing kind and which I am going to define next before I define this let me just say one more thing that when we specify to 2 plus 1 dimension that is for n equal to the topology will be such that the earlier characteristic k kai which is the earlier characteristic of the the earlier characteristic of the manifold must be less or equal to zero which means not s2, so all topologies but s2 and I never remember if it's here or here I don't know probably here 50-50 ok, not this two sphere it's not the theory we don't have a theory for the two sphere ok, so in this assumption we don't know what is the crashing singularity ok, let me define what is the crashing singularity yes it is I would say it's the renormalized cosmological causes the matter satisfies your right, I mean there is a bit of the cosmological causes being subtracted by the matter because otherwise it violates strong energy condition good question yeah, thanks also please ask questions ok, it's good in fact at the level these energy conditions exactly because of your questions become a bit tricky exactly because I think your question is one way to say why it's complicated at quantum level it's ok, so we need to define to be able to talk about the singularity, so let me give a definition ok, so basically what we are dealing with, we are dealing with a manifold ok, it will evolve it will evolve in some place it will be able to call, maybe here but in other regions things are fine ok, things will be fine ok, we want to be able to describe the situations like in our universe, this is a realistic situation like in our universe so so first we need to define a future crashing function a future crashing function t tilde is is a function defined on the full manifold mn plus 1 ok flash that in a global hyperbolic neighborhood which we call n then in n times intersected the region where this crashing function is bigger than some constant then t tilde is a crashing time function we range from c0 all the way to infinity ok, a crashing time function it simply says that the surface of constant time constant t tilde are crashing slices, so basically there is around the singularity there will be some surfaces ok, let me do in this way imagine this is the singularity ok, so there is an hyperbolic neighborhood of the singularity where there are these crashing slices of constant t tilde time identified by t tilde from some constant all the way to infinity ok and the level set ok, if you want this is called sc the constant, the level sets sc such that t tilde is equal to a constant on the surface there there is an extrinsic curvature the trace of the second fundamental 4 extrinsic curvature is less than minus c ok, what does it mean physically means that we want to say that here there is a singularity how do we say this mathematically we say ok, first of all we define a crashing time function that is in the neighborhood of the singularity there must be this this function which is a foliating because she surfaces around the slice around the singularity and as you approach the singularity the level set of this time function have more and more extrinsic curvature negative that is the volume is correct is going to zero because extrinsic curvature is how the volume changes minus infinity means that the volume is just going to zero in observer they are surely dies if you were in the formal lecture that really means that it is not good very bad for Schwarzschild, for example these surfaces would be naturally the equal inside horizon constant radius surface and as you change the radius the constant radius surface is a crashing function ok so we should say that the cosmology which is our manifold has only singularity of the crashing kind which is what we want to deal with if there is an open set open set N in this picture this is N this set here is N if there is an open set N such that outside of N the lapse lapse of the time function which is called N N is such that N to the minus 2 is well, if you write the metric is the mu t the mu t ok, so outside of this of this region the lapse N is bounded is bounded and N contains a crashing function and for t tilde less than c so if you take a certain level set and you look other than that then the lapse is bounded ok this is a mathematical way to describe what is the black hole singularity ok, it is shielded by a crashing function which means that if you go to hit the volume shrinks to zero and as soon as you are outside of this a given level set the lapse is bounded so this means that the singularity can be reached at finite time genetically this space time will be geodesically incomplete so this infinite proper time it can be reached but as soon as you are at finite distance everything is fine ok so this is a way that we have to describe the singularities and physically we think that these are the way the singularities happen probably one can find mathematical solutions where the singularities are not shielded shielded by a crashing function but in a in physical situation we expect this to be the case and then ok, with these hypothesis ok the theorem is the following ok, given all these hypothesis the special volume goes to infinity ok at late at late times where times is going to be defined in a way that you will see next and the same late times and the space time converges on average to the city space ok, the s city space ok and there are and and there are arbitrary regions where physics where observers observers feel that is any experiment that give the same result as in the city space ok so, this manifold goes on forms a singularities but reaches infinite volume on average everywhere looks at the