 Okay, so I'm going to spend the week talking about various things related to mostly mean curvature flow. I'm going to, so today's lecture will be ODE methods. There will be a lot of ODE's during the week. So in mean curvature flow there is, there's a lot of general theory, which is for a large group of people the goal is to take a smooth initial surface and to flow to see what kind of singularities it forms, to do something with those singularities to flow on and maybe prove topological theorems. Other applications with mean curvature flow come from mathematical physics. You have some sort of model and you're trying to construct a more robust theory where the initial surface can be smooth or it doesn't have to be smooth and there will be singularities but you want something, you want to have a nice way of constructing generalized solutions that pass through these singularities. And so there are various theories, so Takis was talking about those this morning. What I, much of what I'm going to do will be constructing examples of solutions, examples of what singularities can look like, if you have formed a singularity, how to continue the flow in some sort of smooth way, what kind of things can you run into. And so I will not be producing general theorems but a lot of examples and many of these examples are easier. It is, so in a sense if you've ever talked to a number theorist or an algebraist and you're an analyst, after a while you will get a sense of jealousy because they have a problem and they say from us problem, which was a very hard problem. But is it true, well you just do examples. You calculate the first, you test it for the first thousand integers, the first million integers. So you can do all these computations and they give you an idea of whether something might be true or not. With something like mean curvature flow, all these geometric flows, you cannot just try the first ten surfaces. That doesn't exist. So constructing examples is a more intensive, more labor intensive enterprise. And so what I want to show you this week are a lot of examples. So first let me begin with, so a lot of the examples will be elementary. There's going to be a fair amount of calculus. I'll try to keep the amount of the number of actual calculations and computations and formulas to minimum but they will show up. So let me start with, begin with mean curvature flow for graphs. And let me start with the simplest version for curves in the plane. So I have a curve, I have a curve in the plane and I want to know what equation, so it's the graph of a function of, and so time is not on the picture of course. What equation should this thing satisfy for this curve to move by mean curvature flow which in the plane is curve shortening. So mean curvature flow is the velocity is equal to the mean curvature which for curves in the plane is just the curvature of the curve. Now we have from calculus, we have formulas. What is the curvature of the curve? It is the second derivative and it's that formula with that strange three halves in it. So that's the curvature of the curve at this particular point. So that point actually looks like an inflection point, so the curvature there should be zero. And now what is the normal velocity? Well, if this thing is moving a little later, it'll be there. Then the normal velocity would be this, but if you differentiate u with respect to time, you get that. So the normal velocity is the time derivative of u, but then you have to look at this little triangle that has the slope of the graph in it. And so the udt has to be equal to, so that's the equation for mean curvature flow for curves. So then you can cancel this against that. So that's the PDE that solutions to curve shortening satisfy if they are graphs of functions or so in higher dimensions, there's a similar calculation where I'll do the calculation, I'll write the. So if a graph of a function is moving by mean curvature flow, then mean curvature flow is equivalent with an equation like this thing. And so it's a parabolic equation, the udt is derivative show up and then there are coefficients and the important thing for what I'm going to say today is that these coefficients, there's a formula for them. They only depend on the gradient of the function u. Okay, so now unlike the equation that Takes wrote for the level set flow, this equation as long as ux is bounded is non-degenerate. It's a non-degenerate parabolic equation and there are off the shelf theorems that you can use to conclude short time existence and if you can bound the gradient, you get long time existence. And there is, so there is a theorem of Ecker and Huskin. So I'll write the equation like that. So this PDE has a unique solution for arbitrary elliptic initial functions. And the solution exists for all positive time. It's elliptic constant is bounded. And then they prove further theorems. You can reduce the condition of Lipschitz continuous to being arbitrarily continuous. And there's some interest that the function doesn't have to be uniformly continuous and it doesn't need to require any growth conditions at infinity. Which you should contrast with the standard linear heat equation where you can start the linear heat equation with really big initial data. But they do have to be bounded by some e to the a x squared at infinity. So