 Okay well thank you for all being here. Okay so let me begin. I'm going to talk more about solutions with the same symmetry SOP cross SOQ symmetry in Rn. I talked about self-similar solutions and I have a summary here of what I explained last time. So these are the conclusions from last time. We're looking at solutions hypersurfaces that you get by taking a curve in the first quadrant and rotating it around one axis and also the other axis in a certain number of dimensions. And when you do that you get a certain hypersurface. If this curve is the graph of a function that hypersurface evolves by mean curvature flow if the function satisfies this equation with lambda is zero so without those terms. So mean curvature flow is that equation without the term lambda is zero. Then in studying singularities you can do a parabolic rescaling and study expanding mean curvature flow or shrinking mean curvature flow. So before the singularity you study shrinking mean curvature flow and the substitution is this one. So let me write both of them. So for shrinking so for t less than zero you assume your solution. This is a solution to mean curvature flow. You assume it is of the form squared minus t times n. So if you let n be fixed then you have a self-shrinker. If you allow n to depend on time then it is good to choose this as your time dependency. So and then for t positive if you're looking for solutions after the singularity you could again try to write them like this and if you write them like that they are self-similar. You get expanding solutions and so you can look for things that are not self-similar by allowing this n to depend on time. Okay and so these two quantities will both if you call that quantity tau then mean curvature flow for m t in both cases is equivalent with an evolution equation. Autonomous evolution equation for n as a function of tau and that evolution equation is this one. Velocity equals mean curvature plus lambda times x dot nu where for lambdas plus one half you get expanding mean curvature flow for lambdas minus one half you get shrinking mean curvature flow. What are x and nu here? Here's my hyper surface n. X is the position vector of any point on the surface and nu is the outward unit normal at that point. Okay and so if your hyper surface is of the form that special symmetric form this equation is equivalent with that equation so these are for for surfaces with symmetry these are equivalent equations and this one provides more insight but if you want to do calculations this thing is more useful. Okay so then the the theorem about expanders is you can look for self-similar solutions to this equation so these are these would be here I should have written tau so these would be solutions stationary solutions to this equation so self-similar solutions shrinkers or expanders are solutions to this equation where you replace the right hand side left right the left hand side by zero either here or here so for expanders you said this equal to zero and you said lambdas equal to plus one half you get an ODE and the theorem the analysis that I showed you has this conclusion that if you started a certain shooting height here a then there's exist there's any there's always a unique local solution and for in the case of expanders the solution always extends it is asymptotic to a cone with slope a capital A and so that's what this says and then as you let a go to zero you can analyze what the slope of that cone is by dividing the region into three pieces of which I showed you only the first two last time but the last one is kind of simple and the conclusion is that for small values of a because near the origin the solution starts behave like that minimal surface that was due to Alencar and that oscillates around the stationary cone that analysis show that this opening slope has this expansion it's one slope of the limiting of the stationary cone plus some constant times this that goes to zero times something that oscillates so it's a dam that exponential oscillation as a function of log a so if I could make a movie here on the spot and if I could take this point and drag it down what you would see is that as I drag this point down the asymptotic slope of that thing starts it just oscillates like this up and down and every time it crosses the stationary the stationary cone it picks up one extra intersection so in the limit you get a solution that has many many intersections and then it goes off and is asymptotic to some cone and the smaller you choose a the closer it is to the stationary car okay then for shrinkers which you get the same equation and you just flip change the sign of this lambda you get a different a similar but still different conclusion so instead of having so for most values of a that you choose here if you draw the solution you will find something that goes close to this and then does that and so it's not does not give you something that is asymptotic to a cone but there are and then if you choose another value you'll get something that is some similar to that and then it does this right it turns over but in the other direction in between those two values there must have been a transition from one to the other and that turns out to be a value of a at which the the thing is stationary to a cone so what you can prove is that there's a sequence of a's converging to zero such that if you start at that height you get self shrinker that is asymptotic to a cone so I'll write them as you minus here and the asymptotic slope of this cone satisfies the same equation plus something that is little all of this term so the only difference is that for shrinkers you only have a discrete sequence for expanders you have them for all a okay so this leads