city space and there are big arbitrary large chanco spaces where experimentalist will always find themselves in the city space this is the statement that there is probably one then somewhere the city space will be formed ok, now one thing I realize I forgot to say is that the physical motivation for this was the inflation was to prove that inflation start always start with some topology so there is no tuned initial conditions in order to have a universe that goes inflation but now I am talking about the cosmological cost so, this is the following I forgot to say sorry remember that inflationary potential is like this and inflationary phase is when the inflaton, this color field that drives the met, dominates the met during inflation sits here well, if the potential inflation really means that the potential is very, very flat happens only if the potential is very, very flat and in this region the potential is very similar to this which is the cosmological cost and so, to have a simple treatment we will approximate the potential of the cosmological cost so, for us, saying that inflation starts, means that the universe is dominated by the cosmological constant and becomes the city space ok so this is the problem, our hypothesis is that is what we want to prove yes yes initially they have finite finite volume then they go at infinite time the volume, of course at each time the volume is always finite but there is no bound and later given any finite volume there is a time when it will be reached it grows unbounded yes, that's a good question no, in the true, very good so this is the difference between the model in the mathematical model and the true inflation so, it's true in the true universe now until a million years ago a million years ago we were not inflated so inflation could not last better not last forever in our universe in fact, if you take this calafil potential it will roll very, very slowly because the position is very flat but inflation will last a finite amount of time and then it ends typically I last the universe is spun by e to the 60 which is a big number, very close to infinity e to the 60 is big so, to show that inflation starts it's enough to so, of course, the difference is that one of the difference important is that if you have a cosmological model it will never end but that's not really it's good, so this is incompletes of the model but it's not the question we are trying to address the question which was interesting is how it starts once it starts we know how to stop it because once it starts the universe quickly becomes homogeneous inflation is the city space it's very homogeneous we can do calculations in 72, people established how to end inflation people didn't know how to start so, the question was how do we start that's why this modeling approximation is good is good enough for our purposes thank you ok, so there is this modeling and that's what we are trying to prove how are we gonna prove it again, there is an initially expanding slice this is the curvature is positive everywhere this is what is called a big bang but it's very homogeneous very big, very small completely homogeneous you will evolve and maybe somewhere as I said we will form black holes and somewhere else we will evolve we assume that the manifold satisfies the generative equation and how do we know we have some intuition that tells us the numerical experimental evidence that tells us it will go to the city space how do we establish this we will use a probe to explore what is the manifold so the manifold satisfies m plus 1g which is obtained by solving generative equation and we will try to learn what is this manifold by using a probe the probe will be a special slice but it could be the initial special slice we will take a special slice and we will try to foliate space using this slice so we will take a special slice and we will define a flow we will try to foliate the space so we will define a flow with this special slice trying to cover the whole manifold and then we have foliated all the space and we will see what it is so we will define a flow and the flow is called we will use a flow called mean curvature flow the flow is defined in the following way take a slice and imagine that there is maybe at some time there is a bit bent slice maybe this visual is going to form a black hole so maybe here this curvature is negative so it is going to contract and form a black hole maybe here is positive we are trying to see that there is the seated space so we will try the surface to go where there is the seated space the moment we know that there is the seated space maybe somewhere else there is black holes but it is fine so there is a deformation that you can define say that you move the surface forward in time if the seated curvature is positive or backward in time if the seated curvature is negative by an amount proportional to k so you can define a flow that the variation with respect to the flow parameter lambda of the flow is given is proportional to the seated curvature of the manifold of the three manifold along the normal direction to the manifold ok so this is what the flow is it is the flow of the surface proportional to the volume it is a well established field in mathematics but it is in the context of Riemannian geometry in the semi-Riemannian geometry like in our case it is less well known but for our case it is very nice because you see that it will be very helpful for our purposes because the trace of the seated curvature is the variation maybe there is a factor of 2 of the relative variation of the volume of the special volume of the surface if h is the induced metric on