for mean curvature flow, there is no such growth condition. You can take an arbitrary continuous function. It can grow arbitrarily fast at infinity. All that is required is continuity. Okay, so a consequence of this is so much of what I, so my topic today is ODE methods for studying expanding and shrinking solutions to mean curvature flow. The Ecker-Husskin theorem, so a consequence of this is, so consider the following situation. Suppose the graph of our initial condition is a column, okay? So in other words, U0 is homogeneous of degree 1, okay? Then the solution has to be self-similar. And why is this? Well, so for two reasons, one, U is a solution. This implies that lambda times U, sorry, one over lambda. So this is the usual parabolic rescaling. You're allowed to trade one space for two times. So for all positive lambda. And so for the graph equation it's, so for either that equation or this equation if you like, it's a matter of just substituting. The important thing here is that this is not the formula for these coefficients. The thing is that these coefficients only depend on the gradient of U. And if you take the gradient of this function, that lambda comes out and we'll cancel that lambda. So rescaling here does not change the gradient, okay? And then you have here one time derivative is equal to two space derivatives. And it's the usual parabolic scaling, okay? So that's also a solution. On the other hand, if your solution has that cone as initial value, then my rescaled solution, its initial value is one over lambda U not lambda X. Our rescaled solution has the same initial value. And then the uniqueness part here is important, okay? So the uniqueness tells me that the solution that we get satisfies that for all positive lambda, which is the definition of being self-similar, okay? And now this means, so this is an example. It's an example of an expanding self-similar solution. And it's not one example, it's a really large class of examples for each for each Lipschitz function whose graph is a cone for each homogenes of degree. One function, you get one of these things, okay? So in particular, if I let recurve shortening, if I take a V as initial condition, then there exists a self-similar expanding solution. This one is called, was found by Brachy. So it's in his book on verifold solutions of all the stuff in this book. This is the most trivial thing in the book, but it's one of the pictures in the back, so there's a self-similar solution that comes out of this, okay? So the forward evolution of this cone is self-similar, and it's given by a particular, so the graph of this function, it's a solution to an ODE, there's no explicit formula for it. You should think of this as an analog to the self-similar solutions to the heat equation. This solution, so to sort of continue what Takis was talking about. If you're thinking about evolving in forward time, a pair of crossing lines, then what can you do? You can say, well, this thing, it's the union of this and that, and those should evolve in forward time by just expanding outwards in a self-similar way, or you could say it's the union of these two. Then in forward time, it should expand in this way. Or you could say they're both self-similar lines and they have curves you're zero, so they don't move and they sit like that. That's another solution. Or you could say it's a network, so there will be talks about this week. You could say it's a network of curves, so these are smooth curves that come together and the boundary condition, so each smooth part evolves by curve shortening. The boundary condition at the points where they meet is that these have to be equal angles, so this looks more like it, that one is a bit too small. So there's another evolution forward from this as a network. There are at least two of them and the viscosity solution is the most robust thing, it contains all of them. And in this particular case, the boundary of the viscosity solution is everything contained inside these smooth, self-expanding solutions. So all possible other solutions that you might imagine have to lie inside there, and the question is, what is there inside there? If we translate. Yes. Yes, not uniformly. No, no, it's right. Okay, so another example in this vein is, suppose that you start with an actual cone, how does this evolve forward in time? And so using this construction, we can come up with at least one way of evolving this forward in time, namely, I think of this as the union of two Lipschitz graphs, where you have to look at it this way. This part is, it's the graph of a Lipschitz function and there's going to be some self-similar way of evolving this forward given by this construction, and so there's one way of evolving this forward, so this thing expands out self-simility in that direction. So at least there's at least one forward evolution, namely, this thing splits into two parts and you get two caps and each are moving independently of the other forward in time in a self-similar way. You may ask for the cross, the two crossing lines, we had more than one way of evolving it forward in time, namely, you could decide how to split it up, can you do the same thing here? And so here there's an asymmetry between the inside, what is inside the cone and what is outside the cone, those are topologically