 OK, zelo smo začniti. So, pilično je hrabe, da povedemo Prof. Frans Fračnerič iz Univeristija Fjubljana, pilično je se odnočil te nekaj kompleks analitici v rlješku poznatnih sorceri, če pletima, vsak z nekaj terbe. Če neznam, da so imamo, Klaudija, Aretzov, za invitacij, je to pilič, da sem pripravil, da sem regična. had one very pleasant experience here three years ago for two weeks we had a program a conference in a PhD school. Okay so today this is on, right? It's okay. So today I'll give a rather easy and non-technical talk about a subject that was new to me in 2011 when I first met these two people from Granada, Spain, Antonio Lercón and Francisco López and then we became collaborators and whatever I know about Riemann surfaces, I mean about minimal surfaces is basically they taught me and maybe I taught them some complex analysis we discovered that putting these two things together was extremely fruitful. So the talk is, as I say, it's going to be very non-technical kind of survey. The first 10 or 12 slides are kind of promotional. If you've never seen the stuff before then it's okay. If you have seen it then please just skip this part and then later I'll tell you about some new things which we did together using complex analytic methods. So first this subject of minimal surfaces is quite old. So in fact the first time when the notion of minimal surface was mentioned was by Euler in this year 1744 and this was on a kind of intuitive level. He thought of just as we think today that a minimal surface is a surface with a property that if you take some domain with boundary inside such a surface so the surface is sitting in some Euclidean space, maybe R3, maybe higher and you take a small piece of it and then you vary it in a normal direction, you perturb it and if the area can only increase but cannot decrease then you call it a minimal surface. And so Euler in fact gave an argument that the only area minimizing surfaces of rotation meaning you take a curve and you rotate it and you get surface then these are planes and catanoids and here is a picture of a catanoid. Then this topic was picked up by Lagrange who did first analytical work on the subject and he actually came up with this equation. So this is an equation for a function. You look at a graph of a function over a domain and you'd like to know when is it critical point of the area functional and this precisely means that if you take one per meter family of perturbations and you keep the boundary fixed, you look at the area each time so you find area as a function of t then you differentiate the time 0 and you ask that it be 0. This was the first time when variational approach was used so Lagrange came up with this equation which has to be satisfied which characterizes the stationary points of the area functional. This equation if you look at it, it looks like Laplace, but in fact it is the metric Laplacian, in fact we'll come to it later. And then we see a few years later showed that he was the first one who made a connection with this main point namely that the mean curvature function has to vanish identically. So we say today that a smoothly immersed surface in R3 is a minimal surface, if it's mean curvature function is identically 0. So what is the mean curvature? It's simply at each point of a surface you have two principal curvatures which means you look at the curvature of the curves going through that point in different tangent directions and there are two special directions, one is along which the minimum will happen and one is along which the maximum will happen, they are orthogonal to each other and those are principal curvatures then you take their average and that's called the principal curvature. The product is of course called the Gaussian curvature. So the connection is that the surface is a minimal surface even if and only if it's mean curvature function is 0. If you look at surfaces in higher dimension it's still a little bit like this except that then you have more normal vectors, not just one. If the surface is in R3 you have only one, you need normal vector or actually two, you can take depending on the orientation but if you're in dimension four or higher then you have N minus two independent normal vectors fields and you have to actually choose one. One of them you fix it and then you can calculate again this quantity is the principal curvature, principal curvature is the mean and the Gaussian curvature with respect to that normal direction but the theorem is still the same that all these curvatures, mean curvatures have to vanish in order that the surface is a minimal surface. Here is another picture of a minimal surface, this is called a helicoid. So you see it's essentially what you do is you take a line in the plane and you rotate it but then at the same time you lift it at a constant speed so it's just a screw, a hem at a screw helicoid. That's another minimal surface and then it was proved by Katalan in 1842 that the helicoid and the plane are the only ruled minimal surfaces. Ruled means they're actually unions of straight lines. Plato introduced a new topic in the theory because he started looking at the question when you take a curve in space, a closed curve, image of the circle, can you find a minimal surface in such a way that this curve is its boundary. This became known as Plato problem and in fact this Plato problem was solved in 1932 independently by Douglas and Rado and they showed that the answer is yes. That every continuous Jordan curve spans a minimal surface. This minimal surface might have some singularities. It's not immersed in general but still. Then Riemann also made a contribution to this topic. He discovered some interesting family