the special slice ok the seated curvature is the variation of the volume so if you are moving forward the surface where the case positive that I am letting the volume grow here the volume was decreasing moving forward so if I let the volume go back the volume is grow so the flow will tend to make the volume increase and if we are talking about a three dimensional manifold in 2 plus 1 dimension that is a co-dimensional one manifold where the extra dimension is time in the Lorentzian constant if you try to make a surface grow you have to smooth it out for example in the Euclidean constant in the Euclidean constant if you want to make a path very very fast very very long you do this so a very singular path is very long ok so if you try to make the minimum flow in the Euclidean in the remaining geometry it becomes very singular very quickly but if the path is co-dimensional one if you start with a space like a path and you try to make this you go through null and null has zero length or zero volume so you don't want to do that you just want to stay flat so the minimum flow makes the volume grow and also is very smooth so this flow has a lot of regularity problems that I will mention so this flow will not become singular on its own it could be singular if it hits a singularity if it lives in a space which is bad which is not great and the intuition is the following whenever there was a singularity it was shielded by a crashing time function which means that the volume there was going to zero and so it's entreated that if the flow is in front gets somehow in front of a black hole it will try not to go there because the flow wants to maximize to increase the volume and here the volume will go to zero so the flow will not hit the singularity and on its own it stays calm and it stays smooth so this flow exists for some times and will allow you to explore in particular trying to maximize the volume will try to go will try to fully a space where there is a lot of volume and the city space produces infinite volume so will try to go where the city space is that's the intuition behind why Vikovo to flow is useful and all these statements we're going to prove them next ok in fact let's start by proving the existence of the flow ok, it's a very very simple exercise that ah, yes, thank you yeah, yeah, sure yes, yeah, sorry yeah, yeah, yeah that is, yeah, yeah, thank you t is bounded by a c-dependent constant that is here yeah, thank you it's bounded, then you go forward it's still bounded by larger constant but the past of the Cauchy slice is is well behaved nothing happens until you hit the c-infinity slice ok, thank you yes sorry, thank you very much this is sc, defined by the level set tt equal constant ketilde is the mean curvature of the slices of t tilde equal constant yeah, very good yes, and fortunately yes, good ok, the open set this is script n is the set and then there is the lapse yes, I apologize, this is terrible the n square is the lapse, you know what is the lapse ok, given a time function t and d mu ok, you will never believe that the theorem is true now I do so many mistakes ok, check the theorem because maybe it's wrong ok, so thanks for the question if something is unclear please ask questions because we can fix them quickly unless we can fix them and then it's a problem ok, but we'll try ok so, it's a simple exercise to derive the question that if you define the flow in this way or in this way how the mean curvature evolves it's in the reference I gave you a pedagogical derivation a physicist derivation rigorous but understandable and ok, so because if you look at the math paper many people don't like me cannot understand it it's very simple to derive that the change on the intrinsic curvature under the flow is following this equation is similar to the Ricciaduri equation or in math settings also Riccardi equation but is not quite the same ok, so this is the equation I want to, some of you might be familiar with the Ricciaduri Riccardi equation this equation which tells you how the convergent jodesic move along the jodesic flow this equation is not that because as we are moving the surface we are the normal vector which the intrinsic curvature is the gradient of it as to stay normal to the surface which is not the case for the jodesic flow so it's not quite the same it's similar ok, in particular there is no this is the Laplacian on the surface and Riccardi Ricciaduri equation doesn't have that Laplacian surface and then there is K lambda square is basically the cosmological constant the intrinsic curvature that you have in the seated space with that cosmological constant ok, and Armonu matter is the combination of Timonu that satisfies the strong energy condition apart from 8 by g in fact, sec implies that Armonu matter and nu bigger or equal to zero in fact, this combination apart from the K multiplication is the combination that enters also in the in the Okin-Perro's incompletest theorem is the thing that tells you that jodesic tends to converge ok, in fact another way to say anti-Perro's Okin theorem it says that the Perro's Okin tells you that there will be a singularity somewhere a correlation of this theorem tells you that there cannot be a singularity everywhere somewhere the universe will keep expanding ok, so there can be many singularities but somewhere the universe will go on so no global crunch theorem ok, yes ah, thank you, yes this is the next thing, yes sorry K mu nu is 1 over N h mu nu h mu nu is the induced matrix plus mu nu is the traceless part of the traceless the traceless part of the traceless curve ok there are interesting properties that I will