 Okay, for this final lecture, I'm going to continue the same theme, but I'm going to come back to three dimensions, and I'm going to talk about some aspects of what we've done which work in the general case, that is, for a general initial data set, satisfying the dominant energy conditions. So we'll have dimension three, and so remember the constraint equations, I'm going to leave off the eight pi just for notational convenience, so the scalar equation is mu, which is the energy density of the matter, is one half the scalar curvature of G, so we have a triple, a three-manifold Riemannian metric and a symmetric zero-two tensor, k, and so the first constraint equation is that this is minus the norm of k squared plus the trace. And then the second one is that, we'll write this as a one form, so j sub i, which is the momentum density, is the divergence, dj of, and I'll call this tensor pi ij, where pi ij is this tensor algebraically equivalent to k, so it's k ij minus trace k g ij. And then the dominant energy condition, which is the condition that will replace, or which reduces to non-negative scalar curvature in the case when k is zero, is more generally the condition that mu is greater than or equal to the norm of j, so that's the study. So we want to look at three-dimensional manifolds, which have a Riemannian metric and a symmetric zero-two tensor, k, satisfying this inequality where mu and j are given above. So from a geometric point of view, it looks sort of strange, but of course these equations arose from the Einstein equations, so that gives them some nice structure. So the three basic questions we were looking at were related to geometric properties, in particular, gravitational energy or mass associated to, so the complete case, so the asymptotically flat case was the first one, asymptotically flat, and so that was one. The second one was finite regions, omega with boundary omega, which is smooth, and this in the scalar curvature case was the Brown-York mass, which we discussed, focused on, and the third one was the polyhedral setting where the boundary of the domain is polyhedral, so it has dihedral angles and then we proved a sort of weak angle comparison theorem for this setting for non-negative scalar curvature. So you can ask, well, which of these theorems generalize in a reasonable way to the, it's called the space-time case, and so the answer is the asymptotically flat case, just the positive mass theorem does generalize, and actually the Brown-York mass has a very interesting and intricate generalization, which has been studied over the last seven or eight years by Yao and his collaborators, especially Mu Tao-Wang and several other people, one recently is poning Chen, has also been a contributor to that, and it's a complicated enough generalization that it would take me another lecture series to really explain it, but just generally the idea is that, so for the Brown-York mass you made assumptions on the boundary surface that it could be isometrically embedded into R3, and so in the general case, general constraint equations, what they do is they take the boundary surface and they isometrically embedded into Minkowski space, so this has something to do with the space-time, not just the spatial part, and so it turns out, though, if you look at that isometric embedding problem, there are many, many, there are an infinite number of ways of doing that, and so one of the hard things that they do is they formulate the notion of an optimal embedding in a precise sense, and it's very hard to compute the optimal embedding, but I mean to check whether an isometric embedding is optimal, but it's based on choosing good isometric embeddings into Minkowski space, and then to compare the Minkowski geometry to the geometry in the curved space-time, in particular to the induced metric, which of course is the same on both, but the mean curvature, and also, yeah, so it's a sort of complicated story, but nonetheless it's in some way quite a satisfactory space-time extension of the Brown-York mass, so I'm going to just reduce it to a check mark for my talk and not go into the details anymore, but actually it's still very much an ongoing project to define various quasi-local quantities and to check their properties. Now the third case, the polyhedral case, I'm just going to put a question mark here because I thought about it in preparing the lectures, and in fact I'm going to give a proof of the positive mass theorem, which looks a lot like the one we gave for scalar curvature, so actually you could, I mean I sort of did motivate the polyhedral results for the cube by looking at a large cube in our asymptotically flat manifold, right? And so I'm going to give a proof which looks the same, and you could try to do the same thing, but it doesn't appear to actually work, so at the moment there hasn't been anything done on that in the space-time case, so I consider that to be an open question. It could end up being interesting. I would assume you'd have to look at polyhedral, which don't necessarily lie in R3, do that, but again I don't know how to do that. Okay, so what I'm going to do in the talk today is I want to start by, okay so I'm going to discuss the first case, and then I'm going to talk about some other aspects which are not exactly in this direction but closely related, which are related to trap surfaces and moths. And so the basic, so remember we started in two dimensions and then we said well when we go to three dimensions we should replace curves by surfaces and geodesic curvature by mean curvature and so it comes down, so we ended up using the theory of minimal surfaces, which are surfaces of mean curvature zero, so that's not the right thing to do here with a K, but the correct thing to do is to replace minimal surfaces by what are called moths, and so we're going to study, so the idea is to look at, we're going to be interested in surfaces, sigma contained in our M, which are what are called moths. And it turns out that there are, there's a notion of stability for moths and there are some existence theorems, and I'm going to also point out the difference between the two cases, so actually this case I think should be considered to be much more flexible, more complicated than the non-negative scalar curvature case, so I expect the geometry to be much less restrictive. In fact, actually one of the questions that came up in the discussion on Tuesday was the question of what topologies arise in the three-dimensional case for asymptotically flat three manifolds with non-negative scalar curvature, and actually the topologies are, they're an infinite number of possible topologies, which can be classified, and they're somewhat restrictive though, I mean there are lots of cases that can't occur. If you look at the general case, just the dominant energy condition, then it's known, it's a theorem of Maseo and Pollock, that every topology occurs, and so you expect, you don't expect to get any topological information from the dominant energy condition. So, and so that suggests that, you know, something is quite quite a bit different in the two cases, and so let me first describe Mott's, okay, and so, and so if we look at surfaces, we could take a closed surface or non-closed one, so this would be a surface sigma, so think of this as in the space time contained in M, which is contained in some four- dimensional space time, then we can measure what's called the expansion, so at each point of sigma