 Tako je s tem knukovamo za max, kdaj buri nekaj nekaj pridane. Če imamo nek Najma edit pro max, naredam prijeznizo nekaj pridane, da je zelo, da je lepo taj pridane, There is also something important that has to be said about the manifold itself, that is we have the following result. We have not only comparison for different functions on our hyper surface, but also for different hyper surface. Tukaj imamo izgledan result, kaj je izgledan prinsipel izgledan. in zero in zero in a n plus one hypersurfaces. Smooth, for the sake of simplicity closed, smooth closed. If they are disjoint at the initial time, then they stay disjoint. I denote by mt the evolution of the mean curvature flow, by mean curvature flow of m0 and the same for n. So, mt and t are the corresponding evolutions at mt. For t greater than zero, let me specify such that mt and t exist. We'll speak about smooth evolutions, so for t up to the first singular time of the two manifolds. But this principle holds in great generality, so any reasonable definition of weak solution and any good solution also satisfy the same principle. So we have already seen by maximum principle argument a special case of this result when one of the two is a sphere. So we have seen that if m0 is an arbitrary surface and then zero is a sphere inside, for instance, then the two evolutions remain distinct, or also if n0 is a sphere outside. But the result says that the same holds for any pair of hypersurface. Or if you have one hypersurface inside the other, then the evolution preserves this inclusion, but also holds if they are just in closed region or disjoint. Also in this case, the evolution will stay distinct. And this is a very useful principle because in many cases you can say something about the behavior of a given hypersurface by using barriers with something which has a known evolution. So how can one prove this result? I will not give all the details, but there are basically two ways of proving this. So one way is, one says, suppose that the result is not true, then there is a, sorry, if they are disjoint at initial time, then they stay disjoint all following times. If at later time they are not disjoint, you can argue the other way around. If they are not disjoint at a later time, then they are not disjoint at all previous times. It's not true that if they are not disjoint at initial time, they can become disjoint afterwards. This is possible. So if they start with an intersection, this intersection may either persist or be lost, but it propagates in the past. So disjointness propagates forward and having an intersection propagates in the past. And so one way is to say, suppose that we have a first time where they touch, then we have a picture like this. If you have empty and empty, they continue in some way, they may touch at one point or at more points, but at any point where they touch, since it's the first time that they touch, they have to touch tangentially, then they have common normal, then one can have a common tangent plane. So one can locally write both hypersurfaces as graphs over a given plane, hyperplane, and then write locally the evolution of the main curvature as a graph, as you saw in the previous lectures. It is a parabolic equation, and you can argue that you have a bounded domain where two solutions of the same parabolic equation have a touch for the first time in a point in the interior of the domain, which is impossible by the strong maximum principle, and therefore this cannot happen. But there is another way to prove this result, which is probably more interesting, which is the following. One can say that actually, not only the state is joined, but we can be more precise and say that the distance increases. The distance between the two hypersurfaces is increasing. And how can we see this? Again, I just give the intuitive part of the argument. That is, let us draw our two hypersurfaces and P at a certain time. Let us consider a pair of points where the distance is attained. So, take one point, there is at least one pair of points such that you have P and Q, so P on MT and Q on NT, such that the distance between the two hypersurfaces is equal to the distance of these two points. Let me call G the immersion, which gives the hypersurface N. Then I claim that the distance between these two points has a non-negative derivative, so these two points are not becoming closer. So, there are standard arguments that show that if you consider the evolution of a minimum of many functions like this, if at the point where the minimum is attained, if at any point where the minimum is attained, the derivative has a certain sign, then also the infimum has a derivative, possibly weak derivative with the same sign, so it has the same monotonicity. So, how can we study the derivative of this distance? Well, since these two points minimize the distance, it is easy to see that the segment joining them must coincide, must be parallel to the normals at both points. So, the normal at these two points must coincide. This means that the normal at P must be equal to minus the normal at Q, plus or minus, depending on the orientation, but the product of the mean curvature and the normal is independent of the orientation. So, this vector divided by its norm. So, then what's the derivative of this? Well, it is simpler to consider