of examples. Here is a picture. So this example is supposed to suggest, these are called Riemann's minimal examples. So they suggest properly embedded minimal surfaces which have countably many parallel planar ends. You see them here. End is what goes to infinity. So they are countably many, parallel to each other and every horizontal plane intersects each of them in either a circle or a straight plane. Topologically they are actually just plane domains. Now, let's now do some mathematics. If you assume that M is an open Riemann surface, how do Riemann surfaces come in? Riemann surface is coming as follows. If you have a surface in Euclidean space embedded or immersed, then what you can do is put the Euclidean metric, you pull it back, you restrict it to the surface. So you have a metric on the surface. Now let's say that the surface is also orientable. We know that metric, it determines conformal structure and that also determines the complex structure together with orientation. It determines the, because you have a unique, for each vector you have a unique orthogonal vector of the same length and so you determine the J operator and that gives you complex structure. So it's very natural to look at these things, at least if they are orientable as Riemann surfaces. And if you do this, if you take this metric, which I told you that you just pull back the Euclidean metric to the surface, then actually by the very definition your embedding or immersion becomes conformal from this structure which you defined on the surface by this metric to the Euclidean space. It's conformal. So it's natural to look at conformal maps. And then it turns out that if you look at the smooth immersion from a Riemann surface to Rn, we define it to be conformal if it preserves angles, the usual definition. Now, if we denote by H its mean curvature vector, I told you before what is the mean curvature if you have a surface in R3. In that case, this is a scalar function but you multiply it with a normal direction so you get a vector field. In higher dimensions you have also a vector field which is called mean curvature vector field. And in fact, there is a crucial relationship that the Laplacian of your map is simply twice the mean curvature. Here this Laplacian has to be the metric Laplacian. So it's this metric. You take the Euclidean metric, pull it back call it G. But you see, this simplifies actually this formula because usually you like to work in what's called an isothermal coordinate. There is a classical theorem going back to Gauss where the metric on the surface the simplest coordinates you can choose so the metric becomes as simple as possible are called isothermal coordinates and they are coordinates in which the metric is simply a multiple this is a function of the Euclidean metric. And if this is the case then it turns out that the metric Laplacian is related just by this factor. This is a positive factor of course with the usual Laplacian. So it doesn't matter really which Laplacian you use either you use the factor or you don't but here basically you have this formula. So the main conclusion that we now have is that if you look at a conformal immersion then it's minimal if and only if it's harmonic because h, minimal means h is zero and harmonic means this function Laplac is zero. And again it doesn't matter Laplacian or the standard complex Laplacian because it's only the scalar factor in between. Good. Now what is the connection with complex analysis? Of course as soon as you talk about harmonic functions or harmonic maps it's somehow clear that complex analysis is involved but it can be made a bit more systematic. So if x is a smooth function of a complex variable now I'll call it zeta which in fact I think of this zeta as some local coordinate on a Riemann surface. Then we have these two derivatives. This delta of x which is written like that this is the one zero part of the external derivative so this is the c linear part and then you have the d bar of x which is the zero one part the anti linear and in fact holomorphic functions are characterized by this equation the d bar so the second derivative is zero this is the standard Cauchy Riemann equations and that's equivalent to saying that the exterior differential is just del. Then x is harmonic if Laplacian of x which is really the same thing as this operator 2i d d bar is zero this is just a little calculation and so this means that another equivalent condition is that del of x is holomorphic because del of x is a one zero form and here you see that del bar of it has to be zero this is just equivalent to saying that this is holomorphic so upshot the two things together is that if you look at an immersion no that's not an upshot this is discussion about holomorphic and harmonic that point namely that an immersion from a Riemann surface to R to Rn is conformal meaning angle preserving if and only if this is true you take partial derivatives of x on u and on v u and v are the real and imaginary part of the local coordinate and conformality means that these two partial derivatives which are vectors are orthogonal to each other and that's clear this is just conformality going from u plus iv to the euclidean space but now the good point is that we can rewrite this condition as follows that we take this delta of the components of x so this derivative the holomorphic derivative and we square them and we sum them up and we get zero and this is very easy to see x is a real map so x is real so this is the real part this is the imaginary part what happens when you square it you get these are vectors so what you get is a dot product of this vector with itself so you get this quantity x on u square and then you