 So let's go on with our study of rule surfaces, we interrupted, it took a bit of time speaking about spaghetti, but now let's go back. Remember, which are the data of such a surface. We have a curve, the directrix, and a one parameter family of lines, so we call it alpha tw of t, and we are looking at the map xt v equal to alpha plus tw, plus v w. Meaning w is working for us just to give a direction, and for such a direction we put on the surface the whole line, the whole straight line in this direction, while something is moving, the point is moving on a given curve alpha. We saw a few examples. Let me add just one example, which will turn out to be important later, which is called the tangent surface to a curve. I mean, this is adds to the list. Last time we made many examples of this type of surfaces. Tangent surface to a curve. That means just, so it's a one line example, in the sense that given any curve alpha, for any alpha, I can take w to be the tangent vector to alpha. In this case, of course, my surface will look like alpha plus v alpha prime. Geometrically is a particularly simple situation. You have a curve in space, and you add for every point of this curve the whole line tangent to it. We are going to give us a classification of surfaces where this type will be one type of such surfaces. Let's keep it in mind, but of course it's a very special example. In any case, given this type of surfaces, so now let's go back to the general case, alpha in w. Let's compute first, second fundamental forms and curvatures. We know how to do it. X, X, XT. In this case, the variables are not UMV, but TMV. The first one would be XT is alpha prime plus vw prime. I drop the dependence on what they depend. It's clear, it's written there. XT is this one. Xv is just w. Xtt is alpha double prime plus vw prime. Xtv is just w prime. And Xvv is equal to zero. So then what is the normal to such a surface? Remember, actually one general comment. We are performing the usual theory without worrying too much if this is the parameterization of a regular surface. Remember, there is the problem of, for example, of injectivity of this map, which is highly false in general. So we perform it formally, and the computations we are going to do will work on the parts of the surfaces for which this is a good map. So we have to be careful where to make interpretations. But let's go on. Now n would be what? Would be Xt wedge cross product. Xv normalized. Well, there is not much more that we can say. I mean, we have to do the cross product of these two, and in general we have no information. Of course cross products simplify if you know that something is parallel to something else or something is orthogonal to something else. In this case we have no information in general about properties, mutual properties of these vectors, alpha prime, w prime and w. They are almost free to be whatever they want. So let me, I leave n in this form and then depending on the situation we can make further analysis. So how much would be e? OK, so I don't write it because I would just write the general formula. OK, so what is the, which are the coefficients of the first fundamental form? Well, I have to take Xt scalar product n. OK, sorry, first fundamental form Xt, Xt. OK, so, well, in this case, in this case I made a mistake here, but that's OK. OK, this is the norm squared and again I cannot really say anything. OK, so it's just the norm squared of this vector. OK, how much is f? F, again, would be the scalar product between this and this and I have no information to use. So it would be alpha prime scalar w plus v w prime scalar w. And what is g? And g is just the norm squared of w. OK, how much are the coefficients of the second fundamental form? Well, let me not even write little e because little e would be the scalar product between this and this, but OK, I have no formula to use. So besides rewriting the definition there is nothing I can say. Let's see what is little f. Little f would be the scalar product between n and Xtv. OK, so let me introduce a notation. If I take the scalar product, so some notation, if I have three vectors, a, b, c, OK, in R3, whatever you want, in Rn, but I mean I call the bracket a, b, c is by definition the scalar product a scalar product b cross c. OK, this is a standard notation in many books that you will find, so better get used to that. I mean it's not a great simplification. It's almost the same number of symbols anyway because now what is f? F, you see, it's the scalar product a vector and a cross product. So I can use immediately that notation there and that becomes alpha prime, alpha prime ww prime divided by the norm of n. OK, so now I can add the formula for the norm of n, which is alpha prime cross w plus vw prime cross w. This is nothing but substituting the norm of this. I'm not cheating you. I mean this is just what I would get here immediately. OK, and then how much is g? Little g in the second fundamental form? Well, this is the best one because of course it's zero scalar, whatever, so it's zero. That's why I don't have to compute little e. Because in all formulae, I mean for example in the Gauss curvature I would get eg. So if g is zero I don't care about e. OK, so having done this how much is the Gauss curvature? Well, it's eg minus f squared. So eg minus f squared divided by eg minus f squared. That becomes, so eg is zero, so minus this object here, alpha prime squared divided by alpha prime cross w plus vw prime cross w squared. This would be just f squared, minus f squared, so times one over eg minus f squared. And I don't even write it. OK, divided by eg minus f squared. You substitute those almost generic functions and that's it. Because this is enough to make some