to so this leads to an example of non uniqueness from smooth initial data for mean curvature flow so how do so I said that very quickly last time so let me go through that argument for again so here in this picture I have plotted the asymptotic slope of the self shrinker or expander versus the parameter a the neck you could call this you could call this the width of the neck if you imagine that it's only being rotated around this axis so for shrinkers there's only a discrete sequence of a's for which you get self similar solutions so and for expanders there's there's a solution for each a and the asymptotic slope is a function that for which and I didn't write this here but if a goes to infinity so here here a is infinity when a goes to infinity so if you start up here somewhere you'll get a solution that just goes out like that and the slope will become larger and larger as you choose a larger okay so this starts up here it crosses this line the horizontal line where a is big a is one and near near this point we have this expansion so we know it crosses infinitely often and converges to one okay so let's pick so in this picture so and I don't know the numerical values of these a's there is an expansion so from this computation you can also get an expansion for large values of n what the a's are and they go to zero exponentially in that and they're quite small numbers so if you just start if you were doing numerics and you were solving the ODE numerically and just plotting solutions here the the pictures would not look like this at all they would so I've exaggerated the oscillations the pictures if you have this the thing would look like that so I think numerically I would expect that it takes some skill to to detect these things numerically okay so but let's say that so if you choose so let's say I picked the second number here and imagine that that it's over here and that it is in the second oscillation of this pink curve so then we know that there is a self-shrinker and there is a self-shrinker which is asymptotic to a particular cone and the slope of this cone is this height then I can read off just by drawing this horizontal line I can read off them in this picture there are one two three other three expanders that have that are asymptotic to that same cone so I can now pick any of these three expanders so what happens this is my self-shrinker so let's say this is t is zero what what is the evolution of this self-shrinker it shrinks to a cone like that and it is the cone with exactly that it's the cone with exactly that slope and now you have three choices you can you choose your you choose any of the three expanders and your evolution for positive time is square plus t times and one and two or in three right so there are three ways of there are three three forward evolutions coming out of this same con so that's an example of of a solution to mean curvature flow that starts out being smooth forms a singularity and then after that there are three different ways of continuing it and so if it didn't work with a two then you just increase n and if you choose a large n over here the aperture the of the of the cone that you get of the shrinker cone will be so close to one that there are many many many expanders with that same aperture and therefore you can have there will be as many forward continuations as you like finitely many and so one general comment so I wasn't here last week but I think many talks were would have been about mean curvature flow for mean convex surfaces where H is positive and so none of this happens so there has been an enormous amount of you know beautiful results in the for the case of mean convex mean curvature flow and there's a very strong theory very good compactness results in particular the fattening so this was already known by Evans and Spruck and at that time so Takis explained that for mean convex flow there are the fattening this kind of example does not happen so the viscosity solution is is in this case the viscosity solution forward evolution would be a set with an interior because it would have to contain so there it's still just a hyper surface afterwards it's it's the union of this this and this and everything in between and then there are also others coming out in this direction so these non uniqueness results in the examples so the further examples that I'm going to show you today you should think of these as examples of what can happen if you drop the hypothesis that your mean convex and so the theory is not as nice although I like the examples that come out so yeah right so a very natural question is these examples that I gave you are none of them are compact so I and of course that allows me to make some allows me to make the simplifications just looking at self-similar solutions so now you could say can I make does it really depend does this this property does it really depend on the on the compactness condition or is it really a local phenomenon if the if if I start with an initial condition that is like this and that forms a singularity here would the same thing happen and I so the answer there is yes so but it is not yes in this yeah so you should be able to take so for me encourage your flow how would you even in this context of symmetric surfaces I take something like this and then I cap it off like that okay and now it's a compact surface if you wrote this this around the axis it's a compact surface you could figure out its topology let's not if if this part is large enough and this is far away large enough and this part is also close close enough to forming a singularity and it looks like that thing then you should be able to adjust the number of parameters here and cons and show that it does the same