the derivative of a square. So, the derivative of times f pT minus g qT, then we have the mean curvature pT, so minus h pT times nu pT, plus h qT, but so this vector, this vector, and this vector are all parallel, and so let's choose the orientation as in the picture. This means that this is equal to, we have, let's say this is, with this orientation, this is a multiple of f p minus f q is a positive multiple of minus n p, so let's, the norm is the distance itself. Then we have, this is minus n p, and also this is minus n p, so this is minus pT. Well, maybe it's more clear if I change the orientation here, let me call this the normal of q, so that they coincide because the argument becomes clearer, so we have minus curvature in q, so we have this two minus sign that become one, nu times nu is one, so this is two times the distance times the difference of the mean curvatures, which is intuitively clear because every point is moving in normal direction by the mean curvature, so the distance evolves like the mean curvature here times the mean curvature here, minus the mean curvature here, and now what is easy to see from the fact that these two points minimize distance is that they have either to turn in opposite direction, that is, in this case, this would be positive, this would be negative, then this would clearly be positive, or if also, otherwise they could also turn a bend in the same way, but we could also have n of t doing like this, but the curvature here has to be smaller than the curvature here because if the curvature is larger, then this mean that we could find a nearby point, which is closer to m than this is, so this is greater than zero by minimization property of these two points, greater than or equal to zero, so this is greater than or equal to zero, and so we see that at this point that the distance increases, so the same holds for the global distance of the two hypersurface, so we have this avoidance principle, which is actually not just forming curvature flow, it just requires that the speed is monotone in the curvatures, which is the usual parabolicity condition for this flow, so also for the kind of flow Totidas kalopulos is speaking in her talks, and by similar argument, one can also prove that if m zero is embedded, then it remains embedded. Roughly speaking, embedded means without self-intersections, so if you start without self-intersections, in curvature flow cannot develop new self-intersections, intuitively speaking, different parts, so the singularities can only be local, the nearby points can develop a corner, a cusp, but you cannot have faraway points that come together because we would have behavior similar to two different hypersurfaces coming together, which is excluded. Let me make a comment about this, since you have seen interesting examples of in the Tony Gower's lectures of flows with intersections that have peculiar behavior, there are two possible ways of considering objects with intersections, depending if you interpret them in curvature flow as an evolution of emergence or of an evolution of sets. We have, let's say for simplicity, a curve with a self-intersection. If you have the first point of view that you consider the evolution of the parameterization, then this is not a singularity. This point is just, I mean, the two times that you pass through this point are independent from each other. They don't see each other, so you don't realize that in the evolution that here you don't have a local homeomorphism on the image. One point will smoothly evolve in this way. The other point will smoothly evolve in this way. You just counted this intersection as two distinct points of a revolving object which at that time occupy the same position, and evolve independently. Then in this context these results just says that if you don't have something like this at the beginning, then you don't have it also at later times. But you can also have the point of view that you don't regard this as a parameterization, but of a subset over n or the boundary of a subset over n. Then in this case this counts as a singularity and then it is non-trivial to describe what is going on. Then you have this phenomenon that you have heard of in the other lectures. So don't be confused by this. It's two different ways of looking at the same object. OK. This was the avoidance principle. The next thing I want to tell you about is something I've already mentioned that is a version of the maximum principle for tensors, so not just for real-valued functions on our manifold, but typically it applies to functions which take place in some bundle of bilinear forms or linear operators on the tangent bundle. So typically we will see the first application will be to show that convexity is preserved by the mean curvature flow. Convexity is equivalent to the positive or non-negativity of the second fundamental form. So second fundamental form is a bilinear form on the tangent space. So we have to use a principle which ensures the preservation of positivity on time-dependent bilinear forms on our manifold, evolving by suitable pt. And there was some independent result in the PDE community, especially in the context of reaction diffusion equations. There are results about invariant regions for reaction diffusion equations for parabolic systems of PDE which are somehow related. But