 Andrew Pessoa from UFPI and he's talking about detecting the completeness of a Finslay manifold via potential theory for its infinity elapsion. So we ask you to keep your microphones muted in the talk and if you would like to ask a question you can unmute yourself or either write it in the chat. So please Leandro go ahead. Thank you Alejandra. Thank you very much for the invitation and it's always a pleasure to talk to the STP is my second time and I selected this work because I didn't talk about this before of course because the content. So I will talk about some ideas to detect completeness of some non-symmetrical space like Finslay unfolds and this work is a joint work with Damian Araujo and Luciano Mari. They are both here. Thank you for the participation and before to talk this specific theme I will introduce this by talking about some previous work that we have done together with Professor Luciano Mari and our previous idea was to approach in a unified way some maximum principles mainly at infinity and to use potential theory to characterize these maximum principles at least in a viscous sense. So the ideas here are very technical but I will avoid to talk in a very technical way. So I will explain roughly some themes and at least in a situ case maybe. So a classical maximum principle defined as a maximum principle is what everyone knows is the classical for the calculus. If you have a maximum point so the gradient is zero at that point and the hashing is negative or maybe the laplacian if you take the trace. So to extend these ideas for a non-compact manifold or maybe subsets you need to think about sequence the maximizing sequence. So we have three kinds of maximum principles for us that may be the relevant ones for us the echelon the Amore maximum principle and the Amoreal maximum principle. So we rephrase these maximum principles using sequence maximizing sequence and we stood along the sequence the the behavior of the gradient and the hashing and the laplacian. So this kind of maximum principle is very fairly studied. So for instance the echelon we are going to see that is related to completeness of a metric space in the general space and the Amore maximum principle is related to martingale completeness and the yaw's principle is related to stochastic completeness somewhat but we will not talk about that. Our main goal here is to study a maximum principle that involves the completeness so we select the echelon maximum principle. It's a very well well known result that a metric space is complete if and only if it satisfies the echelon maximum principle and I choose to I choose this form to talk about the maximum principle like this but I know the the precise statement of the maximum principle is is not in this way but I select this way just to to get easy the way to understand the relation with the previous maximum principle that I showed before. So the sequence maximizing we can see this in the first statement and the gradient goes to zero in some sense here along this sequence but this the echelon maximum principle is very strong the true statement is more strong and a very nice remark to do is that the the proof of this theorem do not use the symmetry of the metric so we just use that the distance is positive defined and it satisfies the triangle inequality. So our idea is to rephrase in some sense this this kind of maximum principles in terms of some potential theory. So the first the first idea is to rephrase using a kind of Alphos maximum principle let's say if you take an open subset and take a bounded from above and post you somewhere function satisfying some inequality for a differential operator then the supremum of this function holds in the boundary of the subset. So we can rephrase the echelon maximum principle using the the icon of our equation but here I put a dual equation the gradient the norm of the gradient minus one is greater than zero and there are more maximum principles somewhat more complicated because we need to take the maximum between two sub equations I'm talking very hopefully because the true way I need to talk I need to take two sub equations and get an intersection between them and this is very not not so easy to talk in a in a meeting like this but the idea is always we take a subset then the the supremum of a sub solution for some sub equation of course in the the boundary of the subset this is the idea for the Alphos maximum principle and related to the just just a comment here this is the greatest I mean I'm taking the other eigenvalues of the hashing so lambda m is the biggest one okay just a little comment okay it's complicated to use this so a more usual property is the Louville property so I also would like to talk about the Louville property for in the case of the echelon I I mean the non-existence of a bounded framble function satisfying this sub equation for the gradient and some subset and for the almore we have the same the same equation before it's more than zero and we have the non-existence uh I use I use case I mean if you think about the Laplacian equation just think about the sub harmonic functions bounded from above so uh we cannot have this kind of function parabolic manifold for instance okay so this is a very general set but the main idea here is we can rephrase think about the the the analytical maximum principles using potential theory using potential theoretical ideas and that that was the the main idea in the previous work that I have done with Luciano and another property very important in this kind of study is the is the the kasminsk property so the idea of the kasminsk is a potential that in this case decrease for minus infinity but we can control this is the the way that the it decrease and it satisfies some sub equation outside a compact set so we take a compact analyze the the the behavior of the the potential outside the discomfort and they at the same time the we have some kind of proper characteristic for the the function the this goes to minus infinity or