there's a function, which is theta plus p, and it's the expansion, so in normal to the surface there are a pair of null directions, null lines, and we're in a asymptotically flat setting, so we can distinguish between the outer pointing and inward pointing, and so theta plus p is the logarithmic rate of change of the area element on the surface when it's moved in the forward outer direction, so you can think of taking the surface and moving it along, and pushing it in a particular null direction, and then computing the rate of change of the area, so generally speaking in any Riemannian or Lorentz manifold, if you take a normal vector field, and you look at when you deform the surface in that direction, and you look at the rate of change of the area, that's related to the mean curvature of the surface in that direction, and so in particular this theta plus is a null mean curvature, and because if I take sigma to be in M, then I can write this normal vector, so I haven't drawn my picture very well, but so I have here M, and here's sigma, then M has say a forward pointing normal vector, call it E0, and then this particular null direction I'll choose to be, sorry, I'll take the other vector which is the unit, outward unit normal, let's assume we have an orientation, so we choose a unit normal on sigma, and if sigma bounds a region, which is often the case, then we take the outward unit normal, and then this null direction we're interested in, we could take to be the sum of these two, E0 plus nu, and so when I write the mean curvature, the space time mean curvature in that direction, I can break it up as a sum, I get the one term involving the mean curvature of the surface sigma in M, and then plus another term which involves the second fundamental form of M in the space time, and so that second fundamental form is given a priori, that's K, and so what this is equal to, and again there's always the question of the sign of the mean curvature, this is actually the wrong sign, but I'm going to use it anyway, so this is equal to the sum of the mean curvature H sigma, and this would be the mean curvature of sigma in M, and then plus the term involving the second fundamental form of M is the trace of the tensor K restricted to M and traced along sigma, okay, so K is a symmetric 0, 2 tensor in the ambient space, if I have a surface in M, I can restrict K to the surface and I can take its trace, it restricts to a symmetric 0, 2 tensor on the surface, and then the theta plus then works out to be that, and so the condition that theta plus at every point of sigma is negative is what's called outer trapped, and outer trapped surfaces of course play a very important role, I think Bob mentioned it in his first lecture in the Penrose Singularity Theorem, so if you have a closed outer trapped surface in your initial data, then that implies that the evolution of the data will be at least incomplete, you expect it to form a black hole, but of course we don't know in general, but the rigorous argument says that initial data with a closed trapped surface will not evolve to a complete space time, okay, and so this is the outer trapping condition, and so the basic picture of the Schwarzschild initial data set is sort of like that, and if I want to I could you know fill in with something there, and so this is infinity is up here, then there's the Schwarzschild horizon which is here, and then if you take surfaces or spheres behind that horizon, these are examples of outer trapped surfaces, and if I take a sphere outside the horizon, then this is a situation where this is strictly positive, so I'll call that strictly untrapped, and then between those two, between an outer trapped surface and a strictly untrapped surface, there'll be this very special surface which is the horizon of Schwarzschild, and that will satisfy point-wise theta plus equals zero, and so this is the Mott's, so if it's equal to zero, then that's called a marginally outer trapped surface or a Mott's, okay, and so the Mott's equation is again related to the mean curvature of the surface in M, and it also involves the specified K, the tensor K, so it's a kind of a sort of unusual kind, at least mathematically unusual kind of prescribed mean curvature problem. Instead of prescribing a function, you're prescribing the trace of the restriction of an ambient tensor, but it turns out to be a very, well, I mean obviously in relativity it's important, it turns out to have very nice properties in certain nice properties, but also some drawbacks compared to the minimal K, so if K is zero, of course, it's just the surfaces we've come to know and love, the minimal surfaces, which we've been studying. So what we're going to do is we're going to look at the Mott's and we're going to try to use them in an analogous way that we used minimal surfaces in the time symmetric, or K equals zero K's, okay, and so it's very important to understand the properties of Mott's, and so if you remember the proofs we gave, so I mainly focused on one and three in the talks, if you remember the proofs we gave, there were sort of two parts, so one part was under, so the arguments were indirect, so under the opposite conclusion that we're trying to draw, we were able to construct minimal surfaces with very special properties. In the asymptotically flat case, they were area minimizing and asymptotically planar. In the cubicle case, they were free boundary surfaces, assuming the angle was small, and so part of what we need, so there are really two key ingredients that we need, so the first key ingredient is existence under appropriate conditions, and the second key ingredient is, so once we had this Mott's, which isn't supposed to exist, how do we show it doesn't exist? Well, we use the fact that it's, for minimal surfaces, it's variationally stable, and the stability condition gave us a, we were able to put in the information from the ambient geometry, scalar curvature non-negative, to say, show that it was impossible in cases we considered, so the other, the second part you might call non-existence, or vanishing, so in geometry, things don't exist, then they're called vanishing theorems, so for, in hodge theory, vanishing means it's zero, the form is zero, in this case it doesn't mean it's zero, it means the surface doesn't exist, so this is non-existence, and this is the part that uses the geometry, so in particular it has to use the dominant energy condition. Okay, and so, which of these go through? Well, it turns out the most analogous one is this one, which is perhaps not obvious, or not, not surprise, not, not what you would expect, but it turns out there's a pretty big difference between the existence theory for the minimal case and the mot's case, and I'll say more about that in a minute, but I'm going to focus now on, on two, and so, and so how did we, how did we do the non-existence? Well, we use the, we use the stability inequality, we use the fact that the, the, the curvature condition, the positivity on the ambient manifold tends to make minimal surfaces unstable, okay, and we, that came in precisely through the, through the stability inequality, and so it turns out there's a, there's a natural notion of stability for mot's as well, and I can describe it here, so, so what does it mean for, for a mot's to be stable? Here I have the mot's, so what's a stable mot's? Well, it doesn't, so, so in the minimal case we could, we could say it's variationally stable, that means the second