thing as this and it is so this is the procedure that you would use a similar to things that Velazquez has done and also there's a paper on the non-generic the degenerate neck pinch for me encourage a flow that I worked out with him and I looking at also he has an example of type 2 blow-up in for me encourage your flow so the techniques that you would use are very similar to to those and I'm very convinced that it would work having said that no one has written it up okay so so far we have non uniqueness you start smooth there are three or ten or a million but a finite number of forward continuations but different and the examples what I want to tell you this hour I want to show you that actually the number the set of forward evolutions is it's not it's not finite it's it's infinite and in fact it's it's uncountable in fact you can it has a topology and it contains sets of arbitrarily large dimension okay so how do we get that and so the ideas is the following I'm going to just going to go back to this equation the forward evolutions that I described here are self-similar and the idea is well what if you look at non-self-similar solutions how many are there then and so if you look at those the answer is you find many more so I just I want to show you how those come about and so to make the story short because it's Friday I'm going to look at non-self-similar solutions and so from now on we're only concerned about what happens after the singularity so let's say we have we've been in the we've been put in the position of having a cone and we want to know what forward evolutions there are either because we had a self-similar solution that shrank or some four-dimensional being produced an oil drop in a four-dimensional water tank and with electricity made a cone like this what forward evolutions are there so from now on we're not going to think about t less than zero it's a Wisconsin thing to do every state in the U.S. has its own motto and the motto of the state of Wisconsin is forward always forward so that's what we're going to do now so I yeah again because it's Friday I said p equal to 2 and q equal to 2 and this so for which dimensions does this work let me keep those here all the calculations with the coefficients that I'm writing is for these and it works if pq is greater than or equal to 2 and p plus q is less than or equal to 7 yes yes so yes yes so the Simon's cone is stable but they're so there do exist expanders for other cones so so the Simon's cone has a particular aperture if you change so you can still solve those ODE's and take a shooting height the one thing that you don't see if if the dimension is eight or more and you do you're not in the CMOS particular case is that the aperture is a monotone function of the shooting height so for each expander there will be for each aperture there will be a unique expander so there will be only one way forward and for the Simon's cone the the forward evolution is is just the thing is the Simon's cone itself yeah so we're in these dimensions today okay so what we want is I want a solution for mean curvature flow for t for some sort tiny time interval after zero okay so I'll I'll substitute okay so if you do this substitution then what we really want is a solution and tau to expanding mean curvature flow and so how does the for for which time interval do we want this well tau is log t so as t goes to zero tau goes to minus infinity as t goes if t becomes equal to epsilon okay so this epsilon is not so important what what this shows is that we want the solution to expanding mean curvature flow to this equation where I've said lambda equal to plus one half so we the lambda is gone this one is a minus originally I had written sorry I made a mistake I wrote this and so this plus should have been a minus unless you are new skin colding millis mini-cosy and many many many other people by now pretty much everybody except me then you would have chosen the opposite sign convention for the mean curvature and then this one has to know this stays the same anyway okay so there we want a solution to this equation that has existed for all tau going back to minus infinity okay so now you can do two things you can try to study all these you could say suppose I have a solution and then study its properties and that's that's a very valid thing to do but it's not what we're trying to do here we don't we don't have a solution we want one so we can let's let's let's look so we can impose further conditions and as if there are simplifying conditions try to look for solutions that have that satisfies those conditions so the symmetry will be one of them let's assume that n tau converges to n star as tau goes to minus infinity so let's assume that these n tau's have an initial value at tau's minus infinity so what will n star be you could look at either so and I'm going to assume all the time that that the hyper surface is of the form some graph of why is u of x and tau rotated around these two axes although part of the part of the story doesn't require that assumption but you so you're welcome to think about either this equation or that equation for now so if you have a solution that converges as time goes to minus infinity it has to converge to a stationary solution and a stationary solution is an expander so it has to be one of the expanders that we found okay so suppose that we that our n star is an expander and so it's one of these and let me not include the a in the notation all the time so it's y is u of x and that capital U is a solution of the od that you get by setting this equal to zero okay so it's one of these things and in particular let's assume that if we have an initial value here so if to make sure that the solution