Closed g2 -structures
Fino, Anna
I will review known examples of compact 7-manifolds admitting a closedG2-structure.Moreover, I will discuss some results on the behaviour of the LaplacianG2-flow starting from a closedG2-structure whose induced metric satisfies suitable extra conditions.
 So first of all I want to thank very much the organizer for the invitation and the opportunity to give it all here. I will talk about closed G2 structure so I will quickly remind what I mean for first G2 structure. Maybe many people already know but let's introduce again. So we will work on a seven dimensional manifold and for G2 structure I will mean three four seven manifold which is stable so that means that if you take this orbit under the action of the linear group this is open but this is not sufficient to have a stabilizer for the free form G2 because in general there are two open orbit and what we ask is some non-degenerate condition plus non-degenerate that means that if you take any vector field and you take the contraction a x phi wedge a x phi wedge phi this is non-zero. This is actually the condition to be non-degenerate and that means that then you have a non-zero seven form and up to a factor you have defined a positive being a four given by a very dangerous with a chalk I lose the chalk I take given by a x phi wedge a y phi wedge phi and here of course we have also the volume four. So that means that phi on the same time the term the metric plus the volume four. The stability for free form on seven dimension are similar to the stability of two form on let's say vector space of even dimension. In fact this is a part of what you have when you have a symplactic form and we will see some difference between closed to structure and symplactic form. So let's put here the name closed. Why? Oh this is too high for me. I am sorry. Okay. Okay. It's okay I will not I will not move again so that's fine. So the idea is to follow why G2 is interesting because it's one of the possible possible. Alonomy get up by the release of a compact manifold compact manifold when you know symmetrical force and when we have a long of it contains on G2 this is actually criminal and to ask that phi our free form is closed and co-closed. So this is zero plus the condition of the star where the star will be all the time the star odd with respect to the metric induced by phi. This is also zero. Okay. It was quite hard to find compact example with this with a long of equal to G2 in the talk we will forget about this condition and we will consider only the condition for the free form to be closed. That's what I mean for closed G2 structure. So the condition is weaker than the other one but we see that in fact is anyway strong condition because for instance there are no topological of suction and in the talk I will talk about such close G2 structure especially on the groups so only algebra. So the idea is to consider phi will be left in volume G2 structure on a group G and that means that we are consider G2 structure on the algebra G of G and we will see also some condition that one can put some restriction that one can put on this metric induced by phi like high sun condition or more general the condition related to the rich tensor what is called extremely rich pinch that was introduced by Robert Brian and we will see a little bit how is the behavior of closed G2 structure especially in relation to this last condition when you evolve closed G2 structure because there is Brian, Robert Brian introduced also a flow, a geometric flow that allow you to evolve a closed G2 structure and again we will see how the groups have special role for that. So let's where is here so let's start so the idea of the talk is I will review a little bit what is known about closed G2 structure in general and then we will see what happened on the algebra and we will see many different from the case for the algebra and the last part I will talk about Laplacian flow and this condition about it. So I already mentioned what I mean for G2 structure so we have a phi4 phi and in any point the stabilizer is isomorphic to G2 as I said before and as an explain then you have automatically a metric and the volume four that's a nice thing of G2 structure then we test Raffin and as I already mentioned was showed by Fernandez and Gray that the condition for the free form to be parallel is equivalent to the condition for the free form phi to be close, the point that is here okay phi to be close and co-close and that is that equivalent to the fact that the volume of your metric is as a morphic to a saguolo pojic two. Okay and a nice thing of course when I mentioned parallel or torsion free G2 structure I essentially mean that case when phi is close and co-close in this case the metric automatically which flat that was proved by Bonin. Okay so we saw that G2 structure is close it's called close or also calibrated and if phi is close and of course we when we talk about closed G2 structure I will always mean that phi is not co-close and you can see from from that expression so in this case we have that the differential of star phi which is a phi form you just get by tau wedge phi and then if phi anything will be encoded by this two form tau and tau belongs in fact in general if you take the space of two form on your standard space and you deform irreducible component with respect to G2 you have two reducible components one is a seven dimensional which is isomorphic to R7 and the other one which is 14 dimensional is isomorphic to the Lie algebra G2 so in this case tau is isomorphic to the space to the irreducible component on the space of G4 which is isomorphic to the Lie algebra of G2 and that is actually the situation that we have that means that if you take tau wedge phi algebraically that means that is equal to minus star tau and also you have these relations so the wedge of tau with star phi is zero but the interesting thing you can see from here since tau is essentially the cut of your phi then tau is co-close but the interesting thing here that if you take the this ultralaplation of phi with respect to the metric induced by phi then is exact so it is exactly the differential of tau and that's an interesting point that allow Robert Moran in fact to define this geometric flow evolving closed G2 structure is actually using that because you evolve and you evolve in the center and class of the starting closed G2 structure okay and the name calibration comes from Arby Lozub because phi define a calibration so it means that if you take any tangent oriented plane and you restrict phi to that then this is less than equal than the volume and then you can talk of course about calibrating some manifold and so on but I will not talk about that in this talk okay rich tensor since phi the terms the metric then one can expect that the the the rich and that's what's the general formula that Robert Moran found for more general G2 structure but in this case you can see essentially that the rich tensor sorry the rich tensor is completely determined by tau you can see from here you have an a part which is a multiple of the metric this is just the norm of tau squared and then another part which is obtained by this free form the free form is this tau minus 1 over 2 star tau and tau and you use the G2 map between volume up very natural again by contraction this is a map of from free form to symmetric to tensor and just given by you take the contraction as before the two terms are exactly the same like here you wedge by beta and then you take the star and then you get something that is symmetric and that's the first problem that you get in this case because you don't get any restriction on compact case of the existence of such structure because they as you can say from this expression the scalar curvature is less than equal to zero so that means that the complement for you don't get any restriction on the existence of such close G2 structure okay before I say let's see what happened if you impose some condition on the metric induced by phi and the first thing is that this is dangerous the first thing that one can see is what happened if phi the metric induced by phi is Einstein and in the compact case was proof in two different way by Clayton even though and then in different way by a rubber Ryan that if you impose a the metric is Einstein then automatically tau is zero and phi is far but there is a nice inequality that one as that was showed by a rubber Ryan that involve you see the the integral of the square of the scalar curvature with the integral of an M of the north of the Ricci square so one can ask okay it's possible to have the equality we'll see that it's possible and the condition to have this equality it means that if you take your branch of a relation you have this equality if it only if the differential tau is exactly equal to this expression so you have a part again multiple of phi plus a part which is multiple of star of tau which tau and this is what is called a stream a rich pinch and we will see that we'll have a nice behavior if you evolve close G2 structure using the Laplacian flow of course this makes sense this condition makes sense also that even not in the compass case but the level of left invariant group and so the level of the algebra okay so the first thing that one can do always like that at least for me is okay let's work on manifold symmetry right so the first question that one can ask can exist compact homogeneous manifold with an invariant G2 structure close G2 structure and that's what is no so as I said before there are no known result about the distance of close G2 structure on compact compression manifold not restriction at all but what we can do is to see some symmetry so let's consider the automotive group or your clothes or your manifold with a close G2 structure that means as usual that you take the group of the film or feel which preceded before right so they pull back and F phi is equal to phi if we look at the Lie algebra the Lie algebra can be defined in terms of the linearity so adjust the vector phi for which the lead derivative of phi with respect to x is zero and what for the style of the show that in the compact case if you have a close G2 structure which is non parallel so just thousand zero and if you have an element of such Lie algebra of a automorphism group then the two form I x phi is not only close but is also close and then you have a restriction because you have