variation of area is non-negative. Now the thing about the mot's equation is that it's, it's not variational, you can't, there's no, there's no, there's no modified area functional which you can minimize to get, to get mot's, and so, and so the stability conditions a little different, so it turns out there's a, there's a, an equivalent notion of stability for minimal hypersurfaces which does generalize to mot's, and so, so the definition of stability is, is as follows, so I have the picture here already, so let's see, so, so in this picture, this is an example of a stable mot's, so why is that? So this is our sigma knot. The reason is because it is the, it's in a, it, it lies in a foliation, so there's a foliation here, sigma t, where the sigma t, for t negative say, if I take these to be t negative and these to be sigma t positive, so here theta plus is positive, those are strictly untrapped and this is for t positive, and down here theta plus is negative. Okay, so if I, if I have a mot's that lies in a foliation of that type, then, then that will be a stable mot's, okay, and, and so, the, what you should think of is, again, going back to geodesics, you could look at the geodesic on this negatively curved surface, that one's stable, just like this, but you could also look at the geodesic on the two-sphere, like that. Now the geodesic on the two-sphere is unstable, because if I look at a family down here, that one has theta plus, so if I think of moving up this way, the expansion, if I move it up, in, in this case, the theta is, is positive, the area is increasing, and here above theta is negative, and so, when I pass through that surface, so if I, I have this foliation, when I pass through the, the surface, theta is actually decreasing at each point, so if I follow sort of a family of points here, but take a normal to the sigma, and I look at the value of theta along that path, then theta will go from positive to negative, so it's decreasing, whereas in this picture, the opposite is true, so if I, moving up this way, here theta is negative, and here it's positive, and so in this picture, theta is going from negative to positive, so it's increasing, and so, and so a stable mat is almost that, but it's a little bit, it's sort of the infinitesimal version of that, so, so, suppose we had a situation where, where we had a family like that, so, so it was, so theta, theta goes from negative to positive in, in the foliation, then what we could do is we could differentiate, so what we would have, so if we have a family like that, then we could look at, we could look at theta t for a fixed point, so we could parametrize them as normal deformations over the, the value at time zero, and then we could look at theta t dot of zero, so theta, theta zero dot, so we could differentiate at time t, and, and what do we get when we do that? Well, you see the family, if we think of, if we think of the nearby surface as just a normal, a, a normal deformation, we parametrize it by a function where we move a certain distance in the normal direction, and if we call that function, say u, then theta dot at p turns out to be minus a differential operator, l, u, so u times nu, the normal nu is the generator of the foliation, of the foliation, so I have this foliation, sigma t, sigma zero is my sigma, and I think of it as, as given as moving in the normal direction at time zero to sigma naught, and so u is the, u is the, the, the initial, the infinitesimal vector field, which, which tells us how to move to, to stay on the, the foliation, so this is minus l of u, and in particular the condition that, that this be positive, or actually I'm not gonna demand it be positive, I'll demand that it be greater than or equal to zero, and so because I have a foliation, u is a positive function, it means the sigma t's don't intersect each other, so I move in one direction, and so, and so if I had such a, a foliation locally, then I would be able to produce a function which is the generator of the foliation, which satisfies the condition that l of u is less than or equal to zero, and so that's the definition of stability, and l is an operator which we're gonna write down, so in a minute, but so the, the definition here is that there exists a u positive on sigma function with l of u less than or equal to zero, so there's a minus sign here, again it's related to the, the definition of the mean curvature, so, so the, right, and, the important thing is this operator l, so, so in, in the, in the minimal case, if, if we were looking at minimal services, this l would be exactly the operator that we studied, it would be these, the Jacobi operator, the, the linearized mean curvature equation, okay, and so, and so it turns out for minimal services, this notion of stability is completely equivalent to the variational notion that we wrote down, and so, and so let me describe this operator l, so it's a, it's a, it's a bit of a mess in the Mott's case, but it turns out it has some very nice structure, so, so what is l? Well, well, it's a computation, right, you have to, you have to do the calculation, but if you do it, you find that l of u is the Laplacian of u on sigma, and then plus a term that looks very much like the term we, we studied for the minimal case, and so, and so, what it is, so the scalar curvature term is replaced by a term that's positive by the dominant energy condition, so this is mu minus j applied to nu, that nu is the normal, the normal vector of my Mott's, the unit normal, so I have that, and then I have minus the scalar curvature of sigma, and then I have plus the square of the second fundamental form, except now it's the, it's the square of the null second fundamental form, so it's, so it's the second fundamental form of sigma in M, and then plus k squared, it's this, this stuff times u, and so this part, so if this were, if we were in the time symmetric case, this would be exactly one half r, so this term would be, would be the R M, the ambient scalar curvature, and as you can see, the dominant energy condition comes out very naturally in this calculation, so we want this term to be positive in order to make things work, and the dominant energy condition precisely implies that, it's the difference between mu and j, and then there's some other terms which a priori look kind of nasty, so let me write them down, it turns out they're not so bad, so, so it's plus two times the inner product of a vector field w with the gradient of u, and then let's write it as plus norm of w squared minus the divergence div sigma of w multiplied by u, so that's another zero order term, so in this case there's a first order term, and if you know something about differential operators, you know that, that, that makes the operator non-self-adjoint, so if you look at, formally at the integral of l of u times v, and you ask is that equal to u times l of v, the answer is no, and it's because of this term that, that messes you up, and in particular, that's morally a proof that, that, that this is not a variational problem, so, so whenever you, you solve a variational problem, the second variation is always a symmetric matrix, okay, a symmetric operator, and so, and so, that suggests it's not, and this vector field w is constructed from k and nu, so, so let me call it, say w is the, let me call it the, the interior product of nu with k, so in other words, wi is equal to k of ei nu, if I, if I chose a basis, so that's thinking of w as a one form, so, so