considerations. You see, any ruled surface has Gauss curvature less than or equal to zero because whatever this is, this is a positive function because it's the determinant of the first fundamental form. One over. So this is positive, this is of course positive and this is minus something squared. So this is less than or equal to zero and in fact it's zero if and only if this number is zero. Now I don't go through the examples we made. Now you have this nice formula and you can substitute in the one sheet hyperboloid for example or in the cylinder. For example this is another way to compute the Gauss curvature of the cylinder or general cylinders or of a cone. I mean now you pick all those examples you have the explicit expression for alpha, w and w prime. You substitute and you get the result. OK. But now it's clear that the vanishing of this function is measuring something interesting. So let's see if how many times we can reach this position. Now in general, so no loss of generality you see the role of w we said at the beginning is just to point in the direction. So to give me a direction. So there is no arm, I mean I'm studying the same surfaces if I assume the norm of w to be constant equal to one. OK. If I have one parameter family of lines where w is the norm of w is a function of course I can res k and I get the same surface because it's giving the same direction. So since I'm adding the whole line I don't care. It will change the value of v for which I touch a point but I don't care. The image is the same. So no loss of generality in assuming this. OK. But this is technically convenient but before seeing why definition a ruled surface is called developable developable if this function is zero. If alpha prime w prime is equal to zero. OK. Or equivalently if the gauss curvature is constant equal to zero. OK. We have given a name but we could have been giving a name for how many surfaces are developable. OK. Now let's see. So now suppose we have a developable surfaces. So let's see if we can identify it in some form. OK. Well, first and this has nothing to do with the developability we have put this normalization on the direction. OK. If I have a family of vectors of given norm I take the derivative and I get one equation between the scalar product of the vector and its derivative. OK. Nice. So now let's see in how many ways I can get this. OK. Of course this is the scalar product between this vector and the cross product of these two. So, first case when this is zero cross product of these two is zero. Of course, if this is if the cross product of these two is zero whatever is alpha I get zero. So let's first see if I can understand the geometry of such a surface. So, case one if w cross w prime is equal to zero what can I say? Well, if the cross product of two vectors OK. So, this implies that, let's see in which way I prefer w prime of t, now let me rewrite what they depend on is equal to some number but which in principle depends on t times the other one. So they are proportional but the factor of proportionality depends could depend on t. OK. We have this equation here so let's use it. So on one hand I have zero is equal to w scalar w prime but now I have also this equation so if I substitute to w prime this I get what? I get lambda w w but w w is equal to one OK. So this is equal to lambda. So in fact the second equation is a stopable equation forces this factor of proportionality to be constant equal to zero. OK. But then but then the geometry of the surface is almost trivial because that means w prime is constantly equal to zero but that means that w is a constant vector and we have given a name to such surfaces. So that implies w of t is equal to sum w zero if you want independent of t. So the surface is what we called a cylinder. Is a generalized cylinder of course now drop from your mind the idea that the cylinder is only the rotation invariant straight cylinder. So there is a curve and there is a constant vector in the surface which at each point of this curve takes the line in this parallel direction. So cylinder. Very well. But this was of course the first trivial case. I mean this color product is zero because one of this vector is zero. Case two. Of course suppose now that we are not in that situation. Another way to say it is of course zero. That means these two vectors are linearly independent this time. It's the contrary of the situation before. But the scalar product between alpha and the cross product of these two is zero. That means that the triple alpha w prime are linearly independent. Alpha sorry, linearly dependent. Are linearly dependent. Sorry? Sorry, alpha prime, alpha prime. Yes, of course. That's information coming from here. So now it's a little bit like before but it's slightly more complicated because if they are linearly dependent for example, I mean let's assume that alpha I express alpha prime in terms of a linear combination of the other two. So alpha prime of t would be, so there would be now again two functions because the coefficients are free to depend to move in time. But there are coefficients such that something like this holds for some choice of lambda and me for two functions lambda and me. And alpha tilde of t. Let's define another curve by just alpha alpha of t you see, I want to change the directrix of my surface without changing the surface. Remember that the rule surface has many infinitely many directrix. So in this case it's clear that if I take alpha minus this, I take this on this side, I mean minus mu well, before taking the derivatives minus mu w I'm doing something smart because what is alpha tilde prime of t? Well alpha tilde prime of t is of course alpha prime minus of course I introduce a little mistake because I had