in this form suitable for geometric evolution the maximum principle is due to Hamilton. For the way I'm going to write it now it's contained in the first paper by Hamilton on the Ricci flow. And there is a second form of the maximum principle contained in a later paper today just about this first one which just concerns invariance of positivity. So suppose that m is manifold with Riemannian metric g of t which depends on time. So Riemannian manifold with time-dependent metric. And suppose that on this manifold so t goes in some interval 0 capital T and you have a bilinear form defined on the tangent space of m for each point and time. You have some mij of pt is a bilinear form any point on space and time on the manifold which satisfies an evolution equation the time derivative is equal to the Laplacian. Laplacian of course corresponding to this time-dependent metric plus some first-order term let's say some ak pt k derivative of mij and then we have some reaction term we have some bij where so ak is given vector field bij is a function of the metric and of m maybe m is not best choice of letter because it's very similar to the m of the manifold let me write it more calligraphic so let me maybe let me write it component-wise so it's clear that so this is a function of smooth function of our bilinear form and of the metric typically is a polynomial is obtained by taking products of the bilinear form with itself and the traces with the metric we will see some example afterwards then we want to give the definition which ensures that the positive definiteness of mij is invariant in time and what's the condition for the scalar case if m would be a function then the positivity would be invariant if the reaction term is non-negative when its argument is zero but if m is zero which in this case means no longer strictly positive definite but just semi-definite then this is positive now we have bilinear form so we have many possible directions then the right condition is this one it's what Hamilton called the null eigenvector condition so if the i is null eigenvector for m is equal to zero then the reaction term must be positive or zero evaluated when the quadratic form associated to be ij has to be non-negative on this vector the notation here is a bit there is a small abuser notation so in this case I mean an arbitrary matrix so not our specific evolution so if we take a matrix so this is a function of matrices so if we take a matrix and a zero eigenvalue of this matrix then if we evaluate bilinear in this matrix we obtain something that gives a non-negative result when you evaluate the quadratic form on this eigenvalue so you may restrict yourself to the possible image of m if you know that m is taking values in some subset you can impose the condition only on the subset well then the conclusion is so under the above assumption if mij is non-negative definite times zero then let me write it positive then there are also versions with non-negative then at equal to zero then mij is strictly positive for all positive times all the above assumptions so the symmetry somehow included in the statement because we are but probably not I guess I mean in order to speak about mijj since we are talking about positivity then somehow we are semetric and maybe the so the bijj but I mean if the bijj is in the equation so all other terms are semetric so I think bijj should also be symmetric for construction so I think it's not it's it only makes sense for semetric of course one could make other versions one could state it for mijj operator from the tangent space to itself and then one could speak about the positivity of the associated biliner form however let's prove it in this way and for simplicity I will make the proof under a stronger assumption that we have a strict inequality here on the null eigenvalue but this is standard in the proofs of the maximum principle to pass from this case to the case of the equality by suitable perturbation of the mijj so it's really just a technical point which is very standard so how do we argue again we argue by contradiction so suppose the assertion is false so this is not positive definite for all times that there is a first time where it is not positive definite by continuity it has to be positive semi definite so this means that there exist some point p star and time p star such that mij is has a null eigenvector v v i p star t star but is strictly positive for t less than t star and by continuity semi definite at t equal t star then the idea is to reduce the situation to a scalar one we want to focus on this direction v i the null eigenvector we know that there exist at this point but it is convenient to continue this vector all over the manifold in some way so define v i as a vector field on so for all points and all times we only need t star we can do it in such a way that at the point we are interested in p star t star the derivatives the covariant derivatives of v i are 0 and also the time derivative we cannot have it so it is not trivial that we can have it everywhere but it is easy to see that we can have it at a given point we can continue it at the point where we are looking the derivatives are 0 and then we reduce our situation to a scalar one by using this function which is just define f of p t equal to the m i j of p t evaluated in this