a restriction on the dimension of your automotive group is less than equal to the second but in armor of M and also less than equal to six but the important thing that is also a billy and with this you can conclude that there are no compact homogeneous example not only that but well by compact homogeneous example I mean that you have a group acting transitory and this is a subgroup of the automorphism group of your manifold with the close G2 structure they conclude more even for commogeneity one in fact you have only the seven hours but you don't talk about that okay first relation with the groups are given by example because the first example of compromise for which amit a close G2 structure but no parallel G2 structure was found by Marisa Fernandez and in fact it's a compulsion so when I mean there that you have G your group you say gamma a lattice so a dice creates a group such that the quotient G over gamma is compact and that's what is actually the first example of such manifold and later Diego Conti and Marisa Fernandez classify when I mean classify I mean here I mean that they work at the level of the algebra okay and they classify up to isomorphism then importantly algebra amiting such close G2 structure and they found only 12 classes there are also example later constructing the solvent case like almost a billion groups so it means that these are group of semi-dark product on this form or semi-dark product of our six and that one were found by Marco Freiber and later we Marisa Fernandez is also Robert Ryan the laureate found more example about commogeneity one there is also a construction of closures to structure not in the compact case but in the metric is complete and they are commogeneity one of a compass in polygroup so it means that compactness is much stronger than completeness for commogeneity one and recently with Marisa Fernandez and Kovalev and we said the whole lot so we construct an example which is different from the previous one is not locally homogeneous but is constructing adapting the Joyce construction not when when you use so you use R before metal you resolution of R before but instead of starting of T7 of this flat dollars like Joyce said we start with an in manifold so a compacution of an imposter group by lettuce and then again we take a final group and then the nice thing here that you see all the previous example are not form up in the salamium case and this is form up and we are still working on that I think we can get maybe an example we be one zero that would be nice because it is completely open that but also this case is some sense new okay so we saw before compact plus if I sign is not possible right we cannot have a compact homogeneous but then one can ask okay what happened that was something that I studied with Fernandez and Victor Manero what happened in the non-compact homogeneous case that the first thing that is still open to find a non-compact example for which which I made a closest structure for which the metric is I saw this is completely open even non-complete is not clear if there exists anyway we work on non-compact homogeneous science a manifold and that's fortunately in in dimension seven they they conjected by Alexie's kid that the answer and soul manifold exhaust the class of non-compact homogeneous science a meaningful was show by a royal and La Fonte sorry is still open in some sectional case and also in I get dimension than 11 as far as I remember but in seven dimensions that's exactly the case so here I use soul manifold soul manifold is not compact in this case but just the terminology that they use so I mean simply connect this solvable Europe with a left in body a match and what we show that in fact if you have such soul manifold I sense or manifold in some sense if the metric is I sign then this implied that the metric is flat so you cannot have that in fact we show more general even if you impose instead of fee close this is still true even if fee is co close which is quite straight because co close is much weaker than then close and the idea of proving that use nice theory that was show before very strong theory in fact because one strong result that will show by your galore it's that the ice and soul manifold are standard so that means that the the algebra or your group solvable group has is a splitting of these four and it's just a new radical plus orthogonal complement a which is a billion and so another thing that one can use that one can always reduce to let's say rank one rank is a dimension of this a billion complement of and and everything in fact is determined by it we coincide also with the commutator of it and that's that's important for us because essentially by a result by Haber that prove before the result by Loret was just for standard I sense or manifold but we know now by Loret result that the standard the standard is a metric is unique up to isometry and scale and then we use we reduce to find more of section we reduce the dimension from seven to six just using the tricky that anytime you take an a in the complement of n or togola or togola sorry or normal unit or number one then I will say in the order of complement you have a natural ratio free structure and then using that you you you can arrive to the to the results so it's sort of reduction dimension six okay so as I said before simple t structure and closest to structure can have some similarity let's say that simple it doesn't induce a metric you need to to impose an extra metric but we can see that they are completely different on at the level of the algebra especially for the unit modular case because we'll show by two and Lichnero which you know that if you have a unit modular the algebra so the trace of the joint at x is zero for any x and you with a simple structure then it's to be solvable and this is the first difference with a fellow we show that in fact it's possible to have non-solvable only modularly algebra with this property and the courier thing that you can see from here okay it's not surprising that you have this pre thing free plus four so this is the radical maximum ability unsolvable here and this is a semi simple part and the semi simple part the strange thing of this we didn't prove this geometrically but the semi simple part can be only as two are and then we also classify up to is almost how many algebra we get we get essentially four algebra a only one is really not a direct sum or two pieces okay now I show another difference okay this is why I am looking for exact closely to such a because it's completely open in the compact case it's not clear okay when you have a simple to form you know the simple to form on a compound manifold give you a non-zero there on class but when you take a copper manifold seven manifold we're closely to such you don't know if the class the class determined by fee is the exact or not and there is no nothing about that for instance one doesn't know that if the seventh sphere amines close you should such or not in particular you don't know so the motivation for the AC is okay let's look in this class where it was g over gamma all right when you have that what's happened if you have a fee which is exact team for him which is exact that's quite a natural question and okay in the simple the case was show by the uptime manga that if you take a remodeling Europe then it cannot omit any left in body exact simple to form right so the same question you can put in the two for G2 structure case okay we were happy because we found an example but not so a pillator because okay we found a unimodally algebra we satisfy this condition so the second and the third chevrolet island better number of G are both zero and I miss an exact G2 structure and then you say okay now we have example no we don't have this example is solvable is also called G2 trivial because unfortunately there is not only the unimodally of section but there is also another of section that you can have for solvable Europe for a meeting let you see and was found by Godland is a paper around 60 before the middle paper and show now explain what I mean for a strongly only model I show that if you have simply connected solvable group really a job and if it's not strongly only model of them cannot omit any lefty so strongly only modularity is a necessary condition to admit lefty's what does mean strongly only model is only for solvable the group and the condition is like that I didn't know before this example in fact yes so let's think this is our G which is a solvable the algebra let's say can be radical okay which is in particular important and let's say the central this ending serious as we say to end so we have any zero and then and no I is and and I minus one and so on okay and the condition is that if you take for any x in G if you take a cab acts and you restrict to the quotient and I over and I plus one the trace of this should be zero for any I bigger than equal than one so that's a condition that you have an extra condition that you have for in the solvable case for the distance of lattice and this example is not is not strongly more than and then okay we thought okay what's up and in general if we take b2 b3 zero it's possible to have only modular plus strictly only modular the algebra of this condition and the answer is no this is still a work in progress we have also an example with b3 zero and b2 zero but the again is you know the modular but not strongly module so probably one can prove in general that is our but we don't know yet so we see is he open the question okay let's go oh how many minutes I have 10 minutes I look that time well okay then I can okay la plush and flow so okay again we will come back to the lead group and the algebra I will see the role but that's general so the idea of la plush and flow as I say was of a right and then the idea was to use a geometric flow we'll see in a moment why is a geometric flow because it's a gradient flow in fact of the volume functional and to the form close it to structure and the idea is to obtain if this flow is converging a parallel you just actually for the moment they we are all the people working on that are far to find that we there are some result if you impose for instance that the your closest structure is also far for a parallel one that you have some convergence but it's still there are no zero result in that direction and so the flow is defined these ways so the idea is to involve the free for 15 in this