get if you look at the real part of these squares you also get because you have this i here so what you get is the real part of this quantity you get precisely the difference of this length length of the x on u square minus x on v square this becomes the real part of this expression and the imaginary part really becomes this expression because the imaginary part is you multiply this vector with that vector and once more the other way around so you can encode these two real equations into this complex equation that is crucial because now so now the upshot is that the smooth immersion from my Riemann surface to Rn is a conformal and minimal both at the same time which means conformal minimal surface if and only if first of all del of x so the one zero derivative this vector this is a vector valued one form is holomorphic and the sum of the squares of the components are zero okay so this is an analytic characterization of what is a minimal surface now I can fix maybe I don't like to work with forums instead of working with one forums it's easier to work with functions so what I do I can fix no over vanishing holomorphic one forum on M and you see such always exists because M has to be an open Riemann surface M cannot be compact by maximum principle you cannot put it a compact thing inside Rn by such a map so if M is an open Riemann surface then it's a very classical theorem going back to 1967 it's the so called Gunning and Erasm can theorem which says that in fact you have such a forum actually Gunning and Erasm can tell you more they tell you that every open Riemann surface admits a holomorphic function which has no critical points so this holomorphic function gives you an immersion of your Riemann surface to see and then you can simply take the differential of this function for this forum so you even get an exact forum that has no zeros and if you use any such forum then you can actually divide you see if you divide del x by such a forum then in fact you get a map into this into this quadratic variety so you define this is called a null quadratic for some reason this is the quadratic of all vectors in Cn where the sum of the squares of zero is following this equation and then the upshot is that how do we represent conformal minimal immersions everyone is of this forum so you start with some map f holomorphic map from your Riemann surface to this null quadratic except I take away zero to take away zero basically means that I will look at immersion solely if you allow value zero then you allow branch points so you go to puncture null quadratic this gadget without zero take a holomorphic map you multiply it with this one forum so we have f theta this is now a candidate for this del x but what is the condition that you can integrate it well the condition is this one that the integral of the real part of this one forum has to be zero over all closed curves otherwise the integral is not well defined and if you indeed know that your thing comes as a differential then of course this is satisfied differential when you integrate it on closed curves it's zero okay so this is called a vajerstels formula and this vajerstels formula which tells you everybody you are interested in all your objects are of this forum so they are essentially integrals of products of a map holomorphic map from m to the null quadratic with your fixed one forum the condition that you can integrate is that that the integrals over closed curves we call these periods real periods they have to be zero then you can integrate and you get a conformal minimally merchant because it's derivative del of this is actually this thing underneath it's f theta okay so this forum is the link so as we see now what is our task if we want to produce minimal surfaces from parameterized by remote surfaces well we have to create maps to this null quadratic without zero such that this condition holds okay and this as soon as I was told this I said well this is something I know about because I had been working the previous decade a lot on something called orca theory and orca theory is a theory about about the problem when you want to know let's say that you want to construct holomorphic maps from open reman surfaces or more complicated objects in higher dimension called stein manifolds stein manifolds into some complex manifold so you want to know when can you construct a lot of maps and the question is having a continuous map when can you change it by homotopy to a holomorphic map and you see this question is rather classical and was studied first by orca already in 39 then by Grauert in 57 and 58 and in fact the main theorem which I need for this minimal surface business but before I get to that let me finish the story anyway let me finish the story so here we were talking of maps to aren we were looking at this condition that the real periods have to vanish but you see you can also look at holomorphic immersions to see n which whose differential satisfy the same vanishing condition and again you conclude that every such is of this form now it's you integrate f theta without taking the real part because now you really want a holomorphic object and what now f should be it should be a map which satisfy now the full period to be zero not only in the previous slide you see we had a real period real part now we have a full period that should be zero so the conclusion is that if we call such curves holomorphic null curves just because they are directed by this null quadric this means the tangent vector at each point is in the null quadric that's why they are called null curves and now we see the connection immediately that the real and imaginary part of a null curve are conformal minimal surfaces in Rn and not