to take the derivative also of this function w minus mu w prime ok but then again I use the equation defining lambda me and this is equal to what? This is equal to me me minus me prime w of t because alpha prime minus me w prime lambda sorry and where do I put it here? Alpha prime minus so this is lambda minus me prime ok now so this case too must split now into sub cases so case if you want 2a sub case a suppose that alpha prime is constant alpha tilde prime is constantly equal to 0 ok well what does it mean? It means that alpha tilde is a constant vector let's say some kind of alpha not so it's a given point of ok but then what? then alpha well this is of course the trivial case alpha is where is it here? alpha is equal to alpha tilde plus me w is equal to alpha not plus me of t w of t ok but you see that we have not changed the surface so what is x? x of tv now let's write everything just to be sure that we are not making mistake is alpha so then it's alpha not plus me of t w of t plus v ok now we could go on with analytic interpretation but now the geometry is clear what is the geometry of such a surface? you see w of course the only thing I would like to do let's do it but I mean it's me plus v w ok but that means what? that the rulings of this surface they all pass through one point alpha not so in geometric terms this is a cone ok this was again trivial case 2a now let's do 2b ok so now suppose that alpha prime alpha tilde prime is non-zero alpha tilde prime different from zero so what can I say now? well let's take let's see how x again looks like so this is alpha plus v w and now let's put the alpha tilde into the game we found alpha tilde ok so alpha becomes alpha tilde plus me w ok so this is alpha tilde of t plus me of t w of t plus so again w of t ok but now I know that w if I want I can divide suppose for a moment generically I will be able to take this thing and put it here so w is equal to that one so this becomes alpha tilde of t plus me of t plus v lambda of t minus me prime of t everything times alpha tilde prime ok now finally I have to use the equation otherwise ok but then you see why I gave you the definition at the beginning as an example because now my surface looks like alpha some alpha if you want call it alpha tilde beta or however you want plus this one which you can call if you want v tilde times alpha prime so this is a tangent surface to a curve to the curve alpha tilde ok so this is the tangent surface to alpha tilde ok so this was very simple but what is the moral of this remember that the question of understanding what is the geometric meaning of the gauss curvature and also of the mean curvature in 15 minutes ok so the first thing is curvature should distinguish between flat and curved ok so naive question above the big question in general you have a function you want to understand what is the meaning of this function easy level zero question suppose this function is constant equal to zero what does it mean I mean is the surface special and then of course the answer is well yes depending what you mean by special but certainly is not too special so I hope I convinced you now that there are huge families of surfaces with constant zero gauss curvature because all cylinders or cones and all tangent surfaces to curves they all have gauss curvature equal to zero and in some sense they are the only one ok so if you ask my four years old baby he would say ok now that these are flat but this seems to contradict openly our intuition of curvature because remember not even this one is flat but you can find some strange argument but I mean now a cylinder over any curve is flat a cone over any curve is flat so the tangent surface to any curve is flat so now it looks desperate now you say well that means the gauss curvature maybe is not that interesting if it's not able to distinguish ok so we have the only point is that we have to work harder ok now let me enter the problem of mean curvature I would like to play the same game with the mean curvature now in fact let me take a long route and we are going to learn something which is as important as the final point in fact probably even more important than the final point because we have treated the first fundamental form you see for example in the consideration of gauss curvature equal to zero the first fundamental form was not really entering because it was always e g minus f squared divided by e g minus f squared first fundamental form but if you ask is this zero or not you don't care about the denominator so it seems a property completely determined and we also said when we define the first fundamental form we almost said well ok it's just to do something since we called it the other one second let's call something the first because it's not really adding any new information it's the scalar product between tangent vectors but now let's go back and let's throw away this this is not bish attitude because the first fundamental form in fact measures many fundamental things and at the end in a couple of weeks you will see it will measure all fundamental things so for example let's ask the following question remember we start with a regular surface and suppose we have a local chart ok that means we have our usual picture we have some domain we have a map x and that's the image ok now suppose you take a curve on the surface of course inside this chart I'm going to make local computation so I will restrict myself so suppose I have an alpha of t which lies on my surface s so of course since we are doing metric geometry because everything which comes from the scalar product has something to do with the metric ok the simplest question in metric geometry is what is the length of a curve and in fact for when we started