way so the derivative of 50 is just as I say the la plush of 50 okay and the size is the correct one because fortunately this flow is weakly parabola so you can you can show a short time existence only if you look at flow in the direction of close you to structure if you don't look at the clothes on clothes to structure this is nothing more true and the idea is as I say since this is the exact in fact because this is just a differential of tau t which you form tau induced by 50 then 50 is in the center on class of a zero and if we look how evolve the metric you can see that if you forget about this term this is just like the rich you flow the user rich you flow for the money but you have an extra term this lower order term depend on tau depending on the social so it makes the life very different from rich you flow case and okay I just say few results about that and as I say the stationary point on this flow are just a parallel you to structure the social free to structure and it's really dramatic flow because it's a gradient flow what is called the itching volume functional that's quite natural because you take fee so you have a fee zero the starting point let's say all your flow and then you take fee in the ground class of zero and then you take the natural volume that you have associated to fee which is fee wedge the star of it that's give you a volume form on M and then you integrate on M and what is surprising that this functional is monotonically increasing along the Laplacian flow and in fact the critical point are maximum I don't think maximum and then that's what's not so easy at all they show that in fact you have short time it is at the level of the group this system in fact it should work in the variant case the PDE system become or the E system so in fact the short time is this you have you don't have such problem and in fact what you know about solution at the end is a solution for for this flow are no league groups there are many many results of that about that and also for the seven tar with commodity one close each such the one that I mentioned before okay as for the itching for flow the natural solution are the self-similar solution which are the itching soliton one can look also in this case for self-similar solution that what does mean you have your fee which is the initial one then you have a FT the film or fees you take the pullback and roti which are the function of T and then if you ask this is a solution it means exactly this is very similar to the case of which you flow it means is actually this condition that the watch the plush and of fee is a multiple of fee plus the lead derivative of fee with respect to x and everything is a journey by lambda but that situation is completely different from the itchy flow why okay of course one can talk is actually the same way and shrinking when land is negative steady when land is zero or expanding when land is positive what happened if you take a compliment if all any laplacian soliton which is not often fee has to be expanding lambda has to be positive and that's different from zero so that's completely different the picture from a rich flow and other diversity is that for each flow in the copper case all the rich is soliton are gradient in this case is still no no if you can have a gradient solution is still open okay so in fact this is that exists as a non-trivial expanding laplacian soliton is still an open problem on Compton manifold but there is a solution as we can see by paper by Laurette Niccolini also Niccolini sorry and also by my paper on the solid case one can and there are example of all the type steady shrinking expanding the steady condition is related to in fact the extremely rich pinch condition that I introduced at the beginning and let's see what happens so what I did wait for a fellow that's a natural thing is okay let's start with a similar rich pinch condition and let's evolve the laplacian flow what happened to such condition is preserved is preserved we show this is the term that we show this is general for Compton manifold if you take a Compton manifold with a closeness structure which is extremely to pinch and we show that the 50 so the solution of the flow is exist for any t in fact you can see we know ready that 50 minus 3 is an excess 4 right this is still from what I say before but the difference of the two is you can see it's just a function of t this function that says an expression that depends on the normal tau I would say that in the similar rich pinch the interesting thing dramatic thing that the normal tau is constant that's a geometric property in this case and the other part is just the tau tau is the two form associated to the original to the starting point okay and so we also show in this case that in a plush on flow is constant velocity and the rich remain the same so it seems that this is a similar to pinch have a role similar to some sense the rich is solid on for each flow so we have a different completely different situation for for the solution for anything and okay I I finish the talk saying mention okay because we have this condition and then what happened for example okay and the first example was given by Brian and was given by on the long compact homogeneous space just given by the semi-dial product of SL2C over T2 over SU2 and then we take in a quotient but this gamma is not a subgroup of the group because this M in fact can be identified with the solvable group is not a subgroup but is a subgroup of the automorphism group so it's only locally homogeneous in some sense and then later Laurette found a unimodular example this is a notation that nobody used that but anyway is instead of using bracket I just use the differential so that means that the first one two three are close and here I mean that the e4 is minus e1 where g4 and so on I just use the differential of the level of one form and these are the similar each pinch which is also steady Laplacian solitude with a fellow we proved that this is the only unimodularly group which admits such type of structure so no hope to find out that example in the unimodular case and Laurette and Ecolini show that in fact this is always the case and he left invariant extremely to pinch you to touch around a simple net to the group is always steady soliton so a natural question is okay is to the comer because from here we have that extremely to pinch in prime city so the question is can you we have that with the referral we found an example which is steady Laplacian solitude but is not a similar each of these and this example is very curious because in the case of rich is soliton you have a similar condition and the vector field x is always not left invariant right in the condition for Alex G in this case if you look at x phi such x is left invariant which makes different I don't know geometrically what that means and I think the time is over yes okay so I finish here thank you very much for for the things
 So first of all I want to thank very much the organizer for the invitation and the opportunity  to give it all here. I will talk about closed G2 structure so I will quickly remind what  I mean for first G2 structure. Maybe many people already know but let's introduce again.  So we will work on a seven dimensional manifold and for G2 structure I will mean three four  seven manifold which is stable so that means that if you take this orbit under the action  of the linear group this is open but this is not sufficient to have a stabilizer for  the free form G2 because in general there are two open orbit and what we ask is some  non-degenerate condition plus non-degenerate that means that if you take any vector field  and you take the contraction a x phi wedge a x phi wedge phi this is non-zero. This is  actually the condition to be non-degenerate and that means that then you have a non-zero  seven form and up to a factor you have defined a positive being a four given by a very dangerous  with a chalk I lose the chalk I take given by a x phi wedge a y phi wedge phi and here  of course we have also the volume four. So that means that phi on the same time the term  the metric plus the volume four. The stability for free form on seven dimension are similar  to the stability of two form on let's say vector space of even dimension. In fact this  is a part of what you have when you have a symplactic form and we will see some difference  between closed to structure and symplactic form. So let's put here the name closed.  Why? Oh this is too high for me. I am sorry. Okay. Okay. It's okay I will not I will not  move again so that's fine. So the idea is to follow why G2 is interesting because it's  one of the possible possible. Alonomy get up by the release of a compact manifold compact  manifold when you know symmetrical force and when we have a long of it contains on G2 this  is actually criminal and to ask that phi our free form is closed and co-closed. So this  is zero plus the condition of the star where the star will be all the time the star odd  with respect to the metric induced by phi. This is also zero. Okay. It was quite hard  to find compact example with this with a long of equal to G2 in the talk we will forget  about this condition and we will consider only the condition for the free form to be  closed. That's what I mean for closed G2 structure. So the condition is weaker than the other  one but we see that in fact is anyway strong condition because for instance there are no  topological of suction and in the talk I will talk about such close G2 structure especially  on the groups so only algebra. So the idea is to consider phi will be left in volume  G2 structure on a group G and that means that we are consider G2 structure on the algebra  G of G and we will see also some condition that one can put some restriction that one  can put on this metric induced by phi like high sun condition or more general the condition  related to the rich tensor what is called extremely rich pinch that was introduced by Robert Brian  and we will see a little bit how is the behavior of closed G2 structure especially in relation  to this last condition when you evolve closed G2 structure because there is Brian, Robert  Brian introduced also a flow, a geometric flow that allow you to evolve a closed G2 structure  and again we will see how the groups have special role for that. So let's where is here  so let's start so the idea of the talk is I will review a little bit what is known