only the real and imaginary part but you can rotate these are called associated surfaces and what is the converse well if you have a conformal minimal surface then it is locally in fact on every simply connected domain in M the real part of a null curve what is the problem in general is that as you know with harmonic function you cannot take harmonic conjugate in general you have conjugate periods likewise here this is the same problem you can take a harmonic map so it will have its differential has vanishing periods but it has the conjugate differential has non-zero periods maybe and that's called flux these periods are called flux in this theory so here is a nice example you have seen before the picture of a catanoid and you have seen a picture of a helicoid and it turns out that these two are closely related they are actually called conjugate minimal surfaces because they are the real and imaginary part of the same null curve and here I wrote down the equation this is not the only possible representation of it but it's a simple one cosine z, sine z minus i z you multiply with e to i t take the real part you call it x t so this is a minimal surface and look what it is it's a combination with cosine and sine of these two columns now the first column you see here cosine hyperbolicus this is a catanoidal curve I mean y and cosine hyperbolic y this is the catanoid and this multiplying with cosine and sine means that you rotate it in the space so this is a catanoid and this one is a parameterization of the helicoid so when your time varies at time zero you have this curve at time pi over 2 or minus pi over 2 you have this curve and this is a helicoid it's also easy to see because you have sine and cosine and x so this means you're really rotating a line and lifting it at the same time and now I wish to point out that in fact this connection how to use Weisstrass' formula this has been really known since the early 60s when this particular man Robert Osserman pioneered this approach in modern time so in the early 60s he discovered this and he wrote a very beautiful book which is even today one of the classics in the subject Osserman was actually more like a complex analyst he was student of Ileforce and that's very much prominent in his work and he's a great expositor so this book is really wonderful ok, now I can finally I wish to say that even though this was known in the 60s that complex analysis could be used but for some reason it wasn't used very much so when I first learned this thing this was back in 2012 maybe when I learned this thing from these two people in Spain it was to me incredible how come that these things after all were not so hard to do how come that nobody has done them I was really amazed and also some people from Russia told me that in fact in Moscow they also knew this work of Osserman and they knew that he proposed this approach complex analytic but somehow it wasn't really developed so here is some but let me tell you let me ask you when do I stop so I don't want to one hour I don't want to be too long ok, so here are some topics I'd like to discuss which we found in probably now about ten papers by now together with these people and one two of them also with my colleague Barbara Adrinovac, who is from Ljubljana so one topic is the Runge-Mergelian approximation theorems for conformal minimally merchants so what this means I'm sure you know because Runge and Mergelian these are classical approximation theorems for holomorphic functions but it turns out that the exact essentially the exact analogs don't prove for minimal surfaces next thing we constructed proper conformal minimally merchants so I now abbreviate this as Cmi into Rn and minimally n in some class of domains which we call minimally convex and again this mimics something in complex analysis which is called Levy pseudo convex domains so it's a kind of analog to Levy pseudo convexity this minimal it has the same role for a minimal surface notion plays the same role for minimal surfaces as the Levy pseudo convexity does for complex curves another topic is a very famous one it's called Kalabiou problem but this is not Kalabiou manifolds but it's a it's a Kalabiou problem for minimal surfaces I'll talk about it when I come to it and then we have some new results on the Gauss map and we have other things but I couldn't put everything in one talk so let's start with this theorem here I'm just recalling what is a classical Runge theorem what is the classical approximation theorem in for holomorphic function so we call a set holomorphically convex if in a Riemann surface if the complement has no relatively compact connected components no holes as we say if m is equal to c then of course this is connected in such case and case called polynomially convex so the complement has to be connected and the classical theorem due to Runge is for functions in c so that every so that you can approximate holomorphic functions on compact polynomially convex sets by entire functions and this was generalized to all open Riemann surfaces in this year 49 by Benkenstein in fact they proved the classical result that every open Riemann surface is what we now call a Steinmanifold so it has lot of holomorphic functions and in particular you can approximate holomorphic functions on such sets and here is then our theorem why I give two years it's because Anarkon and Lopez before we actually met they proved this theorem in dimension 3 with some kind of ad hoc methods and after we met we discussed and then we found a much better proof which works in every dimension so the theorem is you take a compact holomorphically convex set in an open Riemann surface and you take a conformal minimal immersion