I'll, I'll think of it as, so it's, it's really the, the vector field associated to that, to that one form, okay, and so, and so that's the operator, so it's kind of a messy operator. On the other hand, there's, it's possible, I won't go through the calculation, but it turns out the structure of these terms is such that, that you can, that, that the, the stability condition that is having a positive function with l of u less than or equal to zero implies a variational, a variational inequality of the usual form, okay, and so let me just compare the two, so, so there's some manipulation here, so, so actually when k is zero, this is just the scalar curvature of m, these other terms aren't here, and the condition that, that you have a u positive with l of u less than or equal to zero is exactly equivalent to variational stability, so that's a little exercise, so it's, it's a property of operators, self-adjoint operators of this type, so, so having a positive super solution is equivalent to the lowest Dirichlet eigenvalue being positive, being non-negative, and you can get that, there are various ways to see that, and so if you remember, so let me now compare the two, so, so in, let, let me first take k equals zero, so this is the expression we used last time, so it told us, so it was an eigenvalue condition, let me write it variationally, so I take an arbitrary function phi, and I write the integral over sigma, I had one half rm minus r sigma plus the norm of h squared times phi squared, d mu is less than or equal to the integral of gradient phi squared d mu on sigma, well for all phi, phi has to have compact support, and if sigma is non-compact, but for all phi, say, and c infinity compact support of sigma, so that, that's a condition that for each compact domain in sigma, the lowest eigenvalue of the operator l is non-negative, okay, that was what we used last time, and we, we drew the various conclusions related to two from that. Now it turns out if you, if you manipulate this, this, if you manipulate the, the condition here of stability in the general case, you produce something very similar, so, so actually this was done in a special case way back by, by, by yawing me when we did the, the space time positive energy theorem, so we did it, but only in, in a particular, in a particular case, and so Greg Galloway and I generalized, did, did the general case later, so this was really in earlier papers, but, but Galloway and I actually did it for general, and it, it's not, not difficult, so, so it comes down, okay, so, so the strategy improving it, so let me first write down the answer, so, so the result, I can make it look very similar to this one, is one half, okay, so this rm gets replaced by mu minus j, so rm is replaced by 2 times mu minus j of nu, it's a positive term, and then the scalar curvature term is the same, and the second fundamental form, again, is replaced by the null second fundamental form, this thing is less than or equal to gradient c squared on sigma, so you see it's remarkably similar, it's a, it's, it's about as good as you could hope, I mean, it's, it's sort of a perfect analogy, so the dominant energy condition implies this is greater than or equal to zero, just like in the other case, that was greater than or equal to zero, and that's what we used, and now this term here, we, we actually in the, in the calculations we did, we threw it away, so it's, again, a positive, a non-negative term, so, so it has the right sign, and, and this is also non-negative, so, so let me just say I won't go through all the details of the deriving this, but, but really, you have this l of u is less than or equal to zero, a sort of complicated thing, and the way you derive it is just to multiply by phi squared, and integrate, integrate by parts, and then you have to handle, handle the various terms that come in, and it turns out the structure of this non-self-adjoint term is exactly right, that you get the, you can, you can get this, the self-adjoint looking, I mean, it is a eigenvalue condition for a related self-adjoint operator, okay, and so in this way you, you sort of get rid of the nastiness coming from the fact that the operator is not self-adjoint, so, so, so that's, that's a key thing, and in particular, it enables us to pretty much do what we did in the, in, in, we did it in the minimal case, it enables us to do it in the Matz case, so, so let me point out two easy consequences of this, which now holds in both cases, so the first consequence, one is that if I took, if I had a compact sigma, if sigma is closed, compact, then that implies the Euler characteristic as sigma is, well, let's, let's say, I want to get rid of the, so, so let's say sigma is compact and u minus norm j is strictly positive, so there is a, kind of a borderline case, just like we, I can dwell on it, but we had, it also came into the non-negative scalar curvature, so if you stay away from, if you had, if you had, for example, a strict dominant energy condition, or if you know that this term is positive, then you get that the Euler characteristic is positive, so in particular, if sigma is orientable, it's a two-sphere, so this is, this is really, it's related to Hawking's theorem that, that a, a Matz in two mentions is, is a two-sphere, I mean, it is Hawking's theorem, really, and so, so that's the first thing to point out, and the second consequence which actually follows from the same argument that we gave last time is that, if sigma is, is, is that if I'm in an asymptotically flat space, is asymptotically flat, then there does not exist a stable Matz, which is asymptotic to a plane, and the proof is really identical. It, it again uses the Gauss-Ponnet theorem just as we did last time, because if you throw this term away and that term away, you, you just get, you get exactly the same inequality that we used to rule out this, this, the possibility of, in that case, it was an area minimizing surface which is asymptotic to a plane, and here, here it's a stable, a stable Matz, and again, there is the borderline case, so you have to, again, assume you have to stay away from the case when the terms vanish, the non-negative terms vanish identically, otherwise it isn't quite true actually, so, so that, so you need to fiddle a little bit with that, worry a little bit about the sort of borderline situation as well. Okay, and so those are, those are two important consequences which hold for both, so, so in particular on the stability side, the, the non-existent side, we're in pretty good shape, it looks like there's a very good analog between, between the two, the two situations, so that's, that's good. Now on the other hand, for the existence part, it's pretty different, and let me explain why, so let's think about one existence, that would be existence for stable Matz, okay, so, so, so actually we can sort of give a proof, because I mentioned, since somebody brought up the topology of the spaces, we can, we can use the fact that the topology is not restricted for the Matz case to, to, to say that, that, that in fact the existence theory is not what you would expect coming from the minimal case, so, so, so actually I mentioned that, that in the scalar curvature non-negative case, there are strong restrictions, so an example of a manifold that cannot occur for, in the scalar curvature non-negative case is the three torus minus a point, so if I remove a point from the three torus, so I have, I can take an asymptotically flat