about  closed G2 structure in general and then we will see what happened on the algebra and  we will see many different from the case for the algebra and the last part I will talk  about Laplacian flow and this condition about it. So I already mentioned what I mean for  G2 structure so we have a phi4 phi and in any point the stabilizer is isomorphic to G2  as I said before and as an explain then you have automatically a metric and the volume  four that's a nice thing of G2 structure then we test Raffin and as I already mentioned  was showed by Fernandez and Gray that the condition for the free form to be parallel  is equivalent to the condition for the free form phi to be close, the point that is here  okay phi to be close and co-close and that is that equivalent to the fact that the volume  of your metric is as a morphic to a saguolo pojic two. Okay and a nice thing of course  when I mentioned parallel or torsion free G2 structure I essentially mean that case  when phi is close and co-close in this case the metric automatically which flat that was  proved by Bonin. Okay so we saw that G2 structure is close it's called close or also calibrated  and if phi is close and of course we when we talk about closed G2 structure I will always  mean that phi is not co-close and you can see from from that expression so in this case  we have that the differential of star phi which is a phi form you just get by tau wedge  phi and then if phi anything will be encoded by this two form tau and tau belongs in fact  in general if you take the space of two form on your standard space and you deform irreducible  component with respect to G2 you have two reducible components one is a seven dimensional  which is isomorphic to R7 and the other one which is 14 dimensional is isomorphic to the  Lie algebra G2 so in this case tau is isomorphic to the space to the irreducible component  on the space of G4 which is isomorphic to the Lie algebra of G2 and that is actually  the situation that we have that means that if you take tau wedge phi algebraically that  means that is equal to minus star tau and also you have these relations so the wedge  of tau with star phi is zero but the interesting thing you can see from here since tau is essentially  the cut of your phi then tau is co-close but the interesting thing here that if you take  the this ultralaplation of phi with respect to the metric induced by phi then is exact  so it is exactly the differential of tau and that's an interesting point that allow Robert  Moran in fact to define this geometric flow evolving closed G2 structure is actually using  that because you evolve and you evolve in the center and class of the starting closed G2  structure okay and the name calibration comes from Arby Lozub because phi define a calibration  so it means that if you take any tangent oriented plane and you restrict phi to that then this  is less than equal than the volume and then you can talk of course about calibrating some  manifold and so on but I will not talk about that in this talk okay rich tensor since phi  the terms the metric then one can expect that the the the rich and that's what's the general  formula that Robert Moran found for more general G2 structure but in this case you can see  essentially that the rich tensor sorry the rich tensor is completely determined by tau  you can see from here you have an a part which is a multiple of the metric this is just the  norm of tau squared and then another part which is obtained by this free form the free  form is this tau minus 1 over 2 star tau and tau and you use the G2 map between volume  up very natural again by contraction this is a map of from free form to symmetric to  tensor and just given by you take the contraction as before the two terms are exactly the same  like here you wedge by beta and then you take the star and then you get something that is  symmetric and that's the first problem that you get in this case because you don't get  any restriction on compact case of the existence of such structure because they as you can  say from this expression the scalar curvature is less than equal to zero so that means that  the complement for you don't get any restriction on the existence of such close G2 structure  okay before I say let's see what happened if you impose some condition on the metric  induced by phi and the first thing is that this is dangerous the first thing that one  can see is what happened if phi the metric induced by phi is Einstein and in the compact  case was proof in two different way by Clayton even though and then in different way by a  rubber Ryan that if you impose a the metric is Einstein then automatically tau is zero  and phi is far but there is a nice inequality that one as that was showed by a rubber Ryan  that involve you see the the integral of the square of the scalar curvature with the integral  of an M of the north of the Ricci square so one can ask okay it's possible to have the  equality we'll see that it's possible and the condition to have this equality it means  that if you take your branch of a relation you have this equality if it only if the differential  tau is exactly equal to this expression so you have a part again multiple of phi plus  a part which is multiple of star of tau which tau and this is what is called a stream a  rich pinch and we will see that we'll have a nice behavior if you evolve close G2 structure  using the Laplacian flow of course this makes sense this condition makes sense also that  even not in the compass case but the level of left invariant group and so the level of  the algebra okay so the first thing that one can do always like that at least for me is  okay let's work on manifold symmetry right so the first question that one can ask can  exist compact homogeneous manifold with an invariant G2 structure close G2 structure  and that's what is no so as I said before there are no known result about the distance  of close G2 structure on compact compression manifold not restriction at all but what we  can do is to see some symmetry so let's consider the automotive group or your clothes or your  manifold with a close G2 structure that means as usual that you take the group of the film  or feel which preceded before right so they pull back and F phi is equal to phi if we  look at the Lie algebra the Lie algebra can be defined in terms of the linearity so adjust  the vector phi for which the lead derivative of phi with respect to x is zero and what  for the style of the show that in the compact case if you have a close G2 structure which  is non parallel so just thousand zero and if you have an element of such Lie algebra  of a automorphism group then the two form I x phi is not only close but is also close  and then you have a restriction because you have a restriction on the dimension of your  automotive group is less than equal to the second but in armor of M and also less than  equal to six but the important thing that is also a billy and with this you can conclude  that there are no compact homogeneous example not only that but well by compact homogeneous  example I mean that you have a group acting transitory and this is a subgroup of the automorphism  group of your manifold with the close G2 structure they conclude more even for commogeneity one  in fact you have only the seven hours but you don't talk about that okay first relation  with the groups are given by example because the first example of compromise for which amit  a close G2 structure but no parallel G2 structure was found by Marisa Fernandez and in fact it's  a compulsion so when I mean there that you have G your group you say gamma a lattice  so a dice creates a group such that the quotient G over gamma is compact and that's what is  actually the first example of such manifold and later Diego Conti and Marisa Fernandez  classify when I mean classify I mean here I mean that they work at the level of the  algebra okay and they classify up to isomorphism then importantly algebra amiting such close  G2 structure and they found only 12 classes there are also example later constructing  the solvent case like almost a billion groups so it means that these are group of semi-dark  product on this form or semi-dark product of our six and that one were found by Marco  Freiber and later we Marisa Fernandez is also Robert Ryan the laureate found more example  about commogeneity one there is also a construction of closures to structure not in the compact  case but in the metric is complete and they are commogeneity one of a compass in polygroup  so it means that compactness is much stronger than completeness for commogeneity one and  recently with Marisa Fernandez and Kovalev and we said the whole lot so we construct  an example which is different from the previous one is not locally homogeneous but is constructing  adapting the Joyce construction not when when you use so you use R before metal you resolution  of R before but instead of starting of T7 of this flat dollars like Joyce said we start  with an in manifold so a compacution of an imposter group by lettuce and then again we  take a final group and then the nice thing here that you see all the previous example  are not form up in the salamium case and this is form up and we are still working on that  I think we can get maybe an example we be one zero that would be nice because it is completely  open that but also this case is some sense new okay so we saw before compact plus if  I sign is not possible right we cannot have a compact homogeneous but then one can ask  okay what happened that was something that I studied with Fernandez and Victor Manero  what happened in the non-compact homogeneous case that the first thing that is still open  to find a non-compact example for which which I made a closest structure for which the metric  is I saw this is completely open even non-complete is not clear if there exists anyway we work  on non-compact homogeneous science a manifold and that's fortunately in in dimension seven  they they conjected by Alexie's kid that the answer and soul manifold exhaust the class  of non-compact homogeneous science a meaningful was show by a royal and La Fonte sorry is  still open in some sectional case and also in I get dimension than 11 as far as I remember  but in seven dimensions that's exactly the case so here I use soul manifold soul manifold  is not compact in this case but just the terminology that they use so I mean simply connect this  