so a minimal surface defined on a neighborhood u of k we can approximate it on k by globally defined minimal immersions here I am outlining a proof I said it will be a non-technical talk I think some ideas are really easy to explain so I am outlining some few things so we start with conformal minimal immersion from some open connected open set u containing k we can reduce to the connected case so now recall the Schwarzschild's formula which says that in fact x is given as an integral with some initial it's an initial value here at some point it's an integral of the real part of f theta where f is a map and this is a mistake it should be u it's a holomorphic map from u into this punctured nullquadric and the real periods have to be zero in fact why I call it d because I have picked now a domain d is now a compact domain smoothly bounded which is between the set k a nice domain then I have a basis of the homology group first homology group is in a finitely generated abelian group so called z to the l I take a curve switch for a basis for this first homology group and I define this so called period map period map takes holomorphic functions on this domain actually maps to cn to this space cn to l where l is the number of these curves and what is the component each component of this period map is simply that I integrate f theta or the jth curve theta is some fixed one forum fixed ones and for all like in the previous slide and then I note that one forum is exact if and only if this map is zero because it just means these integrals are zero and the real part is exact if and only if the real parts of the periods are zero so im gonna use this period map and here is the crucial lemma this is something non technical given a non flat holomorphic map from this domain into the null quadric and non flat just means that it's not contained in a ray you see this thing is really a quadric right so it's a union of rays lines complex lines through zero and those maps which actually are contained in a ray are a little bit a problem in this theory but not really because what they correspond to if you integrate them they correspond to surfaces which are flat which actually lie in a euclidean plane so those are not so interesting so we can avoid those really so if I have such a non flat map then I can find a neighborhood of the origin in this euclidean space and the holomorphic map from the cross this neighborhood for time and this is parameter now I call it t t is a member of this v so for parameter value zero I get my original map and if I look at this family of periods so I vary t I look at the period map and I take the derivative on t at time zero so I take first variation if you like of this period map on this family it's a map from here to there and this is an isomorphism so the thing is that such a thing exists so in other words I'm saying give me any map I can vary as long as this map is not flat then I can vary it through a holomorphic family of maps of appropriate dimension such that my periods vary in all possible directions so periods become period map the derivative of it is an isomorphism and this is not hard to do because how I do such a thing I use vector fields flows of vector fields to vary in different directions so it's a sorry I went too fast okay so this lemma is not hard to prove but now let's go on we can also observe that the null quadratic has convex hull just geometrically convex hull equal to all of cn that's actually a general effect about any algebraic subvariety algebraic subvariety of cn it's convex hull is actually a linear subspace it's not true for entire subvarities but it's true for algebraic and then one can show easily that if you integrate g times theta theta is this one forum over all possible loops c is a circle so we integrate over loops in the null quadratic can you look at what are all possible values of these integrals and actually they are all of cn and this is a simple idea which is the basic idea of Gramov's convex integration theory in cn I can write it as a convex combination of points in my null quadratic and now what I do so I have some weights it's a convex combination some pj times qj times the points then I construct a curve which goes quickly to the first point spends roughly the correct amount I'm there goes to the next point to the next point and so on so that gives me roughly the right integral so anyway it's not hard to prove this so by considering such deformations for all loops in the period basis we can create a smooth period dominating spray so a map like in the previous slide so initially it's only defined over the union of these curves cj is missing here so this is the union of cj which has this map as a core but there is another standard fact that these loops cj these curves can be taken such that the union of them is holomorphically convex so once I created some functions on them I can use Mergelian's theorem to approximate them by functions which are holomorphic on D and that gives me the spray as in the Slema so in other words I get the spray simply by constructing a good thing on the curves themselves and then I approximate ok, how do I use the spray? well, now comes Groward already mentioned him ok, this nullquadrik has a rather important structure namely it's a homogeneous space of this particular group, Lee group it's called a complex orthogonal group so these are complex matrices which satisfy this relation without the conjugate you see, just a at equal to i so identity complex Lee group and this group acts transitively on the nullquadrik, except at zero of course so the punctured nullquadrik is homogeneous space and here Groward's theorem says that anytime you have a Stein manifold Stein