not prove but they are carrying two properties that are intuitive of mean curve of flow is the induced matrix the first fundamental form ok, so if a surface is space like it will remain so because the flow tends to maximize the volume once the volume to increase if you locally become null the volume is zero so it will not become null ok, at most you stay ok, and the second property this all can be proven rigorously and the property K bigger than zero is preserved K bigger than zero is preserved that is if you start with a surface as we will do, that is initially spanning everywhere the mean curve of the flow will have a mean curve positive everywhere again why, because the mean curve of the flow surface why, because again if K is zero then you stop evolving the flow stops wherever K is zero it stops it never becomes negative ok so we want to prove that the flow exists this is the evolution of the flow ah, maybe I should have kept this ok, first we prove a theorem which we can be called the press which basically tells us the following thing that let m lambda be a solution of the mean curve of the flow started from in the interval m lambda solution to mean curve of the flow in the interval lambda one, lambda two ok inside in the full manifold m and plus one ok supose it exists a point x lambda with lambda in with lambda here in the interval ah, such that the specific curvature of x lambda is bigger than than lambda ok then if Km of lambda is max is the maximum of K in m lambda and the maximum exists because m lambda is a compact manifold then the following inequality holds then the maximum of the specific curvature at the time lambda two is less or equal that K lambda plus in exponential is more correction asymptotically 2 over n the maximum in lambda one minus K lambda ok, so this is a theorem that given a slice on the flow there will be a maximum value of the specific curvature on this slice and this maximum has to converge to K lambda so as lambda goes on this is K lambda the specific curvature of the space and there is a bound like this maybe take some time and you will go there ok, let me prove it the proof is a simple application of the maximum principle proof ok, consider w then w is e to the alpha K minus K lambda ok, so it's how much e to the alpha lambda sorry in exponential constant times K minus K lambda ok, then here a simple change of variable gives d lambda of w minus W plus 1 over n K K plus K lambda minus in alpha times W plus non-negative terms it's equal to zero so this is the question for W ok and let's consider the compact interval lambda 1, lambda 2 where the flow exists times m0 ok now W you see that by hypothesis the maximum in this interval is bigger than K lambda so W is so first of all we are considering a compact interval so there is a maximum there is a global maximum for W and at the maximum the maximum is positive max of W so at the maximum if the maximum is not at the initial point lambda 1 ok, then if the maximum suppose at the maximum look at this equation at the maximum suppose the maximum is not at lambda 1 this the first derivative can be bigger or equal to zero because at the maximum if if you are in the interval ok, if the point is containing the interval the derivative is zero but if the final point it could still be positive the only point where it could be negative if it is the first point but if it is not the initial point then this is bigger or equal to zero then this term is also bigger or equal to zero so at the maximum of W is bigger or equal to zero ok, for n small enough you can choose the n small enough that this is bigger or equal to zero ok sorry, this is bigger than zero for n small enough ok, and plus the negative term so you get zero bigger than zero and for n if n is smaller than 2 over n over kelamda square so the maximum is lambda zero is at lambda equal to lambda 1 and then it follows simply the theorem ok, then the maximum then it follows this statement is two ok, so this is equivalent to cannot be quickly, exponentially fast in time quickly we come on this surface the fastest this is equivalent to can be will be over there kelamda ok, and now with this statement we can prove we can see that we can use the following existing theorem this is a prove these are approved by people like asking and asking you skin ok that tells you that there is this this is a unique infinity continuous infinity differentiable solution to mean cover to flow ok in the interval zero lambda zero for some lambda zero this statement is semi trivial given this differential equation with smooth initial condition for some times the solution will exist for some lambda zero it exists now the non trivial part is that if the flow stays in a smooth and compact part of the part of manifold of m and plus one then the solution can be extended so the second part of the statement is what is non trivial it tells you that as long as the flow in a in a in a manifold this is time slices as long as the flow doesn't run away to infinity ok and it doesn't hit some singularity all the manifold then the flow exists so one can keep the solution so as long as this doesn't happen there is a global solution in lambda all the way to infinity time all the way to zero to infinity now because I've been a bit slow I will skip the proof that we can that that in our situation these two requisites, these two hypothesis hold and therefore the flow holds, the flow exists at all times but it's quite intuitive and let me explain you first the flow stays in a comparison of the flow well it's pretty clear why so if I have time I will prove it but I doubt I will have time so you see that