solvable Europe with a left in body a match and what we show that in fact if you have  such soul manifold I sense or manifold in some sense if the metric is I sign then this  implied that the metric is flat so you cannot have that in fact we show more general even  if you impose instead of fee close this is still true even if fee is co close which is  quite straight because co close is much weaker than then close and the idea of proving that  use nice theory that was show before very strong theory in fact because one strong result  that will show by your galore it's that the ice and soul manifold are standard so that  means that the the algebra or your group solvable group has is a splitting of these four and  it's just a new radical plus orthogonal complement a which is a billion and so another thing that  one can use that one can always reduce to let's say rank one rank is a dimension of  this a billion complement of and and everything in fact is determined by it we coincide also  with the commutator of it and that's that's important for us because essentially by a result  by Haber that prove before the result by Loret was just for standard I sense or manifold  but we know now by Loret result that the standard the standard is a metric is unique up to  isometry and scale and then we use we reduce to find more of section we reduce the dimension  from seven to six just using the tricky that anytime you take an a in the complement of  n or togola or togola sorry or normal unit or number one then I will say in the order  of complement you have a natural ratio free structure and then using that you you you can  arrive to the to the results so it's sort of reduction dimension six okay so as I said  before simple t structure and closest to structure can have some similarity let's say that simple  it doesn't induce a metric you need to to impose an extra metric but we can see that  they are completely different on at the level of the algebra especially for the unit modular  case because we'll show by two and Lichnero which you know that if you have a unit modular  the algebra so the trace of the joint at x is zero for any x and you with a simple structure  then it's to be solvable and this is the first difference with a fellow we show that in fact  it's possible to have non-solvable only modularly algebra with this property and the courier  thing that you can see from here okay it's not surprising that you have this pre thing  free plus four so this is the radical maximum ability unsolvable here and this is a semi  simple part and the semi simple part the strange thing of this we didn't prove this  geometrically but the semi simple part can be only as two are and then we also classify  up to is almost how many algebra we get we get essentially four algebra a only one is  really not a direct sum or two pieces okay now I show another difference okay this is  why I am looking for exact closely to such a because it's completely open in the compact  case it's not clear okay when you have a simple to form you know the simple to form on a  compound manifold give you a non-zero there on class but when you take a copper manifold  seven manifold we're closely to such you don't know if the class the class determined by  fee is the exact or not and there is no nothing about that for instance one doesn't know that  if the seventh sphere amines close you should such or not in particular you don't know  so the motivation for the AC is okay let's look in this class where it was g over gamma  all right when you have that what's happened if you have a fee which is exact team for  him which is exact that's quite a natural question and okay in the simple the case was  show by the uptime manga that if you take a remodeling Europe then it cannot omit any  left in body exact simple to form right so the same question you can put in the two for  G2 structure case okay we were happy because we found an example but not so a pillator  because okay we found a unimodally algebra we satisfy this condition so the second and  the third chevrolet island better number of G are both zero and I miss an exact G2 structure  and then you say okay now we have example no we don't have this example is solvable is  also called G2 trivial because unfortunately there is not only the unimodally of section  but there is also another of section that you can have for solvable Europe for a meeting  let you see and was found by Godland is a paper around 60 before the middle paper and  show now explain what I mean for a strongly only model I show that if you have simply  connected solvable group really a job and if it's not strongly only model of them cannot  omit any lefty so strongly only modularity is a necessary condition to admit lefty's  what does mean strongly only model is only for solvable the group and the condition is  like that I didn't know before this example in fact yes so let's think this is our G which  is a solvable the algebra let's say can be radical okay which is in particular important  and let's say the central this ending serious as we say to end so we have any zero and then  and no I is and and I minus one and so on okay and the condition is that if you take  for any x in G if you take a cab acts and you restrict to the quotient and I over and  I plus one the trace of this should be zero for any I bigger than equal than one so that's  a condition that you have an extra condition that you have for in the solvable case for  the distance of lattice and this example is not is not strongly more than and then okay  we thought okay what's up and in general if we take b2 b3 zero it's possible to have  only modular plus strictly only modular the algebra of this condition and the answer is  no this is still a work in progress we have also an example with b3 zero and b2 zero but  the again is you know the modular but not strongly module so probably one can prove  in general that is our but we don't know yet so we see is he open the question okay let's  go oh how many minutes I have 10 minutes I look that time well okay then I can okay la  plush and flow so okay again we will come back to the lead group and the algebra I will  see the role but that's general so the idea of la plush and flow as I say was of a right  and then the idea was to use a geometric flow we'll see in a moment why is a geometric flow  because it's a gradient flow in fact of the volume functional and to the form close it  to structure and the idea is to obtain if this flow is converging a parallel you just  actually for the moment they we are all the people working on that are far to find that  we there are some result if you impose for instance that the your closest structure is  also far for a parallel one that you have some convergence but it's still there are  no zero result in that direction and so the flow is defined these ways so the idea is  to involve the free for 15 in this way so the derivative of 50 is just as I say the  la plush of 50 okay and the size is the correct one because fortunately this flow is weakly  parabola so you can you can show a short time existence only if you look at flow in the direction  of close you to structure if you don't look at the clothes on clothes to structure this  is nothing more true and the idea is as I say since this is the exact in fact because  this is just a differential of tau t which you form tau induced by 50 then 50 is in  the center on class of a zero and if we look how evolve the metric you can see that if  you forget about this term this is just like the rich you flow the user rich you flow for  the money but you have an extra term this lower order term depend on tau depending on the  social so it makes the life very different from rich you flow case and okay I just say  few results about that and as I say the stationary point on this flow are just a parallel you  to structure the social free to structure and it's really dramatic flow because it's a gradient  flow what is called the itching volume functional that's quite natural because you take fee so  you have a fee zero the starting point let's say all your flow and then you take fee in  the ground class of zero and then you take the natural volume that you have associated  to fee which is fee wedge the star of it that's give you a volume form on M and then you integrate  on M and what is surprising that this functional is monotonically increasing along the Laplacian  flow and in fact the critical point are maximum I don't think maximum and then that's what's  not so easy at all they show that in fact you have short time it is at the level of the  group this system in fact it should work in the variant case the PDE system become or  the E system so in fact the short time is this you have you don't have such problem and in  fact what you know about solution at the end is a solution for for this flow are no league  groups there are many many results of that about that and also for the seven tar with  commodity one close each such the one that I mentioned before okay as for the itching  for flow the natural solution are the self-similar solution which are the itching soliton one  can look also in this case for self-similar solution that what does mean you have your  fee which is the initial one then you have a FT the film or fees you take the pullback  and roti which are the function of T and then if you ask this is a solution it means exactly  this is very similar to the case of which you flow it means is actually this condition  that the watch the plush and of fee is a multiple of fee plus the lead derivative of fee with  respect to x and everything is a journey by lambda but that situation is completely different  from the itchy flow why okay of course one can talk is actually the same way and shrinking  when land is negative steady when land is zero or expanding when land is positive what  happened if you take a compliment if all any laplacian soliton which is not often fee has  to be expanding lambda has to be positive and that's different from zero so that's completely  different the picture from a rich flow and other diversity is that for each flow in the  copper case all the rich is soliton are gradient in this case is still no no if you can have  a gradient solution is still open okay so in fact this is that exists as a non-trivial  expanding laplacian soliton is still an open problem on Compton manifold but there is a  solution as we can see by paper by Laurette Niccolini also Niccolini