minus infinity or plus infinity I mean so the the main theorem that we have they've done for these in this those previous work is the following duality theorem the the name duality is just because of this t above the the letters f and for the equations because we have a kind of duality it's somewhat we can think about this like in the the super harmonic function in super harmonic function but I don't like to to do this it's not so true okay but the idea of the theorem is we have a duality I mean we change we pass from a sub equation for another sub equation but related like sub harmonic and sub harmonic and when you pass from this for the order we pass from alphas or reveal because alphas reveal are equivalent for the kasminsk property so we prove that for a very huge class I just call he admissible but I will I will show you what I think about admissible for instance uh some examples of sub equations that we can apply this kind of ideas is the iconal sub equation evolving the the number of the gradients minds are continuous function g dependent on view they are plashing so the iconal is related to the echelon maximum principle so and and the completeness the laplacian is related to the weak maximum principle that you relate related to stochastic completeness and the Hessian is related to the omori maximum principles and martin gave completeness and we can study for the equations or at the branch of the major the k sub harmonic functions and especially the quasi linear sub equations I put an asterisk here because we must take care about the this tensor because it's not not all quasi linear operators so we need to take care we need some bounds for the eigenvalues of the distance and t and for at at the end but we can also treat the case of the finital application I put here the normalized finital application but we can do the same thing for the usual finital application so so these sub equations are all some sense admissible sub equations so the main theory what is our consequence of this duality theory and we have we did this in the remaining setting and involves the involves the completeness of a remaining manifold so the completeness of our remaining manifold is equivalent to the following leviproper for the infinite laplacian so sub harmonic infinity sub harmonic functions that do not grow so fast here I put linear growth but sometimes we just say that it bounds from above so this function must be constant and we put also here an alfos maximum principle so infinite sub harmonic function bound from above and post here somewhere so the supreme of you is obtained in the bottom okay so just a remark this kind of relation between the completeness and the potential theoretical properties of the infinite laplacian were first studied by Stefan of people in alberto set some years ago where they connected these properties using the infinity capacity so a manifold is complete if and only if the infinity capacitor of all or some compact set with non-empty interior is zero and here I describe what I mean for world is infinity capacity the idea used by by then is that I we can connect the infinite laplacian using the p laplacian equation so we have a formal limit defined the infinite laplacian equation and the p capacity of the subsets is very well studied and we have many properties for that so they employ on this kind of limits to to study these these ideas okay just to describe some difficulties that we will find for the for our setting I have said the student equation must be admissible but I didn't talk about this kind of property so two two hypotheses are very important here the first one is this strong maximum principle so a function I mean super super harmonic function cannot achieve a maximum a local maximum material I think I I write I commit a mistake here is a super harmonic function okay I'm sorry so we'll try to do this is like this sorry okay and the other very important characteristic is the hypothesis the comparison so we have a super harmonic function a super harmonic function so we can compare then and the I mean if it's u is from below v at the the boundary of some subsets so it must be below in the interior of the subset so this kind of comparison the strong maximum principle is very important to to study and to do the duality theorem and especially for the infinities finite laplacian what is well known we can relate we can characterize the the the infinite harmonic functions using the this kind of comparisons but we do this using the comparison with cones so in the case of the I mean the the simple infinite harmonic functions we can characterize these these solutions using the comparison of cones and with linear cones that I put here and another property that is very important to study finicky harmonic functions is the absolutely minimizing lip sheets extensions if you want to say that mean that a function a lip sheets function some subset satisfies this property the lip sheets constants of this function for every open set in this set omega coincide with the lip sheets function of this the the boundary of the subset so it's a very strong property the aml property and just recall the lip sheets constants of some functions some subset is given by the here I forgot the infimum of this the subset so I'm sorry again I think here is the infimum the subset okay notice that here I didn't put the norm the models of the x minus the y because I will use this definition for three decades of most symmetric space so this is the the main tools that we must use to prove the duality theory and then the the consequence and what we we're going to want to see is that some tools are like the aml aml property it's not available for the general case that we treat it's just some motivation should work so the first idea of us was to consider general sub equations because in the previous work we just considered the the simplest equation just consider the finicky harmonic functions so we we did the case where the operator is homogeneous the right hand