curves by themselves that was basically the only thing we could do so how much is the length of this curve well this is a curve in space so the answer does not depend doesn't seem to depend at first look from the surface s and you say well I have alpha of t very well the length of the curve alpha let's say between two points of the interval doesn't matter is the integral of the norm of alpha prime so suppose I have it parameterized by some t in dt between some t0 and t1 you see that this doesn't seem to depend on the surface in fact does not depend on the surface ok so it doesn't seem to be treated just as a curve in space but then if the curve lies in a local chart of course there is the usual correspondence that we use everyday there is some curve here beta of t which is mapped to alpha ok so you might ask now a more refined question and this depends on the length of this curve in the plane and the length of this curve on the surface so somehow how much is distorting lengths the map X so the map X was fantastic from the study of surfaces but it will change lengths of course ok there is no reason why in general lengths of curves will correspond ok if I use this trick of course how much is alpha prime so because alpha becomes X composed beta ok as usual so what is alpha prime well alpha prime is u prime X u plus v prime X v ok where of course I am assuming as usual all the usual notation in two functions u of t v of t ok but then how much is the length of alpha well in fact before doing that how much is the norm of alpha prime of t so then I substitute into the integral ok how much is that well it is the scalar product between this and itself well divided by I mean square root so this becomes what it is for the square root this times this so u prime squared E plus and you can imagine twice ok u prime v prime F plus v prime squared G everything to the power one half ok well this is interesting because ok now put it I don't even write it but I mean now I love alpha becomes what is the difference of course is that if I write it in this way I am thinking of beta I have the two functions u and v of t no and in some sense I am comparing an integral here so I am comparing I mean this means that the philosophy the geometric interpretation of the first fundamental form is exactly what I said before because what would be the length of beta as a curve in the plane it would be the integral of beta prime but what is beta prime beta is equal to u and v is just the integral of u prime squared plus v prime squared one half in the t so you see that the second fundamental in the first fundamental form is exactly the distortion of lengths of curves measured on the domain of a chart and on the image of the chart the same curve but once it's here and the other one is transported on the surface ok and here they come E, F and G very important ok we can play the same game as of domain after all so now let me draw the parallel picture but it's always the same I have my ux and now suppose I take some region here ok now now suppose I have some region that I imagine again inside the chart so it corresponds to some region but it doesn't look very different here so let me say this is v and this is x of v ok now again let's ask the same question but now with the area so in some sense I know how to measure the area of v this is not a v prime this was just the area of v compared to the area of x of v ok so let me give you a definition but the definition is motivated by the following classical if you are in R3 now let's say this is a kind of a side note to justify what I'm going to say if you take two vectors which might not be orthogonal of unit norm nothing of course linearly independent in some sense it's a pathological case let's call it xi and eta in R3 now of course two vectors in R3 give me a parallelogram ok I can imagine this parallelogram here how much is the area of this picture ok exactly so the area of this region is given by the norm of this in fact it contains the pathological case if they are parallel you get zero as an area ok this justifies the following definition in fact let me write it here so I save a little bit of space so the area of x of v because I have to define it actually it's not ok of x of v is equal to the integral of ok why I'm going to write this formula because I want to apply this simple observation which are the two vectors which play a role here xu and xv so you see at this point of the surface I have two canonical vectors tangent to the surface and in some sense in this decimal area well generated by these two vectors the norm of their wedge product of the cross product ok but how much is the norm of this strange enough but not strange at all is again the determinant of the first fundamental form so this justifies to define the area of this region the integral over v now as an integral here eg minus f squared one half now as an integral in R2 so I know what to put there as a measure so I put the standard Riemann measure ok now here that now there is a problem because you see I'm defining the area of a set on a surface of a subset of the surface and here I use to define it a chart so now there is the usual problem if I have two charts covering the same region does the area depend on the chart or not because if I look at the formula it does ok but this one it's an exercise in calculus and I leave it to you suppose you have another chart that you want capital Y which now comes from another which covers this the same set but now for example corresponding to a domain let's say capital W ok so this will have another other coefficients of the first fundamental form because the coefficients of the first fundamental form because E measured with X and E measured with Y are different F