different things, so if I connect things up this way, I would get something, you would get something that looks like a one-sheeted hyperboloid, whereas the evolution, forward evolution that we have now looks like a two-sheeted hyperboloid, so the question is could there be exist a one-sheeted hyperboloid evolving forward in time? And here, so let's look at mean curvature flow for surfaces of revolution, so we'll simplify the problem drastically by looking at symmetric surfaces, so I take the graph of a function and I swing it around the x-axis and then I get a surface in R3, so the question now is if that surface is to move by mean curvature flow, what equation does our function, you have to satisfy and here let me, so you have to do a little geometry of surfaces, what is the mean curvature of this surface of revolution? In general the mean curvature is, so the mean curvature is not the average of the, it is the average of the principle curvatures except if you work on mean curvature flow then it's just the sum of the principle curvatures, so there are two principle curvatures, one is, so this surface let's say you're at this point and again I, for some reason I picked the inflection point, there's, let me pick this point, there's one curvature that is this, so there's a principle curvature direction that is parallel to the axis of rotation and this is, so that's k1, it's just uxx over this 1 plus ux squared, three-halves and then there's another principle curvature direction, namely if you go, so the other one has to be perpendicular to this, so there's a principle curvature in that direction and so the radius of curvature would be this, except that's not perpendicular to the surface, the radius of curvature has to be measured perpendicular to the surface, so the radius of curvature would be this thing, so it's this, the radius of curvature is this, is this, the radius is u, but you have to project it onto this line, so it becomes longer, sorry you have to project it onto this line, it becomes shorter, so you have to do this and now you have to be careful with the signs, if I count, the curvature is going down, I'll count those as being positive, for example if you have a sphere, we want all the curvatures to be positive, then that means that this k1 has to be positive when the second derivative is negative and now this is the opposite curvature, sorry, so for me a sphere with the outward unit normal has negative curvature and I'm sort of unique in the literature with that convention, so we get a minus sign here, that's the mean curvature and then what is the normal velocity, the normal velocity is the same as from the previous picture, the normal velocity v is just ut over that square root 1 plus ux squared and now if you set those two things equal to each other, you find, okay now if you have higher dimensional surfaces this equation, so in other words if you take a one-dimensional axis and you rotate in more dimensions around this so that this circle now becomes an n minus 1 sphere, you get the same equation except this coefficient 1 becomes an n minus 1, but for today I'll just stick to the three-dimensional case, so this is the PDE, it is pretty much the same as curve shortening except we get this one extra term and this term is very nice except when u is zero which is exactly when, so if the surface at some point approaches the rotation axis which it can only do by becoming singular, then the equation develops singularity, okay so question, can we find self-similar solutions and so what is a self-similar solution, a self-similar solution is, it'll be one of these, so a self-similar solution will be a solution of this particular form, so this thing is so in other words it satisfies, but if you assume that this rescaling that I just wrote down before, that that doesn't change the solution, if you make this assumption then that turns out that that means that the solution has to be of this form, where this u now, uppercase u, is a function of only one variable, okay conversely if I have a function of this type then it could be a self-similar solution but the function u has to satisfy a certain condition, so the question is when is this actually, when is a solution of this type, a function of this type, when is it a solution of that differential equation, so you just substitute it and you do the calculation and you find, so this thing satisfies mean curvature flow if and only if u satisfies, and there's a factor of one half, and so if you differentiate this with respect to time this expression shows up and this plus and minus is exactly the same as that plus minus, so what is the relevance of the plus or the minus, so for plus you get solutions that are defined for all positive times and these are expanders and minus, these are solutions that are defined for all negative times and these are shrinking solutions, plus and minus, and that plus minus is the same as here, okay so there are these two ODE's, if we find solutions to those ODE's we can look at the corresponding surfaces and they will give us examples of shrinkers and expanders, in particular if we set the dimension equal to, so if you do the non-rotational, if you only look at graphs of functions, so you go back to curve shortening, this term isn't here and this term would be absent, if you solve this