side is homogeneous so we treat the inhomogeneous sub equations for some continuous function non-negative and in this case a good example is the reaction diffusion equation of strong absorption that correspond to this choice of g and that kind of sub equations was started by Damian when they lay down and take shader in the paper don't remember the time but old paper where they they they they started among several results they started some little property for this kind of sub equation so we intend to to generalize the previous work to include this kind of homogeneous sub equations and another source of motivations comes from the connection between the finsler manifolds and the Lorentz manifolds so we can relate a stationary Lorentz manifold like this uh manifold a metric Lorentz and metric like this where the finsler renders type manifold where the finsler tensor is given by this this equation so it's it's interesting because we can start the causal the causality of the the Lorentz manifold using the completeness of the finsler renders type to correspond manifolds so okay but to start the work I will talk about some some ideas from introduce the the finsler manifold so a finsler manifold is a kind of space where we we lose the symmetry of the space so we can think about uh geodesk issuing from a point and uh it minimizes along this geodesk but if you return I mean we connected uh point here and another here but in this way it is minimizing but come back it's not minimizing so we lose the symmetry of the the metric but just for defining uh finsler manifold we we take this function f that should be smooth at least outside the zero section and we have a kind of positive homogeneity and a strong convex for the fundamental thing so so a finsler manifold is also a lengthy space meaning that we can define a distance and this distance satisfies uh the triumph with inequality and also it is positive defined but as I said it is not symmetric symmetric so we lose this this the symmetry of the the metric but uh as I said before it is hope hopefully that we can prove some kind of echelon maximum principle because for metric space in general we didn't use that okay actually the echelon for finsler manifold is well studied all the definitions uh to define the the forward completeness because in for finsler we need to to separate we need to separate the the the the words the forward and the backward the completeness and Cauchy sequence and everything so uh we just with we talk about just uh the ideas of forward Cauchy sequence and completeness but the analogic case for backward is we can can do also so a Cauchy sequence is forward it is a forward Cauchy sequence a sequence if we have this condition but here the order of the j and y and i is important so we we cannot uh state these distance like the distance from xj for xi in this case we are treating the backward Cauchy sequence so a finsler manifold is forward complete i'm sorry if every forward Cauchy sequence converts in n so it's very interesting because uh the forward completeness does not imply the backward one and and the same i mean so some some special things about the the forward and backward distance we define the forward distance like this the distance for a point fixed in this case is x this is my fixed point so the forward distance is like this in the backward i used the sine minus before this is just to can i say well it's just for to have good computations because if we didn't put these minus before the distance we have problems so we choose to do that this is a convention in finsler manifold so for these uh distance functions you can prove a very similar characteristic for for the gradient of the distance and for the hashing so like in the remaining setting where we have for regular points that the the gradient of the distance function has norm one in the hashing i mean along the vectors defined by the distance we have the same properties for the hashing the hashing is zero so we have the corresponding uh properties also in finsler on finsler manifolds from this second we can derive uh that the infinite laplacian of these functions is zero if they are irregular but the problem now always is they they are not regular as we we know okay so uh the main the main issue to to to prove to to improve our the previous theorem for this kind of inhomogeneous sub equations is the lack of uh how to say as a good comparison theory for this kind of of equations and also the AML uh properties because we don't have that so to overcome the this problem we defined some g cons related to our sub equations and the definition is quite technical we we use uh first consider the equation here in finsler-placian equation and uh we consider some solution of the second order uh odd e and define some constants related to the function and finally we can define the cons here the forward cons and the backward ones so we will define the the forward cons on forward poles and the backward ones on backward poles and the idea here is I mean when we were treating the study in the case of the infinite harmonic functions meaning the infinite laplacian of your wake of two zero we use it uh linear cons but here since g is not linear and it's not zero so the cons must must be different must be no linear so this is the way to define uh the that the precise corn cons that we should use just for I will I will give some examples of these cons oh sorry before to do that I would just explain that using that kind of cons we have comparison so uh the main two ways comparison we have cons because using that we have comparison so we first proved this this comparison wave cons for a function satisfying this sub equation of super harmonic functions we have the comparison from above so if if the function is below uh forward cone on the boundary of the subset so it remains remains below in the interior of the subset and if the function satisfies this inequality is a super harmonic function in this sense so if it is above some backward cone