are different G are different but this integral is well defined of course everything boils down to the change of parameters formula ok for integrals in Rn ok and it works so this is a well defined integral maybe I'll put it in a in an exercise sheet so in case some of you have some problems we can do it together ok but now again you see this simple interpretation definition now tells me that the first fundamental form is exactly measuring the distortion now this time is by definition but I hope I convinced you that it's the right definition the distortion of areas here and here ok so the first fundamental form which seems to be just a name is in fact taking care of the two major geometric things that you can do measuring lengths of curves measuring areas of domain ok ok we will freeze this for later use but now let me go back to in the same spirit we did the ruled surfaces thinking to the Gauss curvature zero we studied ruled surfaces now thinking to surfaces with zero mean curvature we do something else and that's what we are going to do now ok in fact it's better to put it right from the beginning the key definition so a surface a regular surface is called minimal this is kind of the corresponding name to developable if h is constant equal to zero but now I want to convince you that this name has some sense minimal in which sense ok and this is nice so first, so the idea is the following you have a surface and again this is just the piece of a surface covered by some parameterization by some chart I'll try to make a big picture even though I will need so this will be x of u and that means here I have some big domain u and the map x ok now, suppose in fact this u in principle could be unbound and non-compact anything it's just an open subset of R2 so now suppose here I pick a bounded domain d ok so now d is a bounded domain so x from u to s is a local chart d contained in u is a bounded domain actually there I drew it closed but I don't want it to be necessarily closed in fact usually a domain is open in fact to make it proper you have to remove the boundary and now suppose you take a differentiable function h from the closure of d to r differentiable from the closure of d ok I will call the normal variation by h or in the direction by h so this is the definition of this sentence the normal variation of x of d closed determined by h or in the direction of h if you want is the map let me call it phi phi from where small interval so for some epsilon positive which doesn't matter into r3 which takes so d is inside u so in u we have the two coordinates u and v so it takes u v and let me call t the parameter in this small interval and gives me what gives me x of u v bonding point of the surface plus t h is again a function of u and v ok so n at the point u v so what's going on nothing worrying in some sense I should draw the graph of h over d and this is kind of difficult now in the picture but you can imagine most domain d bar so imagine a graph here and now I'm taking phi is what is the corresponding point on the surface so here I have a normal vector at this point h has some value for example it's positive otherwise I need to draw it the other way and here I take the point phi of u v plus t u v t so that means geometrically it's simple if you think of what's going on you have a graph here on the flat r2 and basically you are drawing the graph here in the normal direction the graph of h transported here so this is the idea of course the graph at height t it's a family of graphs but of course the role of t is just to push it to translate it to translate it in the normal direction it's not a translation in r3 because n depends on the point so you see for t equal to 0 this of course is not there and phi takes the value x so for t equal to 0 I'm just taking this disk for example in my case and I'm putting it here somewhere so this will be x of d closed and then as t moves I start moving this it's not even a disk anymore but the image of this disk using h as a kind of a graph function ok I'm writing something nearby not for t it will be very close to that one by using h ok is it clear geometrically what's going on so for example if h is equal to 1 constant equal to 1 what am I doing I'm taking really x plus tn ok so this is the subject this piece of the surface in the normal direction ok every point at the same speed so h is a parameter which gives you the freedom that one point is moving quick and one point is moving slow or maybe one point is going up and the other point is going down because if h changes sign of course we have an imagination of what's going on ok now notation and you might say why it's not the initial notation so for fixed so I call this a normal variation ok it's clear why it's a variation because for t equal to 0 it's the old one and it's normal because every point is pushed in the normal direction ok so for t in minus epsilon epsilon I want to think of this of a variation as a family of regular surfaces and of course I can do it because I define x and now I put the t here of uv to be phi you see if I fix t I'm taking the slice ok it's a surface which is very close to the old one it has moved with the function h ok so to each of this surface so if I keep t fixed I can study the geometry that we studied up to now of the surface xt let's do it ok so basically I want to know how much is xt u so now I have a parameter which is t and two parameters umv well how much is xu xu will be xu without the t now plus something xu plus t hu n plus t h nu ok how much is xv something changing ok so how much are the in principle I could compute the coefficients of the first fundamental form for each t ok let's do it so what is and I call it et et will be the first coefficients of