manifold, so that inside here, a little hard to draw the three torus, but, but this, this is, this is a picture of an asymptotically flat metric on T3 minus a point. Now, now this, on this manifold, this is my M3, this metric cannot have Rg non-negative, Rg positive, and why is that? Well, because you see minimal, minimal surfaces are, you know, like geodesics, if you take a, you know, a surface with a handle and you take any curve going around there, then you can always shrink it to a geodesic, to a shortest curve, so you can do the same thing in, in, in, for minimal surfaces in a, in a, well, in a homology class in general, sometimes in a homotopy class, but in this case, if this is a T3, then I have a hyper surface here, which is a T2, is a T2, okay, and then what I could do is I could take this T2 in there and I could just minimize area in the homology class of that T2, and then I would produce a stable minimal surface, which is homologous to this T2, so if, is it true, there exists a stable minimal surface homologous to T2, and, and that's a contradiction, because according to what we said over there, any, any stable minimal surface, this would be a compact minimal surface, has to be a two sphere, right, but this would be, so let's call this surface sigma, from over there, sigma would be homeomorphic to S2, okay, but I'm in a three torus, so, so any two sphere in a three torus bounds a ball, right, I mean, I can take the, the three torus is covered by R3, so I could lift it up to R3 and I could fill in with a ball and then I get, it's zero in homology, so, so this, this is not possible, and so, and so that, that shows that that topology cannot occur. On the other hand, according to the Mosaic-Opolic theorem, it does occur for, for the MOT, so in particular, the homology existence theorem, the ability to construct a stable MOTs in a homology class is not true, okay, and so, and so, so the existence theorem has to be quite different from, from the minimal case. Actually, when Yao and I first worked on these, these things, we, we used these minimal surface arguments in more complicated ways, and we were, we were able to completely classify the topologies which occur here. Modulo one point, which was the Poincare A conjecture, so somehow the minimal surface arguments were, we were never able to show that a possible exotic sphere does not have positive scalar curvature. I mean, it seemed like we should be able to, but we never did it. But, but anyway, after the, the Poincare A conjecture, there's a complete classification. So the question is, what are the manifolds? And so, and so, let's assume they're orientable. So the possible manifolds are closed manifolds minus a point. So, so obviously any, any asymptotically flat manifold I could just close, is, is defiomorphic to a closed manifold minus a point. And the, and the closed manifolds are the three manifolds with positive scalar curvature. And those, those are, so those are connected sums of a certain family. So there are a lot of them, but, but, but they're, they're of a classifiable type. So, so each of the MI in the connected sum is either S1 cross S2. It's ruling out, let's assume we're orientable, so we don't have S1 cross RP2. And the other possibility is S3 quotient by a, a finite group, a spherical space form. So, so you could take, you know, a lens space and you could take a connected sum with S1 cross S2. And you can do that any number of times. And those are all, those all have positive scalar curvature. And so, and so these are, these are all of the possible topologies for in the, in the scalar curvature positive case for asymptotically flat, flat manifolds. And you can see that, as I said, we did that a long time ago using minimal surface arguments. And then for that last step to get rid of the possibility of an exotic sphere, I don't know of any way to do it except using the Ricci flow, the Poincare conjecture. So, so, right. And so, and so that's a sort of interesting proof that the existence theory is quite different, that it, it doesn't, you can't simply prove the same kinds of theorems that you, you do for, for the minimal case. So in the minimal case, you, you prove these theorems by minimizing the area. So, so you have to, you look at a, the least possible area and you take a sequence of, of surfaces which approaches that and you take some kind of weak limit of those and you prove that the limit is, is nice. And so, because the Mott's problem is not variational, you, you simply, there's just no, no formulation like that. Okay. So that's kind of a bad thing. There's, there's, there's no general existence theorem of that, of that sort. But on the other hand, there is, there are, there is one, one existence theorem for Mott's and it's sort of suggested by this picture. And so, I'll, I'll talk about how this is done a little more about how it's done later. But you might expect, in fact it's quite natural in, in relativity to consider a situation where you have a region which has two boundary components. So, let's say we had a region omega and suppose the boundary of omega. So there's an inner component and an outer component. So let's say infinity is out here. Then, then the boundary components are say sigma one and sigma two surfaces. And then if, if we assume sigma one is outer trapped and sigma two is strictly untrapped, then you expect to find, then there exists a sigma contained in omega, which is a stable Mott. In other words, you expect to. So in, in physics language, when I saw this, I didn't know how to make any sense of it. But, but in, if you read physics books, you say the Mott, the Mott's is the, the, the outer boundary of the union of all outer trapped surfaces, something like that. And so, and so, and so this, this is an existence theorem. So there's a sigma in here, which is a stable Mott. Okay. And this existence theorem is, is, has been proven, although not that long ago actually is only probably within the last ten years. And so it is possible to, you know, to get limited results. Of course, the same kind of result is true in the, in the minimal case. The sigma one and sigma two are sort of barriers for the problem. They keep the, they keep sigma inside, if you like. Okay. And so, and so that's an existence theorem. And I want to focus on that because I now want to give the proof of the positive mass theorem. A proof, which is very much like the, looks very much like the, the one that we gave the other day. Let's see. So I'm going to erase this, but I'm going to come back and talk a little more about existence questions after I do this. Okay. So, so then what about the positive mass? So again, this is, we're going to give a proof that looks almost identical to the, to the, to the one we gave the other day. And so we're going to consider M3GK again. And again, the dominant energy condition. And it's asymptotically flat. Okay. And, and, and this argument actually proves a little more than the positive energy condition. It proves that the, that the, so the theorem is, the energy is greater than or equal to the norm of P. The P, E, E, E is the energy momentum, the ADM energy momentum vector. Okay. And so, and so we're going to use the same idea to prove it of, of, and so remember how we did this, the, the, the other day. We said, well, if we had a metric of non-negative or positive scalar curvature asymptotically flat, if the, if the mass were negative, then we would get this slab. Well, that's provided, we