manifolds are like open Riemann surfaces or actually any manifold which is embeddable into Euclidean space maps from Stein manifolds to such a homogeneous manifold satisfy all sorts of properties for example the Runge approximation theorem in the absence of topological obstructions so as long as there is no obstruction to approximate map on a subset by map on the whole manifold you can do it so what I can do now is apply this theorem to approximate my map which is this period map which is defined on this domain in m cross v, v is a subset of Euclidean space nullquadrik by a globally defined map and since derivative of x was in fact the real part of my original form we have that the real part of the period of f is zero this is from the initial condition if f is close enough to this approximation map is close to vf then of course when I approximate periods will change but since the initial period was zero and my family of maps was period dominating meaning that in the family I had all nearby periods so once I approximate I can find a nearby parameter value t such that this map f of dot t which I now call f tilde still satisfies the same equation that the real periods are zero and this is crucial because now I can integrate it and after I integrate it I get a map except there is a hitch here namely I can only do this on a bigger domain d prime which is such that it retracts onto d such that d is a retract of d prime in other words if I change my domain in such a way that there is no change of topology then I can approximate map on the small domain by map on the big domain which is a change of topology then there are new curves and then I know nothing about periods on them so I cannot integrate maybe so that's the moment not clear but at least if there is no change of topology I have this approximation theorem so this x tilde already approximates x and here I explain you a little bit how I deal with a change of how I deal with a change of period with a change of topology but maybe I better draw a picture because it's essentially just like this you have a domain and to change of topology is described by something like this maybe this bigger domain so it's described by attaching an arc to the previous domain but that's one particular change but this is the main one that's when you take a more exhaustion function that's the point of index 1 that's exactly what happens that you have to attach an arc so what do I do well, I have my map already defined on this domain now I have to extend it smoothly across this arc in such a way that its integral is correct because finally I'll have to integrate so I need to have that the integral of my forum on this arc is actually equal to the difference of values of the already given map in order that the integrals will come out right and after that I more or less repeat the story which I told you so I put this thing into a spray approximate by mergelian here I have to use not Runge but mergelian on this object and finally I find a map on a bigger domain which satisfies the period condition also on this loop you see there is a new loop here this arc is perhaps a part of a new loop and it satisfies the condition also on this loop ok good so now I am in business and then I just have to do induction I'm not gonna read the rest of the slide ok and then after this theorem we said ok how about proper maps, right that's more interesting because proper means that you put the thing so that the boundary goes to infinity so the image is closed much better object if you look at proper maps and in fact we realized that with this technique which I just explained you plus the usual technique how you construct proper maps we could in fact prove this theorem so there are three parts I take any open Riemann surface first part says that if you want to go to R3 there are a lot of proper conformal minimal immersions in fact proper immersions form a dense subset in the space of all conformal minimal immersions in the compact open topology so you give me any any minimal immersion any conformal minimal immersion from a good subset of M I can approximate it by globally defined one which in addition is proper so the boundary goes to infinity ok I'm talking of immersions here and I will show you later a result that in general this is not possible with embeddings in fact on the later slide there will be a theorem about embeddings but this is just immersions now what if you go to R4 instead in R4 the natural thing you can expect and we also proved it is that there are such things which can only simple double points meaning transverse double points ok if we go to 5 or higher then we can create embeddings always so here is a comment which I mentioned before in fact there has been a lot of works but maybe the crowning work is this one Mix, Paris and Ross from a few years ago they show that in fact planes, flat planes catanoids, helicoids and Riemann's examples which you also saw on one of the slides are the only properly embedded minimal planner domains in R3 as far as embedded you are much more restricted of course now these same people I believe are working on further results because one would like to know how about if it's not a planner domain how about if it has some genius it's a domain in some more complicated Riemann surface well I think they have some results already but it's story is not complete and there is a corresponding there is a problem which we post and we don't know how about it go into R4 question does every open Riemann surface embed into R4 properly as a conformal minimal surface we don't know and there is a connection with complex curves because in fact complex curves are also minimal surfaces special cases of minimal