the first fundamental form of the surface xt well that means it's this squared ok imagine you are ok so now let me save three seconds and we are taking the scalar product of this times this plus t h nu ok what can I do and how do I want to do it because of course the only thing I could say is to expand everything and I stop but the thing I'm interested in is this will be a function now I want to see it as a function of t ok so for t equal to zero I want to make in some sense the Taylor expansion in t ok so I want to know what is the t equal to zero or what would be the constant in the Taylor expansion and then what is the linear part in t I want to split it in degree with respect to t ok well for t equal to zero it's easy because of course the only known constant in t now my speaking is referred to the variable t the only known constant object is x u x u so not strange it starts with the old one after all for t equal to zero the surface was the old one so what is there in t plus t what well of course the only thing I don't have to do is a scalar product of something multiplied by t with something multiplied by t because I would get a t squared so for example so this one times what this one times this but x u and n are orthogonal by definition so this one is not there so this is the next one so x u scalar this ok let's write it so plus t times h x u and u ok it will be longer than this and then I am done with this one now what about this well this one goes against x u and then goes against anything because there would be t squared so this one goes away in the linear part and this one gives me t h so t is there h is here and u x u so I get twice this ok plus so let me write it in this form of t squared so the point I want to make is that now I don't care t squared coefficient would be much more complicated but I'm interested in the linear part let's play the same game with f f t is what now imagine we are taking this scalar this so of course it starts with f and then what is t what so this times this zero this times this h x u n v and I am done with this one then this one but this is zero and then everything else is t squared and this is what this is n u x v plus of t squared n g g t is g again so this times this is zero this times this gives me of course by symmetry if you want you know it so it's t 2h x v n v ok plus of t squared ok now I can write in fact I should have left one line because I know what are these these have a name in our language because you see these ones are computed at t equal to zero here there is not a t here and there is no t here so these are the original objects ok so I know what they are for example this one is what is minus e ok so what I found so in fact let me rewrite it, it's a pity so e minus 2h t e ok plus of t squared ok when I write in fact let's invent a symbol ok this means up to second order ok so we save time how much is f, f t would be f ok and now here it seems slightly more complicated but it's not because you both of them are f ok they look different but then you think coming from n scalar x u equal to zero and then scalar x v equal to zero whatever derivative you take you get that these are equal to minus f ok so this would be minus 2 t hf so and this would be again up to second order and g t is up to second order g minus e g ok very well we are halfway through but then remember it's not there anymore but the formula for the area the area of a region the area for the region was given by the integral of e g minus f squared one half d u d v ok so it's quite clear now we are going to speculate on that but for the moment it's quite clear that we want to perform this computation so we want to know how this subject depends on t again with the same spirit first order, first order and higher order ok so e t times g t minus f t squared well the one half bit later now a little imagination and you get this you get of course the zero order is the old one so this is e g minus f squared plus t times what in fact I should have put minus because there is a minus in front of everything minus this object here minus in fact even a 2h is in front of everything no surprise 2h times e g minus 2 f f minus plus e g and here this is again up to second order this is immediate but now you remember I mean you don't remember and you are allowed not to remember even I don't remember but the formula for h in terms of all these functions h of the original surface h of the surface at t equal to zero and if you use it which in fact was actually just telling you that h is this divided by e g minus f squared capital ok so imagine what we are doing we divide by e g minus f squared but that means I can take out e g minus f squared from everything so I can say that this is equal to e g minus f squared times something which starts with one minus 2h 2th and now the only problem is that this is so this divided by e g minus f squared is not exactly h but it's twice h ok so this becomes 4 capital h ok and now let's compute the variation of the area so in you see the spirit is these surfaces are moving but somehow they are all image of the same thing so if I since I define the area of a surface has the area of the coming domain original domain but measured with a strange measure so measured with a measure coming from the first fundamental form because here the integras are always with respect to du dv but there is the function in front square root of e g minus f squared so if I have two surfaces which are image of the same thing I can transport problem of computing the area becomes a problem here on the fixed domain but with measures which are changing ok and that's what I want to do so the area of what of how did we call it of x t of d bar ok so for each t look at this strange graph