assume the metric is in conformally flat form near infinity. And, and, and again, that can be justified by an approximation argument. We would get this trap slab, this slab which is mean convex, and then we could produce a, an asymptotically planar stable minimal surface in the slab. Okay. So it's a little more intricate here, but it turns out there's, there's an exact, exactly analogous statement except with a twist, and, and I want to emphasize that because I think it has something to do with the, the polyhedral case. So, so, so let me give the proof. So, so the first, the first question is, is in order to, you know, construct that slab, we, we assume the metric was conformally flat near, near infinity. And so what's the analog of, of that conformal flatness? Well, it turns out there is an interesting analog, which works at least, I don't, I don't know how physical it is, but it works extremely well for this, for this construction. And this is something we called harmonic asymptotics. So we may assume G and K are in harmonic asymptotics. And so, and again, this is a, this is an asymptotic form which is dense in the sense you can take any asymptotics you like, anyone's for which the energy and momentum are defined, well almost any, I think, and you can approximate by, by something in, by one in harmonic asymptotics in such a way that the, the energy and the momentum change by epsilon, by very little. And so it can be justified. So, so the, it's a little more difficult. These will, these will satisfy the vacuum constraint equations outside a compact set. And so the vacuum constraint equations would say these are both zero. So, so there are, so there are four equations that are satisfied. And so, and so instead of having a single function like we had, we're going to have four functions. So we're going to have a, a function u, so on m minus k, so outside a big compact set. There exists u, which is a function, and I'll call it y, which is a vector field. So u is a positive function, defined here infinity. And harmonic asymptotics means that the metric G is the form u to the fourth times the Euclidean metric delta. And the second, actually not the second fundamental form, but this, I called this expression here pi k minus trace k times G. So, so the condition is that pi is equal to u squared times the lead derivative of y. It's really the symmetrized derivative of y, lead derivative with respect to the Euclidean metric. And then minus the divergence, just the ordinary divergence in the Euclidean metric of y times delta, the Euclidean metric. So in other words, outside a compact set, you will have four functions, a positive function and a vector field. And these two G pi will satisfy the vacuum constraint equations. Okay. And so, again, you have to, it's a theorem that you can achieve that. Actually, the proof I'm, I'm, I'm going to describe is not the original proof of the positive energy theorem we gave a long time ago. It's a proof from 2011. And it's a paper of Eichmer, Huang, Li, Dan Li and myself. So, so it's a, it's a different proof. It gives a little better, it gives a better conclusion in the sense that the originally Yao and I only directly proved that E is non, E is non-negative. And so this is a different argument. And I'm, I'm, I mention it well partly because I don't know if people know it or not, but, but also it looks exactly like the proof we gave the other day for scalar curvature, non-negative. And so the, the point of this, this asymptotics and the reason it's called harmonic is that the constraint, the, the constraint equations which are, which are the vacuum constraint equations which are satisfied to leading order reduced to the Laplace equation both for you and for the components of y. So y is, I can write it in Euclidean coordinates or my asymptotically flat coordinates and it's just made up of three functions. And so all of those functions are harmonic to leading order. And in particular the great thing about it is that the, the, the ADM energy momentum vector appears explicitly in the expansion. So just like we used for the R non-negative. So I haven't actually, I don't know whether anybody wrote down the definition of the ADM momentum, but it's again defined as a, as the limit of a surface integral involving pi and translations, translation vector fields. But if you like you could take these to be the definition of E and P. So what happens is that asymptotically u is like a harmonic function. So it's one plus, I'm, I'm not calling the mass the energy because and the components of yi just in, in the coordinates near infinity are given by minus two times the total, the ADM, the ith component of the ADM momentum divided by mod x and then plus for i equals one, two, three. Okay. And so there's the nice thing about the coordinates is that you can see the asymptotic quantities explicitly in the asymptotics of the, of the, well, of the metric in the second fundamental form. Okay. And so, so what's the, the point here? Let's see, I have till 330, right? Okay. So, yeah, I mean, so the point is so what, what we can do is we can choose, we can just rename our core, our Euclidean, our coordinates near infinity so that the, the ADM energy momentum, so the ADM momentum is, is points in the, in the x3 direction. So I can write it as zero, zero, and for technical reasons, I'll put a sign there. This is just the Euclidean length, the square root of the sum of the squares of P. And so I just choose coordinates so that that's true. And then, and then I look at the slab, like I did before, x3 equals lambda, so lambda is going to be large. So this is out near infinity and this is x3 equals minus lambda. Like that. And then you can just explicitly calculate that, that if you compute the, the, the expansion, the expansion here, you get that, so let me say it this way. So, so let's think of, let's think of moving upward. So along this, what, what happens is, as you move upward, the expansion down here, theta plus, so let's think of the upward direction as the plus direction. On this lower plane, theta plus is negative and on the upper plane, theta plus is positive. Sorry. Assume for the sake of contradiction, so, so assume, assume our inequality is violated. So assume that the energy is strictly smaller than the absolute value of P or the norm of P. Then, so when you do that calculation, you'll find that the, that the, the expansion which is h minus, or h plus trace k is, is just a, a numerical constant times, times e minus norm of P. And so what happens is that if e is strictly less, then, then this slab is trapped in the, in the sense, in the sense that if you, as you move up, the expansion is negative here and positive there. Okay, well, you can't quite apply that theorem, the theorem here, because that was originally for, for, for compact, for compact surfaces, but you can do what we did the other day. You can, you can, you know, look at a big cylinder and then you can, you can apply it in the cylinder and then you can take a limit. And so in the end, you, under this condition, you end up constructing in here a sigma which is a, which, which is a stable mass and, and asymptotically planar. So it's bounded, so it'll be. So, so in other words, exactly what we did before, what we did last time for the minimal surface, right? So you get this, you could construct this asymptotically planar, stable mods, and that again contradicts the second condition, the, the, the stability calculation. So in