surfaces so there is a corresponding open problem which is well known since long time ago called bell forced and a simpkin conjecture actually conjecture that the answer is yes but I'm not sure that it's true so I rather call it conjecture does every open Riemann surface admit embedding into C2 that's also open of course if you could solve this one yes then this would be also yes or if this is no then this is no because there are more minimal surfaces than complex curves in this context I wish to mention that Erlen Wald from Oslo and myself have two papers in recent period one of them is 2013 one is a bit older which gives in fact the two best known results still at the moment about which Riemann surfaces will embed into C2 for example we prove that every circle domain possibly infinity connected in C this means you take from C countably many disks in such a way that you get an open domain such a thing is embeddable but the question is wide open and while doing this thing this I mentioned just as a curiosity there is a famous example of a mix from some I don't know 15 years ago maybe of a immersed minimal maybe strip immersed into R3 which has final total curvature in fact minimal possible final Gaussian curvature ok we found an example where it's an embedded minimal strip in R4 seems to be the first known example very simple map actually just have to do some checking the main problem was not to check that this map is good it's just how to come up with the formulas I'm talking about minimal strip so here I take rimless surface to be C star and this is map from C star but in fact it's you see it has this property that two points are identified under this map if and only if the point is either Z1, Z2 or Z1 is minus 1 over Z2 bar and this map Z send into minus 1 over Z bar is the reflection around the circle is this reflection around the circle so what happens when you in fact no in fact it's not this reflection this would be it has to be the opposite so it's this one here in fact if you look on the circle itself the antipodal points get identified so that's why you get a maybe strip now I promised something about minimally convex domain so what is this minimal convexity I was asking myself you see since I having background in complex analysis it's natural to ask what is a good class of real functions on which your objects in complex analysis maybe complex curves or complex manifolds on which you have maximum principle so the good class of functions for complex analysis are the supharmonic or the pleurisupharmonic functions right? Supharmonic functions are precisely those whose restriction on curves will be actually pleurisupharmonic are those whose restriction on complex curves will be supharmonic and so we identified such a class of functions on the real domain I call a smooth function minimal pleurisupharmonic if the following is true you take all the eigenvalues of the Hessian of rho so this is a real function you take its Hessian you take the eigenvalues you order them and you take the first two the smallest two and you ask that this be bigger equal to zero so it means only one eigenvalue can be negative or the rest non-negative and then basically the main thing is that we showed that rho is minimal pleurisupharmonic if and only if the composition with every conformal minimal surface is a supharmonic function after we came up with this class we have realized that in fact we were not the first ones we found some papers by Harvey and Lawson where they also treated this class of functions and they did some general things but we at least had a good theorem after that I mean we had a theorem where this was used so what is the theorem I am stating only a special case for R3 just to be simpler so if I have a domain in R3 minimally zero convex in the sense that it admits a minimal pleurisupharmonic exhaustion function so one satisfying this condition then every then if I have a compact border R3 surface so this means R3 surface with boundary with finitely many boundary components then every conformal minimal immersion of the surface into the domain can be approximated by proper conformal minimally mergers from the interior so really this means I can if M is entirely together with boundary lying inside I can now stretch the boundary to the boundary of the domain that's what I'm saying I can create proper ones and in fact simple example show that this is the best class of domains in which I can do it and this theorem is actually which says that if you take a hardtox or levi-zero convex domain then any compact R3 surface which you put inside is a complex curve you can stretch the boundary out to the boundary so you can make it proper but ok so it's similar but in fact to prove it was much harder and there is another idea so I'm left so instead of reading through this slide I'd like to draw another picture we introduced a method which is coming from complex analysis called Riemann-Hilbert boundary value problem it's one of the classical subjects of complex analysis since 100 years ago but we found a way of using it in this theory so what we do with this thing so we do the following let m be such a compact Riemann surface with boundary so it means one of these things m is something like that because of few finitely many boundary components now you map it in some space how do I call this map maybe I call it x I think here I mean R3 but I could be in Rn actually we have versions of this thing for Rn so if you think of this picture let's say that this is my x of m so it's sitting as a curve with boundary in R3 yes now what I do is I also have this map which I call f in the slide and what does this map do map says take a boundary point of this first curve and