the advancement the advancement of the flow is detected by k k is the speed of which the flow advances in time but we saw that the maximum of k so there is a limited speed the flow reaches the flow is the most advanced ekelanda speed in particular one can easily show that actually this is the full speed and therefore since the flow advances in time a finite speed for finite time it stays always in a compare region staying in a compare region never go to embed the time function of the manifold it doesn't reach infinity for finite lambda and it cannot because it advances in most ekelanda speed ok, so it stays in this compare region and it's also possible to prove that it stays in a smooth region because as we discussed if this is the manifold the singularities are shielded by the crashing function a crashing function is another some other foliesho with k tilde is negative and it's quite intuitive and it's provable that if there is a Cauchy slice of this neighborhood where k tilde is negative the flow which has k positive will never intersect this region it's a contradiction argument that shows that the flow will not penetrate this region because if it did it would decrease the volume and it doesn't want to do it ok, if there is time I'll prove it but this is intuition but given that these two are very true, then the flow can be extended at all times so now we have a flow that exists at all time and we don't need to worry about it and then we simply see what it does ok ok, so this is our instrument our instrument to explore this manifold which I remind you we don't know very much but we have this instrument that allows us to explore it and now we're gonna see a few facts so, let's start by saying that the volume goes to infinity what does it mean that the volume goes to infinity we're gonna check that the Mikovo to flow slices which are particular slices of this volume go to infinity ok, here we're gonna work in two dimensions ok, we take Einstein equation g mu nu equal a pi g t mu nu our manifold satisfies the Einstein equation and we contract two i's with m mu and nu ok, we take this particular equation now one can use Gasco da C to write this equation as R2 we're into this equation plus 1 half k square minus sin mu nu square equal 1 half k lambda square plus sin pi g t mu nu m mu nu now the Mikovo to flow we can ask how much the volume changes of our surface changes as much as the volume of our Mikovo to flow slices changes with the flow divu in d lambda is the integral of of the change see, there is a k here we move forward proportional to k and now the volume is the derivative we have to do d d lambda of of square root of g but d d lambda of square root of g is d d t of square root of g and now we do the derivative between d t and d lambda that is another k, so this is the k square intuitively the change in volume is positive under Mikovo to flow ok, that's all what we said all the time now let me plug back this equation so this become equal d2x square root of h of now I will stop I will forget the numbers 32 pi g t mu nu m mu nu plus k lambda square plus 2cm squared minus 2r2 now this is positive remember that the weak energy condition says that this is positive this is bigger or equal of k lambda square k lambda square is a constant so the integral of d2 square root of h is volume v jo lambda plus the integral of r2 ok, minus 2 the integral of r2 but the integral of r2 is the characteristic in two dimension so this is equal to k lambda square v jo lambda ok, minus 8 pi kai now you can see that if kai is negative or even 0 then this is positive this is bigger or equal the k lambda square v lambda for all the manifold not s2 ok, so the derivative you see dv in d lambda the derivative is positive at all times so the volume goes to infinity and in particular this equation can be solved and it tells you that v jo lambda is bigger or equal then some v0 times e to the k lambda square lambda plus 1 plus corrections that go like kai over v0 plus correction that is potentially small ok, so so this is the volume goes to infinity ok, how does it goes to infinity goes as faster faster or equal the volume of the slice of the city space ok, so this is faster this is the city space growth of volume ok now it's so the rate of growth is faster or equal to the city space the claim is that we are gonna go to the city space so now we can prove that the flow will grow the volume will grow slow or equal to the city space so this means that it grows like the city space ok, if let me do it for simplicity for a case where the characteristic is negative so for zempo e itoros with some holes ok, yes ah, thank you Sima square is the stress the stress part of the stress curvature Sima mu nu and this is Sima mu nu square ah, thank you Sima mu nu square no, this is not the trace this is the square like Sima 1 2 ok, thank you I wrote Sima and Sima was very misleading now it's fine ok, thank you ok ok, but I'm being pretty fast here so now now let's show that the rate is bounded by the one of the city space from above consider a case where chi is negative for the manifold ok, this means that the integral of R2 must be negative means that at least one point R2 is negative at this point R2 is negative ok, and this is negative ok, and this is positive so at that point k is bigger than kelanda ok, so it exists a point exists xp with k of xp bigger than kelanda ok and this means that the theorem of the press that I showed you before, the theorem that tells you that the maximum of the city curvature goes to kelanda, up to the exponential corrections in lambda time holds and this means that the maximum of the