particular, again, by contradiction, that tells us that e is greater than or equal to, it looks like it's a metric norm. This is just the, the, the Euclidean norm of P. Yeah. And so, and so you see the proof is the same. Now, well, it's the same outline. It's, it's technically more difficult because you have to, you have to prove the, you know, you have to use this theorem and, and take a limit. You know, you use, use it on, sigma's not compact. So there's some technical work involved in doing it. We actually, in our paper, did this in higher dimensions as well. In particular, we figured out how to do this, the strictly stable condition for that. But I'm not going to go into that. I'll just stick to three dimensions here. But, so, right. So that, so that's the proof that e is bigger than or equal to norm P. Right. So then I wanted to come to this idea, this cubicle idea. So you might say, well, why don't we just, instead of doing this, why don't we get a finite theorem just by replacing this with a cube? So we, the problem is that in, in the, in the case we considered the other day, the other day, it didn't really matter which, which direction we chose the plane in. Right. No matter how, no, no matter which plane, which, which coordinate planes we looked at, we always got this, this trapping or this mean, mean convexity, mean convex slab. So we could look at, so all of the faces would be mean convex if we take a large cube. If we do the same thing here, that's not true. It doesn't appear to be true. The, the, it's only the top and the bottom faces that are, that are mean convex. And so, and so the same kind of motivation or idea for, for the cubicle comparison theorem doesn't quite work here. I mean, it's still true the metrics can formally flat, so the angles are the same. But, but the sides, the other sides don't, don't satisfy the boundary condition that you would presumably hope for in order to, you know, to, to, to find motts you would expect the boundary would have to be satisfied, trapping condition. Okay. And so, so, so as I say, the, the polyhedral case is to me is a mystery. I, I don't know whether there should be a result or not. Okay. And so I, I have 10 minutes left. Are there any questions? I'm going to change gear slightly here. Yeah. Say it again then. The limit will be, yeah, I mean, the motts condition is just a point wise condition, right? It says h plus the expansion zero. Curvature, mean curvature or the, I take a derivative. I don't understand the question. So this will be a stable motts. So, so it, it will satisfy that eigenvalue condition that I had written there. The stability condition will tell me that, that on sigma, I, I get some integral inequality involving the Gauss curvature of sigma and the gradient of phi squared. And then I can make the same argument using Gauss Bonet. Yeah. So, right. So you do need, you need a compactness theorem. Yeah, maybe that's what you're getting at. Yeah, that's right. That's right. So again, again, it turns out when you study this analytically, the motts equation is the mean curvature and then there's this other term. The other term is lower order. And so you can, you can prove similar kinds of compactness terms, curvature estimates as, as you could for the minimal case. But you're right. You have to, that's also a little harder for the, for the motts case. But, but it's, but there are analogous theorems. Yeah. Yeah. I mean, so if you do it in very high dimensions, then the motts will have the same problem as for minimal. In, in fact, the, the tangent cone will be a minimal cone. The k term disappears in the, in the limit. So, so it's like a prescribed mean curvature problem, except with a little more complicated because of course the, the k does depend on the tangent plane of the surface. You, you restrict to the tangent plane. But yeah, it's still a, this, the basic theory works. Actually, the, the, the motts, the motts that are constructed by, I haven't given, told you how they're constructed, they actually satisfy an approximate minimizing condition. And, and so in fact you can get the same sort of local regularity theory as for area minimizing, area minimizing hypersurfaces. Yeah. Yeah. So again, in dimension bigger than set, yeah. So up to, up to dimension seven, this, there's an extension of this argument. Our paper just, we only stated it up to dimension seven. So, so, but, but on the other hand, yeah, I mean, I think, yeah. So I, I think the pro, the problem is that in this, yeah, it doesn't immediately follow from, from my paper, my recent paper with Yau and high dimensions, because the sigma that you take, so the way the argument goes is this sigma in higher dimensions, there's this conformal argument and sigma then will be a, a, an initial data set with, with negative mass, right? And so, and so the sigma itself may have a small singular set. And in our paper, we assumed we're starting in a smooth manifold. And so, and so I guess you would have to, yeah, it would require some modification. I haven't thought that far ahead, but I, but yeah, it doesn't, it doesn't, you can't get it immediately. Yeah. Oh, okay. So, yeah, so actually in the rigidity case is a bit of a long story. So there, there, there was, yeah, actually, so there, there, so actually PO has, has done rigidity using spinner arguments, I think. There's a recent following this paper, Huang and Li gave, gave the, a rigidity proof along, along sort of a deformation line. So in the scalar curvature case, you can go quite easily from the weak positive mass theorem to the strong one by doing a metric deformation. Okay. And that's somewhat harder to do in this space time setting, but, but they were able to do it. So there's a recent paper of Huang and Li, which, which does that. But our paper only did the weak inequality that we didn't prove. We didn't show that the, that equality holds only in the trivial case. So the trivial case would mean that the, that the initial data set is arises from a hypersurface in Minkowski space. Right. And so, so yeah, that's, that's a separate argument. I don't know how to, yeah, I don't think this argument could get the rigidity case because there are a couple of things. First of all, there was the, the approximation to put the, to put the, the initial data set in harmonic asymptotic. So that involves changing the energy and linear momentum by an arbitrarily small amount. And then in this case, too, there's, there's all, there's always this issue in the stability inequality that the terms that are non-negative might actually be zero and you might get, so you have to deal with that, too. So, so you can't directly get the rigidity result from this, this argument. Okay, so I have five minutes left and I have about 30 minutes of material I wanted to do. So, maybe I just say, say in words, there's, so in a way the, the, the existence theory for the, the mobs is, is, is inferior to the existence theory for the minimal case. But there's a sense in which it's actually more interesting because, because in, in, in the analysis that, that Yao and I did a long time ago in, in constructing sort of mobs type solutions, we looked at an associated equation which was a graphical equation called the Zhang equation. And we showed that, that the, the, the Zhang equation has a solution, assuming you have, you control things near the boundary. It, it, it has a solution. Well, the only way it can fail to have a solution is if you have mobs in the