ODE you put with the plus over there, one of the solutions that you find is this curve of ODE, so this bracket curve is a solution of that ODE if you delete this one over U-term, and so what expanders can we find, so we look at this differential equation with a plus sign and so there's a theorem, so this was done by David Chop, Tom Ilman and myself, so for any positive A there's a unique solution of the expander ODE, that's this thing with a plus sign, and it's better if I draw it. Our functions are functions where U is positive, so their graphs are in the upper half plane, the solution we're considering here are, and the ODE is fine as long as you stay away from U is zero, if you start at a certain value A here, there is a solution and so we have more things to say about the solution, there is a solution that just follows from the existence there for ODE's, there's nothing singular near that point, so the existence of the solution on a short X interval just follows from ODE existence. If you replace X by minus X in this equation you'll find that this UX appears squared, so that doesn't change, and here X gets multiplied with the UDX, so that also doesn't change its sign, so this equation is invariant on the replacing X by minus X, so the solution that you get will be even, so it does this. It's defined for all positive X, and since it's even it's defined for all X, it's strictly increasing, most importantly it is asymptotic to a cone. Okay, we're not claiming that the solution is convex, for a lot of values of A it is not convex, in fact it crosses that cone very often, or it does occasionally cross that cone, and the proof of this theorem is it's sort of elementary, let me just say one or two things, let me explain why the solution doesn't become singular, why the solution to the ODE doesn't become singular, so you start, you have a solution that exists for a little bit, and now here I'm already drawing that that it's curved upwards, so the basic thing is that if, so the expander ODE, if UX is ever zero, then if the X derivative is ever zero, then this denominator just becomes one, that X UX becomes zero, and you find that UXX positive, so at any point where on the graph that is horizontal the thing is curved upwards, therefore there can only be one such point, right? If you have two of them there would be, there would have to be a maximum in between, and at that point the graph would be curved upwards, which is silly. Okay, so therefore, okay, so then you have to show, okay, so that we know the solution is increasing, then does it go on forever? So then you have to worry about various things, so can it, but the derivative become infinite, could that happen? And here, that doesn't happen because for the expander ODE you have a plus sign here, deleted, so the one over U goes to the other side, okay, so if you step back and look at this equation and you worry, so how big is this, what is the growth in UX for this? So for large values of UX this is roughly, there's UX squared multiplying that one, so it's roughly minus X over 2, UX cubed plus smaller stuff. If the derivative ever becomes large, this will be the biggest term around, and since there's a minus sign here, it's negative, so if the derivative ever tends to become large, if it becomes too large, this will become negative, and the rate at which the derivative grows as you go to the right is negative, so it will go back down again, okay, so this, this implies that UX is bounded as X goes to infinity, okay, so then more arguments of that type show you that, so that means that this can't happen, so that means that the thing is increasing, its derivative stays bounded, and that means that it has to go on forever, so there's a solution that exists for all positive X, okay, so then let me not do the proof of this because it's all elementary ODE stuff, so if you have such a solution, so if you have such a solution then it defines a self-similar solution to mean curvature flow, it is asymptotic to a cone, and it's symmetric, okay, so the corresponding solution to mean curvature flow is it's square root T, we're going forward in time, times U of X, and I'll also press the little A there, this is a solution to mean curvature flow, and so what it does, it is asymptotic to a cone, in particular, right, so I'm letting T go to zero, if I keep a fixed X, and you let T go to zero what happens to this, X over square T goes to infinity, so I'm using this property, right, this thing for large values of the argument is A times that argument plus little of that argument, and so this cancels against that, also against that, so for fixed X, this is just A times X, so this is when X is positive, and now it also holds for negative X, okay, so the solution does start from a cone, and it's like a one-sheet of hyperboloid forward evolution of that cone, I want to, so the following is, these things have asymptotic expansions as X goes to infinity, so you might ask how fast does this thing approximate the cone, and that's sort of an esoteric question, but it is, since there are various solutions of this type, if you have a solution to the expander ODE, and I'm so in the drawing I'm assuming that the solution, it doesn't have to be one of the ones that I was talking about here, if you take any solution of the expander ODE that is at least