we're going to construct it'll it'll start out being in so after this transformation as you as you let t go to zero what you get is the asymptotic cone of this thing that will have to be that will have to be this call we'll remember this okay so it is asymptotic to the cone and at some point I explained that these expanders they all have asymptotic expansions as X goes to infinity so the next two is that they're actually it's it's it's it converges to the cone fairly like what order one of one over X there's no constant term okay so we're going to assume that and now suppose suppose that n tau is given by the graph of a function y of ux tau then then what we're trying to solve so and at this point I don't need the expander theorem or the shrinker theorem or this expansion anymore and this picture is also number not relevant anymore but this equation is what we're trying to find is a solution a solution to this equation with initial condition and we're trying to solve this on some time interval and I I'm just I'll we'll solve up to up to some time town on okay so we don't care how far it goes as long as it goes all the way back to minus infinity so this is what we're trying to solve so one way so how one way to look at this what kind of problem is this there so there are a lot of terms here on the right what we have is u prime is f of u so I'm going to use this notation quite a bit this this f is an abbreviation for all those terms on the left on the right this is an autonomous differential equation I'm going to think about this as if it were an ODE instead of a PDE I'll just think of you as sitting in some space some infinite dimensional space and then it's an ODE on that space and it defines a flow it defines a flow on the space of functions why is u of x and then so all this is terribly vague and it needs to be made precise which is what in the end makes it a long story all right so this this thing is expanding the curvature flow now and this space I'll give this space a name we'll call it x our shrinker sorry our expander u of x the one our chosen expander is is a fixed point for this flow it's a stationary solution because it satisfies the equation with zero on this side okay and what we're trying to find is a solution that at time minus infinity starts at the fixed point and then moves away from it so in ODE theory here's here's a fixed point u this is a cartoon of the space x we want a solution that starts here and moves away okay and there could be other solutions that start somewhere and move towards it there could be more solutions that move away and the way to find solutions that so in other words in terms of in terminology of ordinary differential equations we want the unstable manifold we want to prove that there are orbits of the flow solutions of the differential equation on the unstable manifold of u and the standard way of finding the unstable manifold of a fixed point of a nonlinear system of equations is to linearize at that fixed point look at the eigenvalues and find positive eigenvalues so that's it got the story really short and say that's what we're doing and that's and that works right so the summary of the whole story is that you can carry out that plan and so what so so fee much smaller than you let's say norm be really small and then fee tau is equal to well it's equal to f of u plus fee which is equal to f of u plus the derivative of f at u times fee plus something of order fee squared and then this of course is zero because it's a fixed point so what we get is the linearized equation is this okay so we solve this linearized equation and we need to find so this is a differential operator which you get by differentiating this thing at you and so if you look at this thing there are some unpleasant terms but actually only two of them are nonlinear these three terms are linear so linearizing isn't yeah it's a short calculation but it's given that it's even not as long as you might think okay so it is phi xx and now you differentiate so there's a something times phi x so that gives you that too comes out so this thing gets multiplied with phi x comes from differentiating this then there's another one here and another one there so there are two more terms multiplying phi x over x minus x over 2 plus x over 2 did I put in the beginning this sign was also wrong I'm sorry so the I'm going to reveal my mistakes one at a time so the h minus one half x you had a minus this one also needs to be a minus okay so since I'm not going to do any detailed calculations this won't hurt us times phi x and then stuff multiplying you there only these two depend on you this becomes plus one over u squared so that's that's our linear differential operator it's it's a large formula it is however note that these so it is of the form a x okay and there's lots of theory for differential operators of this type in particular so the difficulties in studying this differential equation are so there are two of them one is there's a singular coefficient that x is zero this one over x that's it's the same singularity as the Laplacian and radial coordinates so it's not it's not a problem and then we're on an unbounded interval that always introduces some complications means you can't use the very standard theorems and one of the coefficients is unbounded this one okay so one has to deal with so this coefficient is unbounded all the other coefficients as x goes to infinity ux converges to the slope of the converges to a so this just goes to a constant these things go to zero that goes to non-zero so this goes away so at and this goes to zero so at infinity really it's just this x over two that is important at infinity ux goes to infinity so this goes away you never gets