sorry and also by  my paper on the solid case one can and there are example of all the type steady shrinking  expanding the steady condition is related to in fact the extremely rich pinch condition  that I introduced at the beginning and let's see what happens so what I did wait for a  fellow that's a natural thing is okay let's start with a similar rich pinch condition  and let's evolve the laplacian flow what happened to such condition is preserved is preserved  we show this is the term that we show this is general for Compton manifold if you take  a Compton manifold with a closeness structure which is extremely to pinch and we show that  the 50 so the solution of the flow is exist for any t in fact you can see we know ready  that 50 minus 3 is an excess 4 right this is still from what I say before but the difference  of the two is you can see it's just a function of t this function that says an expression  that depends on the normal tau I would say that in the similar rich pinch the interesting  thing dramatic thing that the normal tau is constant that's a geometric property in this  case and the other part is just the tau tau is the two form associated to the original  to the starting point okay and so we also show in this case that in a plush on flow  is constant velocity and the rich remain the same so it seems that this is a similar to  pinch have a role similar to some sense the rich is solid on for each flow so we have  a different completely different situation for for the solution for anything and okay  I I finish the talk saying mention okay because we have this condition and then what happened  for example okay and the first example was given by Brian and was given by on the long  compact homogeneous space just given by the semi-dial product of SL2C over T2 over SU2  and then we take in a quotient but this gamma is not a subgroup of the group because this  M in fact can be identified with the solvable group is not a subgroup but is a subgroup  of the automorphism group so it's only locally homogeneous in some sense and then later Laurette  found a unimodular example this is a notation that nobody used that but anyway is instead  of using bracket I just use the differential so that means that the first one two three  are close and here I mean that the e4 is minus e1 where g4 and so on I just use the differential  of the level of one form and these are the similar each pinch which is also steady Laplacian  solitude with a fellow we proved that this is the only unimodularly group which admits  such type of structure so no hope to find out that example in the unimodular case and Laurette  and Ecolini show that in fact this is always the case and he left invariant extremely to  pinch you to touch around a simple net to the group is always steady soliton so a natural  question is okay is to the comer because from here we have that extremely to pinch in prime  city so the question is can you we have that with the referral we found an example which  is steady Laplacian solitude but is not a similar each of these and this example is very  curious because in the case of rich is soliton you have a similar condition and the vector  field x is always not left invariant right in the condition for Alex G in this case if  you look at x phi such x is left invariant which makes different I don't know geometrically  what that means and I think the time is over yes okay so I finish here thank you very much  for for the things
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So first of all I want to thank very much the organizer for the invitation and the opportunity to give it all here. I will talk about closed G2 structure so I will quickly remind what I mean for first G2 structure. Maybe many people already know but let's introduce again. So we will work on a seven dimensional manifold and for G2 structure I will mean three four seven manifold which is stable so that means that if you take this orbit under the action of the linear group this is open but this is not sufficient to have a stabilizer for the free form G2 because in general there are two open orbit and what we ask is some non-degenerate condition plus non-degenerate that means that if you take any vector field and you take the contraction a x phi wedge a x phi wedge phi this is non-zero. This is actually the condition to be non-degenerate and that means that then you have a non-zero seven form and up to a factor you have defined a positive being a four given by a very dangerous with a chalk I lose the chalk I take given by a x phi wedge a y phi wedge phi and here of course we have also the volume four. So that means that phi on the same time the term the metric plus the volume four. The stability for free form on seven dimension are similar to the stability of two form on let's say vector space of even dimension. In fact this is a part of what you have when you have a symplactic form and we will see some difference between closed to structure and symplactic form. So let's put here the name closed. Why? Oh this is too high for me. I am sorry. Okay. Okay. It's okay I will not I will not move again so that's fine. So the idea is to follow why G2 is interesting because it's one of the possible possible. Alonomy get up by the release of a compact manifold compact manifold when you know symmetrical force and when we have a long of it contains on G2 this is actually criminal and to ask that phi our free form is closed and co-closed. So this is zero plus the condition of the star where the star will be all the time the star odd with respect to the metric induced by phi. This is also zero. Okay. It was quite hard to find compact example with this with a long of equal to G2 in the talk we will forget about this condition and we will consider only the condition for the free form to be closed. That's what I mean for closed G2 structure. So the condition is weaker than the other one but we see that in fact is anyway strong condition because for instance there are no topological of suction and in the talk I will talk about such close G2 structure especially on the groups so only algebra. So the idea is to consider phi will be left in volume G2 structure on a group G and that means that we are consider G2 structure on the algebra G of G and we will see also some condition that one can put some restriction that one can put on this metric induced by phi like high sun condition or more general the condition related to the rich tensor what is called extremely rich pinch that was introduced by Robert Brian and we will see a little bit how is the behavior of closed G2 structure especially in relation to this last condition when you evolve closed G2 structure because there is Brian, Robert Brian introduced also a flow, a geometric flow that allow you to evolve a closed G2 structure and again we will see how the groups have special role for that. So let's where is here so let's start so the idea of the talk is I will review a little bit what is known about closed G2 structure in general and then we will see what happened on the algebra and we will see many different from the case for the algebra and the last part I will talk about Laplacian flow and this condition about it. So I already mentioned what I mean for G2 structure so we have a phi4 phi and in any point the stabilizer is isomorphic to G2 as I said before and as an explain then you have automatically a metric and the volume four that's a nice thing of G2 structure then we test Raffin and as I already mentioned was showed by Fernandez and Gray that the condition for the free form to be parallel is equivalent to the condition for the free form phi to be close, the point that is here okay phi to be close and co-close and that is that equivalent to the fact that the volume of your metric is as a morphic to a saguolo pojic two. Okay and a nice thing of course when I mentioned parallel or torsion free G2 structure I essentially mean that case when phi is close and co-close in this case the metric automatically which flat that was proved by Bonin. Okay so we saw that G2 structure is close it's called close or also calibrated and if phi is close and of course we when we talk about closed G2 structure I will always mean that phi is not co-close and you can see from from that expression so in this case we have that the differential of star phi which is a phi form you just get by tau wedge phi and then if phi anything will be encoded by this two form tau and tau belongs in fact in general if you take the space of two form on your standard space and you deform irreducible component with respect to G2 you have two reducible components one is a seven dimensional which is isomorphic to R7 and the other one which is 14 dimensional is isomorphic to the Lie algebra G2 so in this case tau is isomorphic to the space to the irreducible component on the space of G4 which is isomorphic to the Lie algebra of G2 and that is actually the situation that we have that means that if you take tau wedge phi algebraically that means that is equal to minus star tau and also you have these relations so the wedge of tau with star phi is zero but the interesting thing you can see from here since tau is essentially the cut of your phi then tau is co-close but the interesting thing here that if you take the this ultralaplation of phi with respect to the metric induced by phi then is exact so it is exactly the differential of tau and that's an interesting point that allow Robert Moran in fact to define this geometric flow evolving closed G2 structure is actually using that because you evolve and you evolve in the center and class of the starting closed G2 structure okay and the name calibration comes from Arby Lozub because phi define a calibration so it means that if you take any tangent oriented plane and you restrict phi to that then this is less than equal than the volume and then you can talk of course about calibrating some manifold and so on but I will not talk about that in this talk okay rich tensor since phi the terms the metric then one can expect that the the the rich and that's what's the general formula that Robert Moran found for more general G2 structure but in this case you can see essentially that the rich tensor sorry the rich tensor is completely determined by tau you can see from here you have an a part which is a multiple of the metric this is just the norm of tau squared and then another part which is obtained by this free form the free