vector v i v j, v i v j also depend on p t and we want to see what is the equation satisfied by f we can do the derivatives so in general we have to differentiate all three terms but if we only look what is happening at this point we only get this first the term where this one is derived because these other derivatives are 0 at this point the same happens for the first derivatives then we have the Laplace the Laplace is a bit bit more tricky because we have basically to do two derivatives which can fall on these three factors in all possible combinations then we surely have the term where both derivatives are falling here then we have to take care when we have the gradient of v i v j we don't know if it is 0 at the nearby points so all terms with one derivative on either of the two is 0 at our point but there is one term which is actually two but they are equal by symmetry so we need different argument the terms where two derivatives both fall on one of the two so Laplace v i times v j but this is 0 because we have m i j v j so this is v j is an allagan vector so this is basically when we compute the derivative at p star t star the derivatives only fall here and so by the equation satisfied by m i j we conclude that df dt is equal to Laplace f plus ak dkf plus we have what we call v i v j v i v j ok, then we know that this is strictly positive by the null eigenvector condition so in general could be not equal to 0 but for simplicity I am assuming that it is strictly positive and the other terms you have the usual arguments of the maximum principle you have that the function f is 0 at this point but is greater than or equal to 0 for t less than t star and any p so this means that this point the gradient is 0 so it is in space it is a local minimum so this is 0 this is greater than or equal to 0 and the time derivative instead is less than or equal to 0 because it is positive at previous times and 0 there so this is less than or equal to 0 and then you have a contradiction something non-positive is equal to the sum of something non-negative plus something strictly positive this is a contradiction and this concludes the simplified proof of this principle assuming the strict inequality and so let's see an application so I hope in the remaining time to give a sketch of the proof of the classical theorem on the behavior of convex hyper surfaces the theorem that who is approved in the 80s that convex hyper surfaces converge to a round point the way it is something said that shrink to a point but up to rescaling converges to a sphere the original proof is is quite more complicated than the one I am going to sketch here but the one I will sketch here will use what you have heard yesterday with the monotonicity formula and the possible limits of rescaling and some other argument that have been found also later times but the first properties that were already the starting point in Wisconsin's original paper are some invariance properties so h, i, j let's say strictly positive for simplicity on m0 let me just mention in this Hamilton's maximum principle if you start with something non strictly positive then something just positive semi definite the conclusion is that it stays at least a semi definite and there are also some suitable version of a strong maximum principle some rigidity of the possibilities where you have persistence of the semi definiteness that can be ruled out in some cases so it can be shown that all I am going to say would still hold if we have non strict convexity on the initial manifold provided it is compact it is equivalent to say that m0 is convex so convex meaning it is the boundary of the convex region in Rn then what can we say the evolution of h, i, j is given by so let me is this one you have seen it in the first lecture so we have something like that fits in the picture of Hamilton's maximum principle we have seen you have time derivative Laplace you have no first order term and this would be the b, i, j you see it is a function of the second fundamental form and the metric and both h and a square come from the second fundamental form and the metric by suitable products and contraction so the question is does it satisfy the null and vector condition well suppose so if h, i, j, v, j is equal to zero then when you compute this it is clearly also equal to zero second term becomes a square times zero and this first term also you have an m instead of an i but result is again zero it is immediate u so this b, i, j contains as a factor h, i, j so if it is zero and h, i, j it is also zero on b, i, j so it satisfies the null and vector with equality this means h, i, j is preserved by the flow that is if the initial hypersurface is convex then the all the later evolution gives a convex hypersurface as I mentioned in contrast to what's called the Mincovexity that is the positivity of the Mincovature Mincovexity is invariant under Mincovature flow also in a general Riemannian manifold this results instead exploits Euclidean ambient space because you have some extra terms in a Riemannian manifold which would not satisfy in general the null and vector condition there is a related property which is very useful in the analysis of the Mincovature flow of convex hypersurface that is the so-called preservation of pinching pinching curvature pinching is an expression