after some particular X naught and it is asymptotic to a cone, then you can improve this little O of one here, the rate at which it approaches the cone has to be this, and more over these coefficients A, B, C and D don't depend on the solution, so B, C and the coefficients B, C, D and so on, they only depend on the opening aperture, the slope of, the asymptotic slope of the cone, and so this actually has a PDE proof, it has an ODE proof, the ODE proof is long and straightforward, you just, you linearize the equation that you have and you prove that you get one term and then you go back and you prove that you get another term and you keep on going, the PDE goes like this, this thing is, so the ODE that this thing solves is the ODE for expanding solutions to mean curvature flow, and so if I look at that graph and swing it around the X axis, I get a solution to mean curvature flow, it is defined whenever this quantity is bigger than this X naught, it solves mean curvature flow, if I let T go to zero, then this calculation that I did here, this was for the special solutions that we had over there, but all we used was this property, the initial value of this thing is, okay, and this convergence is nice as long as you stay away from X is zero, so if you take an interval, say a compact interval surrounding X is one, so the convergence will be uniform in X is, let's say, in that interval. The U satisfies this PDE, and one over U stays bounded because U converges uniformly through A X, which stays strictly positive, because my solution is converging uniformly through this straight line, and as long as we stay away from X is zero, it's uniformly bounded away from zero. Our UX is bounded. This is a nice quasi-linear parabolic equation. We have a bounded solution that attains, let me call this thing, you know, of mean curvature flow. It's a solution to a non-degenerate parabolic equation, so it is defined over here, and it has a limit as you go to zero, so that means that the solution by parabolic PDE theory, so parabolic theory then says that the solution is as smooth as the initial data, and the initial data is A X, which is linear. This proof is simple because I allow myself to say by-powered parabolic theory and not going to take for granted what is hidden in these words. It is a smooth solution, and it is smooth all the way down to T is zero because the initial data has seen infinity smooth. Parabolic theory does not say that because the initial data is analytic, therefore the solution is analytic. And so just to remind you the standard example, that is the solution of the heat equation, E to the minus X squared over 4T. If you take the fundamental solution of the heat equation and you extend it by zero for negative time, you get a solution that has seen infinity smooth everywhere except at the heat equation that has seen infinity, but not analytic. As a function of time, it is never analytic. It is not analytic at T is zero. So we cannot conclude, even though this thing is analytic, we can't conclude that the thing will be an analytic function of time. However, we do have, at X is one, this thing is just by Taylor. You take the nth order Taylor expansion. The function is the remainder term of order T to the n plus one. And now that is for U is equal to one T. So we are evaluating the solution here. I am applying Taylor forward on this particular segment. So we are evaluating the solution here at X is one. Yeah, at X is one. So that expansion we can apply to this. So that expansion applies to this function U. So what does it say about this one? Well, I said X equal to one. And now I will set having done this, so let's set the square of T is, let this be X. So T is one over X squared. And what you get is okay, I multiply, I divide by square of T on this side, that is the same as multiplying with X. I get times, and here I get U at one zero times plus UT at one zero times T. That is one over X squared. Plus and then I get the second derivative at one zero. And if you multiply this out you get exactly this expansion. So this coefficient is that. This coefficient that's our C. So you can calculate these things explicitly and how do you calculate them? Well, for instance, UT one zero is initially U is equal to A times X. So I use this to calculate the first derivative. Well, to calculate all this, the first time derivative, this becomes zero. I get minus one over minus one over U at zero at one zero which is minus one over A. If you want to get the second time derivative differentiate this with respect to T and just keep on computing. So using this you can calculate all these time derivatives and the only thing you need to know is this number A. So in particular the next term in this expansion B only depends on A and I just computed what it is, it's minus one over A. And then C, the formulas are not very simple. They get complicated as you calculate more and more of them. What this shows is that any solution is so this shows that all these expanders approach the cone concavely. So let's go back to so let's summarize what do we have so far. Okay, so let's assume we have a cone and it's opening angle is given by Y, the radius is A times where A is the asymptotic slope of one of the self-expanders that I found before then which ways forward do we have? So there are two forward