city curvature basically becomes kelanda but the maximum of the city curvature you see that dv in dlanda is equal to the integral of the square root of h k square, and this is smaller than the integral of the square root of h of the maximum but the maximum is of the k maximum ok which is klanda square times the integral of the square root of h which is the volume ok, smaller or equal so the volume grows apart from small corrections which you can get the paper to be slightly more precise but this is less than equal than v0 e to the kelanda square lambda so this plus this implies that the volume grows like e to the kelanda square lambda so the volume actually grows like the city space ok then, 5 minutes then ok, so we got the volume but what was the and the rate of change of the volume is the one of the city space so dv in dlanda is k square is the integral of k square but dv in dlanda is kelanda square since it's kelanda square times v which means that on average now that the rate goes like this means that the average of k square which is the integral d2x of square root of h k square over the volume which is the integral of d2x is equal to kelanda square so on average this risk curvature is the one of the city space this means that in the L2 norm of k the one norm of k square converges to the one of the city space ok this means that if you integrate this equation now if you integrate this equation in space what do we get so let's integrate now let's integrate this in time integrate d2x square root of h ok the first term give us minus 8 by chi then the second term give us plus kelanda square v lambda this is the integral of k square times the volume and this is equal ok, if you want then there is minus the integral of sigma square must be equal to ok, there was one up here the integral of the cosmological constant plus the integral of t mu nu and mu nu ok now the interesting fact that these two cancel ok and the volume goes to infinity so so this means that and this, when you integrate over the whole manifold goes to constant even though the volume is infinite volume ok, so this means that the integral of sigma square is less than a constant ok maybe to normalize it we can put it with respect to kelanda square and the integral mu nu is also less than a constant but the volume maybe with a g newton but the volume is going to infinity which means that in almost this implies ok, that almost everywhere apart apart for a finite volume, a finite physical volume sigma square mu nu mu nu which remind you that positive point wise r over e to the minus kelanda square lambda which is one over the volume that is, if this integral of an infinite volume must give me a finite number it means that almost everywhere the size of these guys is over the one over v and not that these guys are positive so the absolute value of these guys is small so almost everywhere apart and since this is positive also sigma mu nu directly is zero because if sigma mu nu square is positive since sigma mu nu is a special tensor sigma mu nu is zero if sigma mu nu square is zero sigma mu nu is zero then almost everywhere both the energy density and the anisotropic part of the stress tensor goes to zero and notice that the dominant energy condition tells you that if t mu nu and mu that tells you that the dominant energy condition tells you that this part of the stress tensor is the biggest component of all of them so this implies that t mu nu in absolute value is less than t mu nu and mu nu and so all the components of the stress tensor goes to zero this special slices reach in free volume like they see the space one minute and then I'm done then this is the curvature is the one of the space and there is no anisotropic curvature and there is no matter they're empty and by ask an equation now we can finish the proof since we proved that t mu nu is zero almost everywhere we use ask an equation basically tell us that r mu nu r equal t mu nu I mean r mu nu minus minus one half g mu nu t this is another way to rewrite ask an equation with genuto so and remember that there is the cosmological constant so but this is zero almost everywhere we just show it so this means that r mu nu of the two plus one dimensional space time is the one of the city space almost everywhere if you want in this sense here which is the l1 sense in the normal one the one of the city space but in two plus one dimensions the Riemann in two plus one dimension is a known function in terms of the Ricci or two plus one dimension mu, ok, now we get in this round g r nu sigma two plus one so this is equal to this plus permutation sense object like this and since we just show that almost everywhere the Ricci is the one of the city space this means that the Riemann tensor of of two plus one dimensional manifold is almost everywhere equal to the one of the city space ok and this ends the proof so in the l1 sense or l2 sense depends for some quantities there is a one sense of l2 sense this manifold has evolved to an infinity large region of space where there are some singularities but there is a big region here probed by the mean curve to flow and whenever the mean curve to flow reaches the Riemann tensor is the one of the city space for arbitrary large volume for arbitrary large volume there are some regions here which are still singular so if you look at the late time manifold from above there is an infinite volume infinite of infinite full of oval patches there are some black holes here and there but there are infinite large regions here in the middle which looks like the city space in this l2 sense and since I have a bit overtime I will stop here Thank you