put a conformal minimal disk attach to it a conformal minimal disk so this is a disk which center point is here this is my f disk f of p c and c is the variable on the unit disk and I attach these disks around so basically here I have also one at each boundary point in fact I do it on an arc and then I say that in fact what I can do with this figure now I can approximate the central curve by a new curve image of m which does something like this it's very close to m most of the place and then when you come to the boundary it will go almost to the boundary here so this yellow thing now is a x tilde of m so this is a new conformal minimal surface such that x tilde approximates x on as big complex subset as you like of this m could then inside but at the boundary you go almost to the stores ok so this in complex analysis has been well known for a long time and it's not very hard to do not to solve the problem exactly but not approximately but it was much harder to adapt it to this minimal surface case but if you do such a thing then you see now you have technique for doing many things because you see the way you attach these disks this means that you are changing the boundary values from these to those so for instance if you manage to pick your disks in such a way that this exhaustion function is bigger on the boundary than on the center then when you do this game you have actually pushed the boundary of the surface to higher level of your exhaustion function so doing this inductively you push it all the way to the boundary that's one strategy but this thing so I will skip all the rest of this explanation but this strategy was also used and this I'll finish with then with this subject I just want to go through the slide and then I finish this technique turned out to be extremely useful in the subject which is called Kalabiyao problem what was Kalabiyao problem so actually let me first read the theorem ok we take a compact border trimmer surface so one of these we take a conformal minimal immersion of the closed surface inside Rn and then we say that it can be approximated uniformly on M by continuous maps to Rn such that in the interior in M interior this is a conformal minimal immersion and in addition it's complete what does this complete now mean this means that it's a complete metric space in the induced metric from the Euclidean metric ok, what does this intuitively mean that it's a complete metric space this means that this reman surface which I created I mean this minimal surface which I created it stays as close as I like to the original one but it is very, very wiggly at the boundary it has lots of waves it's so wiggly that if you take any curve inside the surface which goes out to the boundary point and you measure the length of this curve in the Euclidean metric ok so these surfaces are staying in a bounded set but they are complete and this has been so this question was first asked by Kalabi in 65 so he asked, actually he conjectured that this cannot be done his conjecture was that if you have a minimal surface in a bounded subset of Euclidean space that it cannot be complete ok, so this was Kalabi's conjecture Kalabi's conjecture was disproved in a very crucial way in 96 by Nadirashvili in fact he found a complete bounded minimal disk in R3 there was a later work there were many works in this area why is Yau mentioned because Yau wrote a big survey paper on problems in geometry the so called millennium paper in 2000 and this was one of the things he discussed and he outlined the remaining problems there was a paper by Martin and Nadirashvili who claimed to have constructed complete minimal disk with Jordan boundary but the proof is not complete the way it's written actually we think it cannot be completed so we introduced a new method by which this can be done you see because what we have is not complete but the boundary is Jordan boundary and you have to work very hard to have this Jordan boundary it cannot be better than Jordan because you see by the isoperimetric inequality the boundary will be infinitely long everywhere it's completely non rectifiable curve otherwise interior couldn't be complete yes I also want to mention that in fact Kalabi was partially right because Kalabi it seems that he kind of had in mind embedded surfaces and in fact it was proved by Kolding and Minikotsi in a series of papers that the Kalabi Yau that actually it's true for embedded surfaces not they haven't proved it in complete generality but for instance for surfaces of finite topological type meaning finite genus, finite number of ends they prove that if it's embedded as a minimal surface then it cannot go to infinity it has to be proper they actually have kind of reverse estimate which says that you cannot increase the length very much in a limited space while what we do exactly is to show that if you immerse or if you go to higher dimension then you can wiggle as much as you like in a very limited space as little as a spot fine and then I have something about Gauss map but I'm over my time and you know what Gauss map is it's just the usual Gauss map of a surface right but turns out that for minimal surfaces Gauss map is in fact a conformal map it's in fact a metamorphic function on your Riemann surface you can think of it like that and what we proved is that actually every metamorphic function can be the Gauss map of somebody so for each metamorphic function you take there is a conformal minimal surface which has exactly this Gauss map and this I was also surprised that after so many years of the theory that such a basic thing wasn't known but anyway this was done with these new techniques and okay I stop here thank you very much for your attention