evolutions, one is split it into two caps has forward self-similar. So the ones I'm talking about are all self-similar. There might be others. There's this thing. So there are at least two forward evolutions of this type. What does this statement say? If you it tells you how close the one-sheeted and the two-sheeted expanders are as they approach the cone. The asymptotic expansion tells us the following. So a surface is a rotation, what do these things look like? So the one-sheeted expander is this and the two-sheeted expander is not all actually it is the graph of a function. It has infinite derivative here. And I say two-sheeted expander because you're supposed to imagine another one over here and this is the one-sheeted. Both are asymptotic to this cone and as x goes to infinity they both have this exact same expansion. So the distance between these the distance between the two expanders is extremely small. It is of order 1 over x to the n for any n. So it goes faster than algebraic. Because they have the exact same asymptotic expansion. In fact they go like e to the minus x squared with some coefficients. So the next question is we have such expanders so for which, so here I assume that I didn't prescribe the cone. I said suppose that I have one of these expanders it's asymptotic to a cone and then for that column we have the situation. Now the question is for which numbers can I put here? So in other words what is this little a? This little a that's this distance. So for each it's the radius of the neck of the one-sheeted expander. If you prescribe this little a then there exists exactly one expander and it will have some asymptotic slope. How does that asymptotic slope depend on this a? So this is also one of the properties of these things. So first of all it's differentiable and second the limit as a goes to zero of is infinite. So the graph looks like this. It actually, so from what I've written here it's a the opening angle depending on the radius of the neck is depending on little a is a function that is it's continuous it's differentiable. It's infinite here and it becomes infinite there and then so a priori it could look like this but then you can calculate these things numerically and it looks like it. I'll write that dotted so we didn't actually prove that it is concave convex or anything but it has there is a minimum value so consequence. There is an a star positive so that this is the minimum value a star then if you pick any a bigger than that there will be at least two and possibly more if the graph is happens to be exotic which really it isn't but so you should be able to, so this should be true if you delete the word a at least right if you replace it by exactly but that's okay so how many forward evolutions do we have then I could try to draw it here in the three-dimensional picture but that would become a big mess no it is not known if it has so it has exactly one minimum that's numerically verified but it's not not proved and I don't know whether it would be hard or not the thing is it's a statement about this one particular ODE so so it's I'm convinced that it is true and if if it ever becomes important then you know if we really have to prove it numerically you know machine precision interval arithmetic program would be able to verify that so what expanders do we have from this cone first of all there is the evolving it forward in the two-sheeted way using the and then now if this aperture A is bigger than that A star there are two self-similar one-sheeted ways of going forward one is this and then the other is this at least two of them going forward the viscosity solution will be the whole region in between the outer ones and this one will be lying somewhere inside the viscosity solution the level set flow this A star is so I don't know the value of A star I know this angle so it's this angle if we call this alpha then A is the tangent of alpha it's about 66 degrees just from numerical computations okay so now from the point of view of just studying the curvature flow as a mathematical a mathematical thing of interest you could ask a very natural question is can I find a smooth surface that forms a singularity and when it forms a singularity its singularity is is a cone like this and the aperture or the opening of the cone is so large that we can then float forward using this and so for mean curvature flow in three dimensions that is I think is still an open problem we have numerical evidence that you can instead of presenting you the numerical evidence I would like to give you a different perspective also so there are these two flows, Ritchie flow and mean curvature flow that are being studied and that have a large number of similarities one difference I think is that Ritchie flow I think it's safe to say it's fair to say that it has been much more successful in actually producing topological theorems the Poincare conjecture and the geometric theories in theorem are enormous successes on the other hand the mean curvature flow I think is closer to mathematical physics and has more applications so there are other reasons to look at mean curvature flow so I want to so there's a paper by Riston part Nature a couple of years ago I think it's 2011 give or take one or two years so they are physicists not even theoretical physicists so they