close to zero because it is you is one fixed expander that we've chosen it starts here it starts at a and never gets below a so there are no other nasty terms in this in this operator the the only thing that you have to worry about is is this thing okay so the to-do list to make this work is we have to use the unstable manifold theorem so we have to find a version of the unstable manifold theorem that finds in this setting or find it you know adapt whatever we have to the best you know the nearest looking version of the unstable manifold theorem that we can find why is df of u so study so as a linear operator in particular find its eigenvalues okay so if you're not going to use the unstable manifold theorem then you could try to prove it by hand and just take as a sort of exercise in homogenization take the proof of the unstable manifold theorem and mix it into the other part of your proof and stir it and then you get one big long proof but I like to separate these things so there was a paper in the 1970s by Erwin who gave a function it gave a an implicit function theorem proof of the unstable manifold theorem on bound spaces so and then there were many versions that follow this improvements the theorem that we that I'm going to use is if you have a map from a banner space and f is c1 f of 0 is 0 df of 0 the spectrum of df of 0 is disjoint from the right so the spectrum of it's all complete the spectrum of a linear operator is all complex numbers such that df 0 minus lambda is not invertible so if you assume that df that there are no eigenvalues on the unit circle so there's part of the spectrum in here and there's part of the spectrum out there right and in finite dimensions these are just isolated points in infinite dimensions these can be they're close subsets and that's all you can say in general so for our operator it's going to turn out that the thing is self adjoint it's going to be a sequence of points accumulating at 0 so in our case df of 0 is going to be a compact operator then the set of points in x such that there exist x minus 1 x minus 2 x minus 3 in x with x minus j image of that is x minus j plus 1 in other words what in a picture maybe it's better if I draw a picture so here's the fixed point the origin here's the origin so a point is on the unstable set if if there is an orbit coming out of this thing so there's an orbit of points so this is x here's x minus 1 here's x minus 2 here's x minus 3 so you should think of this as f inverse of x except I don't want to assume that f is invertible so instead of instead of saying that I'm going to say that this is f of x minus 1 and this is f of x minus 2 note that the hypothesis of the theorem allows 0 to be an eigenvalue these converges to 0 that's that's 0 okay so such a point x sits in the unstable set and the theorem says that this unstable set is a here 0 is a smooth sub manifold and you can specify tangent space let me not do that it's tangent space is the eigen is the invariant subspace of x corresponding to this this part piece of the spectrum so in particular if you count the number of eigenvalues outside the unit circle if that's finite then that number gives you the dimension of this set and if all this seems very abstract you can say that you can just you can summarize it that the the unstable manifold theorem in finite dimensions that you're used to holds in this setting okay so we have to so note that it this is not the unstable manifold for flows but it is the unstable manifold theorem for maps it is much easier to prove that unstable manifold theorem for maps rather than for flows because there are many flows so maps can come about in many different ways once you write down a flow you have to prove so you write down the differential equation and then you want to prove that it has that it defines a map so then you have to solve the initial value problem this version of the theorem just avoids all that discussion if you have so we're going to apply this to the time one map of this differential equation and this theorem says you have no questions asked how you got your map this applies okay so what we have to do is okay so yeah so that's our unstable manifold theorem okay so apply to the time one map of this differential equation so now we have to prove that that time one map exists so we have to prove a short time existence theorem for extended mean curvature flow on first first side that sounds like a lot of work because there are a lot of terms and then you have a happy realization namely wait this thing is equivalent to mean curvature flow and we're talking about smooth initial surfaces and there exists there is exist there is short time existence for mean curvature flow so we can just use that and then transform it back via this right dissimilarity transformation that we had okay so that's a happy realization it is followed by an unhappy realization namely if you write this out what do we have to prove we have to prove that this map is differentiable fresh and differentiable from some bound of space to another bound of space and now you have to prove something about mean curvature flow and if you start if you start to unravel that it becomes a suddenly it becomes difficult again so it turns out that this realization actually doesn't help so it's better to just start with this and say I have this PDE or in general so so parts of what I'm saying really don't use the symmetry at all it would work for this for the general non-symmetric case as well if you had a non-symmetric expander you start with this PDE you prove the short time existence theorem and you your proof has to be so good that the solution depends c1 on the initial initial data so you have to find a bound of space x such that