form is this tau minus 1 over 2 star tau and tau and you use the G2 map between volume up very natural again by contraction this is a map of from free form to symmetric to tensor and just given by you take the contraction as before the two terms are exactly the same like here you wedge by beta and then you take the star and then you get something that is symmetric and that's the first problem that you get in this case because you don't get any restriction on compact case of the existence of such structure because they as you can say from this expression the scalar curvature is less than equal to zero so that means that the complement for you don't get any restriction on the existence of such close G2 structure okay before I say let's see what happened if you impose some condition on the metric induced by phi and the first thing is that this is dangerous the first thing that one can see is what happened if phi the metric induced by phi is Einstein and in the compact case was proof in two different way by Clayton even though and then in different way by a rubber Ryan that if you impose a the metric is Einstein then automatically tau is zero and phi is far but there is a nice inequality that one as that was showed by a rubber Ryan that involve you see the the integral of the square of the scalar curvature with the integral of an M of the north of the Ricci square so one can ask okay it's possible to have the equality we'll see that it's possible and the condition to have this equality it means that if you take your branch of a relation you have this equality if it only if the differential tau is exactly equal to this expression so you have a part again multiple of phi plus a part which is multiple of star of tau which tau and this is what is called a stream a rich pinch and we will see that we'll have a nice behavior if you evolve close G2 structure using the Laplacian flow of course this makes sense this condition makes sense also that even not in the compass case but the level of left invariant group and so the level of the algebra okay so the first thing that one can do always like that at least for me is okay let's work on manifold symmetry right so the first question that one can ask can exist compact homogeneous manifold with an invariant G2 structure close G2 structure and that's what is no so as I said before there are no known result about the distance of close G2 structure on compact compression manifold not restriction at all but what we can do is to see some symmetry so let's consider the automotive group or your clothes or your manifold with a close G2 structure that means as usual that you take the group of the film or feel which preceded before right so they pull back and F phi is equal to phi if we look at the Lie algebra the Lie algebra can be defined in terms of the linearity so adjust the vector phi for which the lead derivative of phi with respect to x is zero and what for the style of the show that in the compact case if you have a close G2 structure which is non parallel so just thousand zero and if you have an element of such Lie algebra of a automorphism group then the two form I x phi is not only close but is also close and then you have a restriction because you have a restriction on the dimension of your automotive group is less than equal to the second but in armor of M and also less than equal to six but the important thing that is also a billy and with this you can conclude that there are no compact homogeneous example not only that but well by compact homogeneous example I mean that you have a group acting transitory and this is a subgroup of the automorphism group of your manifold with the close G2 structure they conclude more even for commogeneity one in fact you have only the seven hours but you don't talk about that okay first relation with the groups are given by example because the first example of compromise for which amit a close G2 structure but no parallel G2 structure was found by Marisa Fernandez and in fact it's a compulsion so when I mean there that you have G your group you say gamma a lattice so a dice creates a group such that the quotient G over gamma is compact and that's what is actually the first example of such manifold and later Diego Conti and Marisa Fernandez classify when I mean classify I mean here I mean that they work at the level of the algebra okay and they classify up to isomorphism then importantly algebra amiting such close G2 structure and they found only 12 classes there are also example later constructing the solvent case like almost a billion groups so it means that these are group of semi-dark product on this form or semi-dark product of our six and that one were found by Marco Freiber and later we Marisa Fernandez is also Robert Ryan the laureate found more example about commogeneity one there is also a construction of closures to structure not in the compact case but in the metric is complete and they are commogeneity one of a compass in polygroup so it means that compactness is much stronger than completeness for commogeneity one and recently with Marisa Fernandez and Kovalev and we said the whole lot so we construct an example which is different from the previous one is not locally homogeneous but is constructing adapting the Joyce construction not when when you use so you use R before metal you resolution of R before but instead of starting of T7 of this flat dollars like Joyce said we start with an in manifold so a compacution of an imposter group by lettuce and then again we take a final group and then the nice thing here that you see all the previous example are not form up in the salamium case and this is form up and we are still working on that I think we can get maybe an example we be one zero that would be nice because it is completely open that but also this case is some sense new okay so we saw before compact plus if I sign is not possible right we cannot have a compact homogeneous but then one can ask okay what happened that was something that I studied with Fernandez and Victor Manero what happened in the non-compact homogeneous case that the first thing that is still open to find a non-compact example for which which I made a closest structure for which the metric is I saw this is completely open even non-complete is not clear if there exists anyway we work on non-compact homogeneous science a manifold and that's fortunately in in dimension seven they they conjected by Alexie's kid that the answer and soul manifold exhaust the class of non-compact homogeneous science a meaningful was show by a royal and La Fonte sorry is still open in some sectional case and also in I get dimension than 11 as far as I remember but in seven dimensions that's exactly the case so here I use soul manifold soul manifold is not compact in this case but just the terminology that they use so I mean simply connect this solvable Europe with a left in body a match and what we show that in fact if you have such soul manifold I sense or manifold in some sense if the metric is I sign then this implied that the metric is flat so you cannot have that in fact we show more general even if you impose instead of fee close this is still true even if fee is co close which is quite straight because co close is much weaker than then close and the idea of proving that use nice theory that was show before very strong theory in fact because one strong result that will show by your galore it's that the ice and soul manifold are standard so that means that the the algebra or your group solvable group has is a splitting of these four and it's just a new radical plus orthogonal complement a which is a billion and so another thing that one can use that one can always reduce to let's say rank one rank is a dimension of this a billion complement of and and everything in fact is determined by it we coincide also with the commutator of it and that's that's important for us because essentially by a result by Haber that prove before the result by Loret was just for standard I sense or manifold but we know now by Loret result that the standard the standard is a metric is unique up to isometry and scale and then we use we reduce to find more of section we reduce the dimension from seven to six just using the tricky that anytime you take an a in the complement of n or togola or togola sorry or normal unit or number one then I will say in the order of complement you have a natural ratio free structure and then using that you you you can arrive to the to the results so it's sort of reduction dimension six okay so as I said before simple t structure and closest to structure can have some similarity let's say that simple it doesn't induce a metric you need to to impose an extra metric but we can see that they are completely different on at the level of the algebra especially for the unit modular case because we'll show by two and Lichnero which you know that if you have a unit modular the algebra so the trace of the joint at x is zero for any x and you with a simple structure then it's to be solvable and this is the first difference with a fellow we show that in fact it's possible to have non-solvable only modularly algebra with this property and the courier thing that you can see from here okay it's not surprising that you have this pre thing free plus four so this is the radical maximum ability unsolvable here and this is a semi simple part and the semi simple part the strange thing of this we didn't prove this geometrically but the semi simple part can be only as two are and then we also classify up to is almost how many algebra we get we get essentially four algebra a only one is really not a direct sum or two pieces okay now I show another difference okay this is why I am looking for exact closely to such a because it's completely open in the compact case it's not clear okay when you have a simple to form you know the simple to form on a compound manifold give you a non-zero there on class but when you take a copper manifold seven manifold we're closely to such you don't know if the class the class determined by fee is the exact or not and there is no nothing about that for instance one doesn't know that if the seventh sphere amines close you should such or not in particular you don't know so the motivation for the AC is okay let's look in this class where it was g over gamma all right when you have that what's happened if you have a fee which is exact team for him which is exact that's quite