normal graph and I measure its area but of the same domain so this is equal by definition of the area over the integral over d bar so it's an integral in R2 so I should put here the square root of this so e g minus f squared 1 half times square order because I dropped the higher order terms times 1 minus for th h 1 half but then everything is integrated with respect to du dv let's compute d in dt so how is it changing so the derivative at t equal to 0 of this function of t for each time I have a surface I measure the area I have a number so I have a function I take the derivative of t equal to 0 simple so you see this does not depend on t so you have just to take the derivative of this evaluated at t equal to 0 and you get this this is minus the integral over d bar of 2 h h times minus f squared 1 half du dv ok, this is take the derivative and evaluate at t equal to 0 ok, there is nothing there oh, but now we are this is a very interesting formula because you see the key point is that here you have isolated so this is kind of a positive function and here you have in front these two objects every time you see something like this this is easy to interpret because the point is if you have a surface if you have played this game starting from a surface which was minimal so h equal to 0 ok so for example there is a stupid simple error here so if the initial surface is minimal then this object, the derivative of the area in any direction well in the normal direction but using any function would be 0 it's a critical point ok well if s is minimal but here the analysis takes care only of what's going on inside the chart x the important thing is that h is equal to 0 on this part ok, so if you want if s is minimal but I mean, what you care is that the mean curvature is 0 on x of d bar ok but now this formula it's not difficult to see that implies also the other way around so, because if this object is equal to 0 for any h you see that this is a huge condition and you can imagine I mean, I'm not going to give you a complete proof because I hope you see it ok if h was not 0 at some point it's easy to build a little h for which this is non-zero you see why I mean, the idea is in fact, this is even the proof in some sense suppose for a moment that we are drawing our surfaces flat otherwise the pictures become impossible for me so you have a surface and at this point h is equal to 1 sorry, this is the normal vector and at this point p h of p is equal to 1 for example so I argue that this integral cannot be 0 for any h that's enough because I build a little so if h is equal to 1 at one point by continuity it will be so if I draw the graph ok, here it's 1 and here it will be whatever it is I don't care I build h as long as it's non-zero I build for example a function which is 0, 0, 0, 0 and then and this is little h ok and I choose this small interval in a way that on this small interval even capital h for example is positive if it's 1 at one point there will be a small disk where it's positive ok, so you build a little h h is a positive function whatever capital h was ok, over the whole d so this is positive this is positive the integral cannot be 0 ok that's in fact the proof so if the variation if the first variation of the area is equal to 0 for any variation then h is equal to 0 ok, now this is a beautiful theorem it's one of the building blocks of the modern differential geometry ok, in some sense but now in the spirit of what we did for ruled surfaces so we wanted to say k is equal to 0 was meaningful I mean, for example was it the plane or maybe just the plane and the cylinder and we came up with huge family of examples so whatever the truth will be it will be more complicated than that now h is equal to 0 now we know a variational characterization of surfaces with h is equal to 0 but in principle it could be the empty set well not really empty because you know the plane the plane is a minimal surface ok but are there other minimal surfaces well first just a language problem the name minimal is very misleading in some sense because it's a critical point of the area functional nowhere is written there that it's a minimum minimum would mean compute the second derivative and prove that this is non-negative and this is highly false so in fact the search of minimal surfaces in R3 it's the search for unstable critical points of a functional which is more complicated of course much more complicated than if your functional has minima you minimize and you try to do something ok here all these objects except for the plane the plane is the only minimum ok so the word minimal but unfortunately it comes from Lagrange and when something comes from such a big guy you shut up but it's kind of remember minimal in this theory it means critical point and not minimum ok so the problem is are there other minimal surfaces or not because maybe we are speaking about the empty set we did a beautiful exercise in differential geometry but well you know another one in fact you know two ok I knew one and you do another so we know two so the first one or I don't know if it was the first or the second but this is one catenoid ok the catenoid is a minimal surface ok but then why this was a kind of a huge problem in mathematics because this is a very natural interpretation in term of if I just find ok so this was the way the theory of minimal surfaces was born kids playing with soap in fact physicists playing with membranes ok take a wire put it in a in soap and pull it up ok now believe that there is a correspondence between the area of the surface that you get and somehow the amount of energy that the soap has to waste to cover to span this wire the soap will make an effort ok you have fixed the boundary and the soap