the solution depends smoothly in the norm of that bound of space on the initial data then you can apply Irwin's theorem so it being Friday I'm going to skip all that so the key ingredient is to use some old semi-group theory so you prove the d-feed this thing this linear operator generates an analytic semi-group as studied by the elder Sinistrari if I'm not wrong on and at which space do we space do we choose and this so here I'm skipping a whole bunch of details these functions phi correspond those are the so in this picture this is our given expander so that one is fixed if you add phi you get something that is going to be close to it it should go to zero as X goes to infinity because we want to perturb things still to be asymptotic to the same cone so phi has to be zero at infinity it turns out that you have to and you can assume that it goes to zero really fast exponentially fast Gaussian exponentially fast and it's all because of this term so let me not work that out so you take continuous functions on r plus appropriate constant here and that constant K is going to depend on the asymptotic slope a that we have here it is if you do the more general version if you take this then so one problem is a I'm perturbing the solution is this is the solution and I have a I have vertical line so I'm perturbing it upward geometrically the geometrically the more natural thing would be to perturb it in the normal direction to the surface okay and if you're in a non-symmetric case and you're using this equation you would have to use the normal perturbation and when you do that then everything becomes a little bit more complicated but other things simplify in particular this way here always becomes either the minus x squared over 8 so now it changes because we've rotated things by some unpredictable angle okay so it generates an analytic semi-group and now there is abstract theory and so there are many names associated to this there's Aman Prato Grifar Nardi actually quite a few Italians I think Sinistrari also the senior this general theory implies that the analysis taken care of so as long as long as soon as we so it boils down to proving that this operator generates an analytic semi-group in a suitable suitably weighted space once you've done that short time existence in that space with differentiable dependence on initial data follows from not automatically but from stuff that's been done and it is it's not just C1 it is it's real analytic because everything in the equation is real analytic so you can expand you can expand the solution in a Taylor series in the in the initial perturbation in the perturbation of the initial value it's a very nice theory okay so all this means that we could we could have skipped all this and just said we're we do formal calculations like an applied mathematician would and and we wouldn't be wrong so the important thing now is to do is to calculate the spectrum of this operator yeah so one more thing is it generates an analytic semi-group on this thing and it is so there's a certain there's certain compactness so DF is itself is not a compact operator it's inverses compact so it's it's one of those things the term the technical term is it has compact resolvent this means that it's it's I it's it's eigenvalues form an unbounded sequence and it is self a joint in some weighted L2 space which I won't write down because the weight is not very nice so and again I won't derive this given the time of the weekend day but it follows from it follows from this expression so and this plus this is a plus note that we are well let me not know that okay so formally symmetric that means that the eigenvalues are all going to be real and it's also bounded from below so there will be a few positive eigenvalues and then most eigenvalues will be negative so so the conclusion of all this is a theorem so if the if DF zero has at least n positive eigenvalue values then along the eigenvectors in those for corresponding to those n positive eigenvalues there will be an unstable set with okay in particular if zero so and if zero is not an eigenvalue so if you can prove that then you can just count all the positive eigenvalues and Irwin's so this is a different version of the unstable manifold this would be the fast unstable manifold so if you know the DF zero that zero is not an eigenvalue then then the unstable set is a manifold and its dimension is just the number of positive eigenvalues that you can find okay so in this case so for each solution so okay so this implies since each of these solutions in the unstable manifold it's a solution n tau of expanding mean curvature flow you can multiply with square root t and replace tau by log t it gives you a solution to ordinary mean curvature flow and at which at t is zero starts at this cone y is ax all these things start at this particular cone so we found an n-dimensional family of such solutions right so the dimension of the family of solutions that comes out of the thing is at least n okay so does this ever happen can we calculate how do we calculate the number of eigenvalues okay so it what we have to do is set this equal to lambda phi and solve it with boundary conditions okay so it's not a pleasant ordinary differential equation fortunately it's a second order differential equation and not going to use this equation anymore so this can go right so the trick to finding to estimating the number of positive eigenvalues of a solution of this kind of differential operator is to use this term comparison theorems if you can find so let me write a theorem so this is sort of a theorem it's a fake theorem it's so it's sometimes true sometimes not like that he said so it is so can be proved in this in this context so