have a lab with so they had the following setup you have a tube in this tube you have two liquids and I forgot what they are so there's one liquid here there's the other liquid is up here and then they somebody throws in a drop of liquid number one so let's say this is oil I'll say they're oil and water but they weren't so not oil and not water just to label them they throw in a drop of this and that drop falls down and at some point it hits this surface and then it gets absorbed and what they did to make this more interesting is that they put an electric field and so they observed the following if the electric field was small then what happened is so let me not draw the whole setup let me just draw this surface and that surface so E small what they observed is that right and these are snapshots in a movie this surface goes up a little bit and this thing they merge this thing smooths out and it just becomes a surface like that when the electric field is really large what happens is that the drop falls it gets close and it bounces off the surface and then they spent they spent time on trying to explain why this happens and I believe I believe they came to the conclusion that there is no way to explain this using surface tension only and it was so surface tension is captured by the energy in the surface so which is proportional to the area if you assume that the surface tension so surface tension would be important at this moment where the two surfaces connect so what happens here do these two surfaces it's like in Tacos's drawing these two surfaces touch do they does this neck does it form a neck and widen or does it separate again so surface tension would so an explanation by surface tension would be to say that the thing tries to reduce its area as much as possible in other words near so at the time that this happens in a short neighborhood in space and time the evolution of the boundary between water is governed by mean curvature flow can you give an explanation for this using that in the answer so Peter Topping and Sebastian Hellmannstorfer came up with an explanation and it's exactly this so what happens in this situation so remember there's an electric charge here and what that does is when these things get negative so a fair amount of positive charge ends up on the surface of this over here there's no charge here this thing develops so this positive charge attracts all the negative charge here and moves the positive charges to the other side right before these things touch there's positive charge here negative charge here and that's why this surface is coming up it's because there's an electrical attraction from the drop on the surface but the attraction is not very strong so when they touch if you when they touch at the moment of contact these surfaces will be more or less conical and the opening angles of these cones will be really large right this angle so this angle and this angle will be really large so that means we are well above well above the 66 degrees required for there to be more to be for there to be one sheet of the expanders okay so when that happens under mean curvature flow this thing would one forward evolution so you would have to check that it's the one that reduces the area the fastest that's a good question for someone to look into the forward evolution of this would be something like that which is what they saw over here if the electric field is really strong then this attraction is very strong and that would cause the shape of the two surfaces when they collide to be also conical but with a much smaller angle if this angle is really small then under mean curvature flow there is only one way forward namely to for this thing to separate okay what happens at this particular moment why do we have mean curvature forward and why didn't that happen before because under mean curvature flow when a neck pinches the singularity is not conical it's more cusp like it's like a cone with a zero degree angle so why did we not have this for negative values of t if this is t is zero it's because right up to the moment of contact everything was being driven by these electrical charges and then at the moment of contact there is a current the charges get equalized and the electrical field does not play any role anymore okay so this is so in mathematics you might want to have mean curvature flow always and start with a smooth surface in nature nature can set things up so that you have one one set of laws up to a particular moment and then once you've prepared the initial value the flow takes over which I thought was a nice application of these ideas okay so I spent more time on this so tomorrow what I want to do is talk about solutions in RN where you regard RN as RP cross RQ so so far we have today we've been looking at R3 is R cross R2 and we assume rotational symmetry in either of these two factors here we had reflection symmetry in R and rotational symmetry in the plane tomorrow we'll assume rotational symmetry in this and that factor and you get a much richer family of examples and I also want to talk about non-self-similar things which are more PDE oriented so there will be examples of solutions that start out smooth form a singularity and then have many different ways of going forward in particular there are we'll talk about that tomorrow okay so