a natural question and okay in the simple the case was show by the uptime manga that if you take a remodeling Europe then it cannot omit any left in body exact simple to form right so the same question you can put in the two for G2 structure case okay we were happy because we found an example but not so a pillator because okay we found a unimodally algebra we satisfy this condition so the second and the third chevrolet island better number of G are both zero and I miss an exact G2 structure and then you say okay now we have example no we don't have this example is solvable is also called G2 trivial because unfortunately there is not only the unimodally of section but there is also another of section that you can have for solvable Europe for a meeting let you see and was found by Godland is a paper around 60 before the middle paper and show now explain what I mean for a strongly only model I show that if you have simply connected solvable group really a job and if it's not strongly only model of them cannot omit any lefty so strongly only modularity is a necessary condition to admit lefty's what does mean strongly only model is only for solvable the group and the condition is like that I didn't know before this example in fact yes so let's think this is our G which is a solvable the algebra let's say can be radical okay which is in particular important and let's say the central this ending serious as we say to end so we have any zero and then and no I is and and I minus one and so on okay and the condition is that if you take for any x in G if you take a cab acts and you restrict to the quotient and I over and I plus one the trace of this should be zero for any I bigger than equal than one so that's a condition that you have an extra condition that you have for in the solvable case for the distance of lattice and this example is not is not strongly more than and then okay we thought okay what's up and in general if we take b2 b3 zero it's possible to have only modular plus strictly only modular the algebra of this condition and the answer is no this is still a work in progress we have also an example with b3 zero and b2 zero but the again is you know the modular but not strongly module so probably one can prove in general that is our but we don't know yet so we see is he open the question okay let's go oh how many minutes I have 10 minutes I look that time well okay then I can okay la plush and flow so okay again we will come back to the lead group and the algebra I will see the role but that's general so the idea of la plush and flow as I say was of a right and then the idea was to use a geometric flow we'll see in a moment why is a geometric flow because it's a gradient flow in fact of the volume functional and to the form close it to structure and the idea is to obtain if this flow is converging a parallel you just actually for the moment they we are all the people working on that are far to find that we there are some result if you impose for instance that the your closest structure is also far for a parallel one that you have some convergence but it's still there are no zero result in that direction and so the flow is defined these ways so the idea is to involve the free for 15 in this way so the derivative of 50 is just as I say the la plush of 50 okay and the size is the correct one because fortunately this flow is weakly parabola so you can you can show a short time existence only if you look at flow in the direction of close you to structure if you don't look at the clothes on clothes to structure this is nothing more true and the idea is as I say since this is the exact in fact because this is just a differential of tau t which you form tau induced by 50 then 50 is in the center on class of a zero and if we look how evolve the metric you can see that if you forget about this term this is just like the rich you flow the user rich you flow for the money but you have an extra term this lower order term depend on tau depending on the social so it makes the life very different from rich you flow case and okay I just say few results about that and as I say the stationary point on this flow are just a parallel you to structure the social free to structure and it's really dramatic flow because it's a gradient flow what is called the itching volume functional that's quite natural because you take fee so you have a fee zero the starting point let's say all your flow and then you take fee in the ground class of zero and then you take the natural volume that you have associated to fee which is fee wedge the star of it that's give you a volume form on M and then you integrate on M and what is surprising that this functional is monotonically increasing along the Laplacian flow and in fact the critical point are maximum I don't think maximum and then that's what's not so easy at all they show that in fact you have short time it is at the level of the group this system in fact it should work in the variant case the PDE system become or the E system so in fact the short time is this you have you don't have such problem and in fact what you know about solution at the end is a solution for for this flow are no league groups there are many many results of that about that and also for the seven tar with commodity one close each such the one that I mentioned before okay as for the itching for flow the natural solution are the self-similar solution which are the itching soliton one can look also in this case for self-similar solution that what does mean you have your fee which is the initial one then you have a FT the film or fees you take the pullback and roti which are the function of T and then if you ask this is a solution it means exactly this is very similar to the case of which you flow it means is actually this condition that the watch the plush and of fee is a multiple of fee plus the lead derivative of fee with respect to x and everything is a journey by lambda but that situation is completely different from the itchy flow why okay of course one can talk is actually the same way and shrinking when land is negative steady when land is zero or expanding when land is positive what happened if you take a compliment if all any laplacian soliton which is not often fee has to be expanding lambda has to be positive and that's different from zero so that's completely different the picture from a rich flow and other diversity is that for each flow in the copper case all the rich is soliton are gradient in this case is still no no if you can have a gradient solution is still open okay so in fact this is that exists as a non-trivial expanding laplacian soliton is still an open problem on Compton manifold but there is a solution as we can see by paper by Laurette Niccolini also Niccolini sorry and also by my paper on the solid case one can and there are example of all the type steady shrinking expanding the steady condition is related to in fact the extremely rich pinch condition that I introduced at the beginning and let's see what happens so what I did wait for a fellow that's a natural thing is okay let's start with a similar rich pinch condition and let's evolve the laplacian flow what happened to such condition is preserved is preserved we show this is the term that we show this is general for Compton manifold if you take a Compton manifold with a closeness structure which is extremely to pinch and we show that the 50 so the solution of the flow is exist for any t in fact you can see we know ready that 50 minus 3 is an excess 4 right this is still from what I say before but the difference of the two is you can see it's just a function of t this function that says an expression that depends on the normal tau I would say that in the similar rich pinch the interesting thing dramatic thing that the normal tau is constant that's a geometric property in this case and the other part is just the tau tau is the two form associated to the original to the starting point okay and so we also show in this case that in a plush on flow is constant velocity and the rich remain the same so it seems that this is a similar to pinch have a role similar to some sense the rich is solid on for each flow so we have a different completely different situation for for the solution for anything and okay I I finish the talk saying mention okay because we have this condition and then what happened for example okay and the first example was given by Brian and was given by on the long compact homogeneous space just given by the semi-dial product of SL2C over T2 over SU2 and then we take in a quotient but this gamma is not a subgroup of the group because this M in fact can be identified with the solvable group is not a subgroup but is a subgroup of the automorphism group so it's only locally homogeneous in some sense and then later Laurette found a unimodular example this is a notation that nobody used that but anyway is instead of using bracket I just use the differential so that means that the first one two three are close and here I mean that the e4 is minus e1 where g4 and so on I just use the differential of the level of one form and these are the similar each pinch which is also steady Laplacian solitude with a fellow we proved that this is the only unimodularly group which admits such type of structure so no hope to find out that example in the unimodular case and Laurette and Ecolini show that in fact this is always the case and he left invariant extremely to pinch you to touch around a simple net to the group is always steady soliton so a natural question is okay is to the comer because from here we have that extremely to pinch in prime city so the question is can you we have that with the referral we found an example which is steady Laplacian solitude but is not a similar each of these and this example is very curious because in the case of rich is soliton you have a similar condition and the vector field x is always not left invariant right in the condition for Alex G in this case if you look at x phi such x is left invariant which makes different I don't know geometrically what that means and I think the time is over yes okay so I finish here thank you very much for for the things
release of a compact manifold compact manifold suction and in the talk I will talk about such will review a little bit what is known about evolve in the center and class of the starting impose some condition on the metric induced by that's completely different the picture from a