interior. So, the interesting thing about the existence theory, which, which is very different from the, the minimal case, the interesting thing about it is, is the existence theory can actually detect mobs in the initial data. So, so you can't get mobs from topological hypotheses, but you can sometimes get them from geometric conditions. So, in other words, you can prove sort of hoop type theorems using this. And I was going to talk about a couple of them. There are two that I know of. Yao and I did one, and there's one that was discussed by Doug Erdley. But you can show in some cases that if the geometry of the initial data set is sufficiently complicated, that it must contain a mobs. So, you can prove sort of quantitative theorems, which tell you that, that under certain local conditions on the, on the initial data, it necessarily contains a mobs. And so the space time will be singular. So, it's a kind of a sort of blow up theorem. And, and it follows, it comes from the method of construction of these, these mobs, which unfortunately I don't really have time to go into. Although I did start a little late, right? When did I start? Okay, I had five minutes. Okay. So, so let me, let me just write down the, the equation. So, so okay, you can ask, you can ask, well, we didn't actually start out this, this project trying to construct mobs. It, it, it turned out that the, the, the equation we were looking at actually became a tool. In fact, the only tool that I know of for constructing the mobs. And, and this was an equation which was written down by a student of Geroche named Jung. And he was also trying to do, trying to give an argument for the space time positive, positive energy or mass theorem. But of a some, somewhat different nature. And I don't think his, his approach actually worked. But, but the interesting thing he did and what really sort of made us zoom in on it is that, is that he gave, so one of the hard points about the, the space time positive mass theorem is that the borderline case is, is sort of complicated. So in the, in the scalar curvature case, the mass is zero only if the manifold is isometric to Rn. In the space time case, the, the energy is zero or, or the mass is zero is supposed to be if and only if the initial data is in bed, can be embedded as a hypersurface in Minkowski space. So there, you know, there's an infinite, infinite parameter family of, you can take, you know, tons of hypersurfaces in Minkowski space. So, so the, the borderline case is a lot more complicated. So one of the things that any proof you give has to do is it has to give a characterization of, of space like hypersurfaces in Minkowski space. And, and Jong's paper explicitly did that. And so he used this, this idea to do that. And so that's one of the reasons we looked at it in, in detail. And so, and so the, the equation, so, so the idea you can describe the idea geometrically in the following way. So, so you, you say, well, so we have our M, G, and K. And then what we're going to do is extend to one dimension higher. We're going to take M cross R. So this is the original M. Let's call the variable here T. And then we're going to extend just in a trivial way, the metric G plus dt squared. And we're going to extend K in a trivial way also. Let me continue to call it K. But K is extended so that K in the d dt direction for anything is zero. So d dt is a null direction. And it's parallel in the del d dt K is zero. So in other words, you just extend it parallel and you take the components of K which, which contain the vertical direction to be zero. Okay. And now what you can do is, is, if you take a, and so you can think of that as, you can think of this as a sort of new higher dimensional initial data set. And what you can observe, so you could again look at the Matz equation here. So you could, you could require that the mean curvature be mean curvature plus the trace of K is zero. Right. And so you can get some obvious solutions to the Matz equation, namely you can take a Matz down here, sigma in M. And then you can take the cylinder over it, like that. So that's a, that's a Matz in the, in the bigger space. And if the Matz is strictly stable, you can actually show that it's, it's, it's an asymptotic limit of a graph. So there's a graph here, t equals f of x. And the graph is a Matz and it approaches asymptotically, it might be minus infinity, you have to get the, again, the signs of the mean curvature, right? But, but you can approach asymptotically by a graph. And the Matz equation for this graph is the Jung equation. So the Jung equation explicitly is, let me write it, g i j minus f i f j over one plus norm d f squared times d i j of f divided by square root is equal to, so now the sign reverses because this is minus the mean curvature. So this is equal to g i j matrix times k i j. Okay, that, so that, that's the Matz equation for, for a graph. So, so this part is, this is just the mean curvature, or actually, as I've taken is minus the mean curvature of the graph. And this part is just the restriction. This is the trace of k taken along the graph, restricted to the graph. And so, and so that's just the, the Matz equation. And so, and so what we did is developed an existence theory for this, for this equation, we sort of understood the, you know, how to construct solutions and, and we understood that the blow up occurs only on, on Matz. And so, and so in particular, you can use this to, to, to give an existence theorem for Matz. You, you, you can, in some cases, you can show, like as I mentioned in these, sort of under your concentration conditions, you can show that the solution, that there can be no solution of the Matz, and that guarantees that there will be no solution of the Zhang equation. That will guarantee there's a Matz in this, at least which intersects this region that you're, you're looking at. And you can also, with some work, prove this kind of theorem using that you can construct a solution of the Zhang equation, which is, like minus infinity there, plus infinity there, and so you can make it blow up along, along, you can produce that, that Matz inside there. So, so, so in a way, the existence theory is more interesting because it also gives these, these, these sort of hoop type theorems, which I don't have time to talk about, but, but, but yeah, so that's the idea. So it's not variational at all. It, it, it comes from solving this graphical problem and, and studying the blow-ups, forcing solutions, forcing solutions to blow up or, yeah, under various conditions. So whenever you force a solution to blow up, we prove it always blows up along, along a Matz. Well, it could go to plus infinity or minus infinity. So that corresponds to either a, just the sign of the trace k term. So it could be a, a future or a past Matz, I guess. It blows up there. Okay. And so that's, that's kind of the story. So, so I think the polyhedral case, again, I don't know whether there is an example of that, but, but some things work nicely, but they're, they're definitely much more elaborate, much more difficult for the, for the space time case than for the non-negative scalar curvature set case. So anyway, I'm finished and thank you all for listening. And it was nice to meet you. And I wish you all success in your, in your career. And I hope to see you again at some research conferences five years from now. Of course, I'll be a really old guy by then, but, but anyway, enjoy the rest of the meeting next week.