will try to minimize the effort to span the boundary so the soap is going to produce a minimal surface it is going to produce the minimal surface when you have fixed the boundary somehow you are telling the soap something I want you to span this piece of wire usually kids play with in fact it is not true anymore I have been playing with round soap wires and then the solution was quite boring because you put it there you peel it up and you get a disk ok but we are mathematicians and we think why if we play to soap bubbles with something like this this is a wire you put it in soap and you pull it up so what is the soap going to do well this is already an interesting example because this shows you that the soap gets crazy there are two minimal surfaces ok in fact one is bigger area so in fact the soap doesn't get crazy this time because it will choose this one ok but you see now you have formulated the problem given a boundary a closed curving space is there a minimal surface with this boundary in R3 this was a classical problem in physics ok it's formulated with soap because it's easy because everybody have played with soap bubbles but it's clearly a very general family of problems ok give me a rule for which area corresponds to energy or effort or whatever you want and you will end up studying soap bubbles ok so that's why you will find many books in mathematics library speaking about soap bubbles because it's the prototype and it's already very difficult ok and in fact you can go in fact the helicoid I didn't print a copy of the helicoid I have a picture of the helicoid but the helicoid is another minimal surface so you start seeing that they look very different but you cannot even imagine how different ok because for example this is another minimal surface it's periodic in space it's contained in a cube and then you start duplicating the cube with this surface inside and you invade the whole of R3 the gray areas are the holes ok so with something like this for example you construct a minimal network of plumbing ok these are plums ok minimizing the area means you are using the least possible steel ok since nature does these things in fact this surface exists in nature it's called the Schwarz surface ok but then you start getting more and more complicated objects so this looks more or less like a catenoid you see for example if you take a cat the stupid thing would be stupid but I mean you take a catenoid and you take a plane it's orthogonal to the axis of the catenoid for example in fact the union of these two things is a minimal surface the only problem is not really a regular surface because they intersect in a nasty way it's not a differentiable surface so the question we have been for many many years is it possible to smooth a configuration like this keeping h equal to 0 because in some sense formally it's either the catenoid or the plane it's ok and the answer turns out to be yes but this is an extremely beautiful and difficult problem which was solved actually in the beginning of the 90s so we are speaking about very recent stuff and there are still people asking in how many ways ok because actually these objects do exist so it's one of those points in which mathematics and nature work together so you really like to have a classification of minimal surfaces because these things exist so we know enough to say that there are plenty of them but we are very very way behind the classification in some sense it's a beautiful research area because of course you can imagine that behind this picture there is also a huge 3D graphic problem because if you think that you are going to prove that this object is a minimal surface by telling me which is X forget it ok you will never be able to write down the explicit solution unless it's the catenoid or the helicoid you are lost either you have some method to prove that something exists so it's proves are very mathematical and then you wonder well there are the pure problems and the applied problems and say now I want a picture ok very well I think but just since we have told you a little bit of history let's go back to the boundary problem because I'm drawing you pictures I showed you pictures of complete surfaces things which go to infinity and so on somehow the kids problem was when you take a wire you put it in soap and now you have forced the soap to span this wire and you want the minimal surface with this boundary ok so does it always exist well this is a very nice I mean some one of the crucial problem basic problems in the calculus of variations in 1930 for any curve and fixing some kind of topological type of the surface that you are looking at by two mathematicians an American and Hungarian Douglas and Rado for some reason which has never been completely clear to me the first fields medal you can dream I'm one and a half two-hold I stopped dreaming one and a half ago so I joined a big group of people who didn't win the fields medal ok I like to be in the majority but so the mathematical community so we don't have the Nobel prize so we invented another prize at the beginning of the 20th century which is called the fields medal which has to be given to one mathematician below 40 so the first fields medal in the history of mathematics has to be given to Douglas for solving this problem which is called the plateau problem plateau was the physicist who has set up some kind of basic physics to speak about this problem in the late 19th century so somehow it's also the beginning of this I don't know why they chose Douglas instead of Rado I mean now we consider the two proofs correct so probably at that time there was a more controversial issue they really came up at the same time with the proof so ok, that's it and we'll see on Thursday