theorem if if ax phi xx plus bx phi x plus cx phi is 0 has a solution so what are the boundary conditions that we're imposing at 0 the derivative has to be 0 so we have norm and boundary conditions at this at 0 and so if if you have a solution of the linear differential equation that satisfies the boundary condition on one side with and if the solution has at least n zeros then then this operator has n positive eigenvalues all right so this expression is our linearization df at you apply to phi okay so we don't have to solve the whole differential equation we only have to solve not for each lambda for each eigenvalue we only have to find a solution to this equation and in these situations there is not a theorem but it's sort of folklore there is always if you have a family of solutions there's always one solution to that equation that you can get and it's so in differential geometry they're called Jacobi fields in PDE is you differentiate with respect to a a particular solution and I'm writing it like this and it's a little bit disingenuous because f was defined on that bound of space x and the which has boundary conditions at infinity and the fee that we're going to get will not be zero at infinity it'll actually be very large at infinity but that's okay I this calculation only is meant in a formal way and here this theorem does not have any boundary conditions at infinity so the name of this kind of theorem is a storm oscillation so it's it's very old certainly reveal proper theorems okay so how do you do that well we have we have a whole family of solutions of the nonlinear equation differentiate with respect to a and apply the chain rule okay so the the fee that we're going to get is will be the derivative of the solution as I change this height a here so how does that solution change well if you're somewhere up here we don't have an explicit formula for the solution so it's not something we can compute easily or at all numerically you could but if you're down here then at least in this corner we know what the solution is so for small a we can calculate we can we can calculate an approximation for this derivative at least near the origin so and if we find some zeros there that gives us a lower bound for the number of zeros okay so it's roughly a times m of x over a it's I'll encourage minimal surface shrank down to the origin and if you differentiate this with respect to a so are we allowed to differentiate this asymptotic expansion turns out yes okay so is this ever zero and if you think about it a little bit then what is this okay so I'm thinking calculus does the following so said z is x over a this is m z minus z bless you it is up to minus z squared up to a factor it's the derivative of m divided by z so anytime that derivative has a zero we have a zero for our eigenfunction so better drawing of the allen-kar solution would be this so what is m over x if you pick a point on the graph it's the slope of this line so anytime that slope has a maximum or a minimum you have a zero for the eigenfunction okay so you can draw all of them so there's one here there's one there there's one here there's one there right anytime so those those are all zeros for this function this thing is a solution to the homogeneous equation it has so it has infinitely many zeros it does not that was too quick this thing has infinitely many zeros but most of those zeros are there at infinity and this approximation is only valid in a large region how large is that region well it depends on how small you took a okay the smaller you take a the larger that region is so for very small a the conclusion is that for very small a this thing will have many many zeros okay so as a goes to zero the number of positive eigenvalues of this thing goes to infinity okay so so the answer is so in this statement this statement applies and the closer our cone is to the stationary cone the larger we could choose and so in particular if you were to choose at bless you again so at the stationary cone yeah you could choose any end so the the not the simons cone because that one is in dimension eight but the analogs of those things the stationary cones in dimensions four five six and seven they have the family the dimension of the set of solutions for me curvature flow smooth solutions of me curvature flow coming out of that thing is infinite okay so then more minutes let me so this is in dimensions four five six and seven let's go back down to dimension three because I want to so you can do this for three dimensions as well the one thing that I don't know how to do is how to form a smooth solution that forms a singularity that then becomes that then it has a wide enough cone to have non-unique continuation so back to our three-dimensional space if you have a cone like this what I want to do is I want to discuss what these solutions look like that to come out of a cone the non-self-similar solutions because one of them looks kind of strange or does does something perhaps unexpected okay so if you remember there is a there is this angle alpha there exist expanders so again we have the same theorem for any a here there is an expander and it's asymptotic to some cone and if you choose this angle alpha large enough then there will be at the always two of those so if alpha is large enough there is one expander that does this and there will be another one that does this and so numerically 66 degrees is enough theoretically if alpha is large enough so bigger than 90 degrees minus epsilon for some unknown epsilon then there are two expanders and now you could apply this whole same story to these expanders linearize at one of these expanders count the number of positive eigenvalues of the linearization and conclude