that appears in many contexts in Riemannian geometry meaning that the eigenvalues of some curvature operator depending on the context are positive and are sufficiently close to each other in the sense that a ratio between the largest and the smallest cannot be arbitrarily large but it's bounded by some constant of course on a compact well this means ratio lambda n over lambda 1 if I call lambda n largest principle curvature lambda 1 the smallest one of course if all curvatures are positive then this object is bounded on the initial hypersurface but the question is as time evolves especially when we reach a singularity will this remain bounded and the question is yes so what we see is the preservation of lambda 1 greater than epsilon h is preserved is invariant this is very similar to this lambda 1 if we have this this is equal to the sum of the eigenvalues again we can say that this is greater than or equal to plus epsilon lambda n so it implies that lambda 1 lambda n over lambda 1 is less than or equal to I think something like 1 minus is less than or equal to epsilon so this gives a uniform bound on the ratio of the largest and the smallest so in this sense a condition like this is called a pinching condition so how can we show that it is invariant it can define a symmetric linear form which is positive if and only if this is satisfied so define yeah yeah yeah we are assuming that all curvatures are positive no no different meanings in the sense of the neck is more pinching in the sense is shrinking and in this case pinching in the sense of stay together I pinch the curvatures in the sense that become arbitrarily far away from each other different different meanings of pinching then one can just define this tensor hij minus so this is no negative if and only if if and only if lambda i is greater than epsilon h for all i which is just this so the to say that the smallest is greater than or equal to and you can check maybe not to the computation but it's easy to check that by a similar argument you can use Hamilton's maximum principle you since using the evolution equations for this, for this and for this using the fact that jij has 0 special derivatives it has only non-zero time derivative you find easily the equation satisfied by this, you see that it's equal to time derivative minus laplacian then some reaction term which again is easily seen that so if you have lambda 1 equal to epsilon h at some point so you have an eigenvector satisfying where the second fundamental form is gives epsilon h the same eigenvector you see that the reaction term vanishes at the null eigenvectors of this tensor so Hamilton maximum principle implies mij greater than 0 is invariant this is typical feature also of other geometric flows of the rich flow that some positivity conditions of the curvature are invariant under the flow which is a very useful property for studying the behavior and then I will state a property of convex set which was found by menendrius so let me first define if if you have a omega in rm plus 1 convex and convex compact rm is the boundary of the set so like our evolving ever surface then I call rho plus of omega outer radius which is the radius of the smallest ball containing omega I consider analogously that inner radius has the radius of the largest ball inside omega so they could be we can regard them as we can extend these definitions to m we call the rho plus and rho minus also the outer and inner radius of m then there is this result that relates the pinching of the curvatures to the pinching of the radii there is this theorem by menendrius I think it's 94 saying that so for any let's say c1 in 0,1 there exists some c2 greater than 0 such that if m equal the boundary of the omega is a convex hyper surface is such that lambda n lambda 1 over lambda n is greater than c1 at any point every p in m then we also have abound on the ratio of these so I can also say c2 in 0,1 then rho minus of m over rho plus of m is greater than c2 ok so somehow this means that if we can control this as we know that we can as I have just shown you then we can also control the ratio of rho minus and rho plus so intuitively speaking both quantities measure how much we deviate from a sphere on a sphere we have this identically equal to 1 so if c1 the closer c1 is to 1 the closer we are to something spherical and also if rho minus is equal to rho plus then of course you are on a sphere so of course this is just an intuitive justification so the corollary is if mT is a convex evolving by mean curvature flow implies that rho plus of rho minus of mT plus over rho plus of mT is greater than some c2 independent of time so it shrinks but it cannot become more and more eccentric so the ratio becomes bounded and then by comparison with spheres ok so let's let us for the moment remember this fact so the now we have yes so rho minus is you consider that the ball which stay inside the hypersurface so in the region close by the hypersurface and you take the largest possible ball which is inside the hypersurface rho plus is the radius of a ball which encloses which stays outside and the smallest possible radius ok then let me explain the strategy of what we are doing we as was mentioned yesterday we want to show that the flow the only singularity that it can have is of type 1 so the we know that t is the singular let us call singular time t t is surely finite because we have said the other time anything compact is by comparison with the contracting sphere has to become singular in finite time so the and we also know that a square so that is the maximum of a square tends to infinity as t goes to t but we don't know so I've already told you everything is shrinking to a point but a priori we don't know so we wonder maybe the surface develops a corner at some time or maybe the curvature can can become infinite but maybe the whole surface collapses to something flat but which is not a point well this estimate we have just proved rules out this because in this case you have rho minus going to zero and rho plus going to something positive so that ratio would would not satisfy that estimate but we have to extrude this and to extrude this there is a trick which was introduced by Kaising Chow for the study of the Gauss curvature flow but applies to with great generality to geometric flows to hyper surface flows which involves the support function so you fix some t0 less than capital T and you know that since you have not reached a singular time then you have something smooth so rho minus is greater than zero then by convexity let define x0 the center of the ball radius rho minus t0 is maintained in our hyper surface and then by convexity we know that p t0 minus y0 scalar the normal is greater than or equal to rho minus of t0 what does this mean and take the scalar product of this vector and this vector the normal this means that it is basically the distance of the tangent plane to this point to this center but since the convex set and for the sphere all lie on the same side of the tangent plane then you have this inequality then the idea of this trick is to consider this function equal to the quotient of the of the mean curvature so the speed of our flow divided by this object here minus one half of this value and one observation is since our flow is contracting if this sphere is included in our hyper surface is enclosed at time t0 then this is also true at previous time so also for t less than t0 so this denominator is never 0 for t less than t0 actually the denominator is comparable with rho minus and well I am running out of time but maybe I just just catch you the how it goes on then maybe I will give you more details at the same time but one can use the maximum principle to obtain a bound on upper bound on w so if you compute the evolution equation of w you see that it has a nice reaction term which becomes negative for large values of w so this means that w cannot become too large so since the our estimate so this somehow cannot be much larger than rho plus this expression this cannot be this is comparable to the diameter which is comparable to rho plus so the denominator is between somehow rho minus and rho plus which are comparable to each other so w is more or less like h over rho minus so this shows that h is bounded above if rho minus is bounded below which means that as long as rho minus has not gone to zero the curvature is bounded on a convex many for have a surface this is the sum of the principle curvatures so if the sum of n positive numbers is bounded above then each of this number is bounded above so this means that you have no singularity if rho minus does not tend to zero the conclusion is that rho minus tends to zero as t goes to the singular time but since the ratio of rho minus and rho plus is also going to is bounded then also rho plus has to tend to zero so this proves that you are shrinking to a point but well I don't have so my time is basically over but using the comparison with the sphere you know that rho minus and rho plus cannot be both bigger or both smaller so you know that the hypersurface is shrinking to a point then you can compare with the round solution at the same time you have your manifold your hypersurface you choose the radius of the sphere so that they shrink at the same point at the same time then you know that the sphere cannot be completely inside or completely outside because otherwise this would contradict that they shrink at the same time so since the two have to intersect for all times this means that the radius of the sphere must be between these two this means that the rho minus and rho plus have the same rate as in a shrinking sphere and then the estimate on w we had before also yields that h so the curvature blow up at the same rate as the sphere so it implies that you have type 1 since you have type 1 of something with positive mean curvature you know from what Wiscan told yesterday that the possible limits if you rescale you find a smooth limit by rescaling which is either a sphere or a cylinder average longer curve times something flat but you know that you have curvature pinching you know that lambda 1 and lambda n have fixed ratio so you cannot have that in the limit you have one flat direction because that would violate pinching so the only possible limit is the only one of these objects which has no flat direction so the sphere so since you have pinching that rescaled empty converge to a sphere so this is a sketch sorry I didn't give the details of this step which is an important part of this argument but I hope I have given you the main ideas of this approach to this result so I thank you for your attention