that there exist that many solutions that you know the family with that dimension of solutions coming out of that expander and hence after blowing down coming out of this column so the top one turns out to have so the number of positive eigenvalues let me call that the index it's so very often it's called the Morse index so the top one turns out to have index zero always this argument that I did here with looking for small values of a and look counting the number of zeros of this thing works again the difference is that is that this picture is different it's a picture that we all know so for surfaces of rotation in R3 what is the minimal surface starting here it's a catnoid okay how many zeros does this thing have well in this picture there's one and no more so we will never get index higher than one but we get index one okay so the index is one it may be that in between here there are solutions that have higher index but I can't prove it and the numeric suggests that it's not true but I wouldn't know how to prove that either okay so this thing has index one that means that the dimension the family of solutions as comes out of it is is one dimensional in particular so one dimensional so there's some redundancy right are they all different so you can produce a one minute if you have a solution to mean curvature flow you can produce a one dimensional family by translating in in time or by scaling it parabolically that gives you typically different solutions but they are equivalent under scaling right so it's a that's a non it's sort of a trivial variation so the one dimensional family that we're getting here these things will be all be equivalent under rescaling so what they look like is this but the corresponding eigen function turns out to be positive so there are two things that can happen so here's your expander and let's look at the equation for for n tau right so let's look at expanding mean curvature flow first what you can do is you can take this thing it's unstable I poke it and push it up a little bit okay and then what it does it's unstable it just moves upward monotonically it moves upward monotonically it does not have it's not mean convex it monotonically means that h minus one half x dot nu is greater than or equal to zero because that's the velocity for being expanding mean curvature flow if you're going to study singularities for this flow and you're in a bounded region then this will be bounded and then if you zoom in on things it'll go to zero since this might be good enough to to use all the mean convex theory and apply it to the solution that we get here okay so that's a suggestion not a not a not a fact okay so there's one solution that just goes up like that and so if you now blow this down to mean curvature flow okay so we say mt is square root t times n log t okay so this thing convert goes upward where does it go well the other solution is up here that's the other one this is the index zero one this is the index one solution so it goes up and an infinite time now it it converges to that one so it is a connecting orbit for the expanding extended mean curvature flow between between one fixed point and another fixed point so if you rescale this what is this what does this look like it's a cone that where the neck thickens it at first slowly and then it changes its mind and it speeds up and then in at the end it starts to expand as if it were this one that's what you would see and you might have to look close to see that happen actually so it's not a qualitative change the other solution is the one that goes down so what does that do right if you perturb this thing a little bit you push it down then because it's unstable it'll keep on going down and now again it is monotone but it'll be going down and what does it do in infinite time well it would have to if it didn't become singular it would have to converge to another expander and there is no other expander of this type because this was the lowest one with one that we have so that means that in finite time now it forms a singularity here and then here you have a neck pinch at this so this neck pinch you could you could analyze as if it were a mean convex neck pinch so you would have to take the who's consider a climber hustle hole for theory and see if if this term is small enough to make everything still work so this thing will separate and it's listen to two so if you take this solution what does this thing look like and mean courage or flow you have a cone so let me draw the stills of the movie it decides that it wants to stay connected so it thickens like that it forms a neck then at some point it changes its mind and starts going down again and the neck pinches and then after that it separates after all so this is kind of so it's this is a forward evolution of a cone where the angle is more than 66 degrees one final comment and then we go for coffee yeah so the initial condition here is a cone so it's invariant under scaling so you can parabolic rescale space and time and you will get other solutions so if for me courage or flow this thing happens at a certain time t then you can rescale it and you can get a similar solution that forms at at any time that you want you can rescale it and you can make the singularity happen at any time that you want which shows that there is no there's no simple lower estimate for how long it takes a solution to become singular once you've constructed a smooth solution right so this there's one initial value and there are you can't predict how long the solution will stay singular because the singularity could happen at any time okay so I'll stop there so the next hour if you're still up for it I'll talk about ancient ancient ovals and go for coffee