on the boundary so it remains above the interior so for for this uh for this kind of cons we can compare the sub solutions and super solutions of the fin and laplacian equation so we we hope these cons are the the precise cons to to study this problem and uh we are not able to to give a precise definition of a absolute minimizing leap extension proper using these g cons or this general sub equation this equation because and also this is not available even the euclidean space just for very special uh function g so we we we cannot do uh a characterization like this like a us a ml if and only if it is a harmonica solution of this equation but uh for our intentions we just need some lip sheets estimates kind of bother to interior interior estimates so we can overcome this problem using uh just estimate that estimate that involve the any one norm of the function g so just to recall uh a functions ml if the cons the lip sheets constant constant or in a subset coincide with the lip sheets constant in the boundary of the subset so uh uh the main problem here is because when we studied uh the the simple case where the the fin key laplacian is more than um zero i mean so the the sliders local uh cone is well defined is just a constant but here uh we should define some ideas for we should define uh what means for this slide slope so we use this definition this constants b dependent a because it depends on the subset the infimum of the positive constant b that we have this comparison on the subset so uh for all points in in the subset we can find the uh a backward cone below the function and the forward cone above the function so this is a kind of slide is low for for our cons just for as in five in the case that g is equals to zero the cones are linear so we reduce to the previous case and also this slide slope is the lip sheets constant of the function so we we define a more general cone but it is a chord length with the the previous one so it's nice for instance if you take g equals to a constant not uh equals to zero but positive and maybe so the cones are quadratum quadratum cones and the this kind of cones and the comparison here was studied for the infimum to laplace equations also in some uh minkov type metrics by mohammed and mibrata okay and a very useful lemma we have here is that we do not we do not have a quality but we can estimate this sliding slope so it is bounded from above the lip sheets constant of the function okay so using these ideas we can give this bounded to interior estimates for the solutions of the infim to laplace equations involve involving these inhomogeneous sub equations so if you pick a solution a lip sheet solution i mean at least in the i'm sorry you pick a solution in the viscosity sense for this equation and assume that this solution is lip sheets on the boundary so it is lip sheets in the whole subset and we can estimate the lip sheets constant by this so we i like this inequality the lip sheets constant of the function in the subset u is bounded from above by using the lip sheets constant in the for the boundary plus some l1 norm of the g function so it's very special and useful for our case okay so this is the the main the main lemma for for the proof of our theorem okay so i will state here a part of the the main theorem because i'm just talking about the infim to laplacian but of course we have some others equivalence involving the echelon maxim principle and but i avoid to put here because the theorem is very big so i just select some items to talk about and and also because i uh i would like you to to emphasize the infim to laplacian equation because the the iconal one is very studied and very well known for specialists so uh even in a finsley manifold we can detect the the forward completeness of the space by using the potential theory and here we we can involve more generous of equations like the inhomogeneous sub equations that i was talking about so here uh a fact that i i i would like to to emphasize is different from the the other kinds of potential theories involving the p laplacian equations and also the laplacian equation here the homogeneous case like these sub equations uh it's not different from this kind of sub equation so for instance uh in the case of the laplacian the the iss case uh a supermold function is related to parabolist of the manifold and if we take a laplacian of u greater than lambda u for some lambda positive so it this kind of sub equations related to stochastic completeness that in the both uh both definitions both both characteristics are very different so parabolist implies stochastic completeness but stochastic completeness is really different from the parabolist so for the infinital laplacian we do not have this kind of separation so the they treat the the same case so they they mean the manifold is complete not uh they have a stochastic property so it's the first remark that i would like to do and a very interesting uh property is that when i treat when i i i studied this supermold case i can use sublinear functions this is the the the corresponding statement for the levied property but if i use uh difference of equation like the reaction diffusion equations like this so we can have the growth of the functions could be more big so it's very interesting because according to the choice of the the function g we have the precise growth for the the subsolution so it's a it's very interesting for this case it's a very nice difference okay so just to state the theorem because i was i'm talking about all the items the first one is a levied we have a levied property for the infinite supermold functions that grow at at most linearly and in this case they are constant so this is equivalence to this to the this kind of levied property and here we have we have the same levied property but involving the inhomogeneous subequations and this is also equivalent to the alpha's property here and i will just write this and as i said we have enough as a particular case this reaction diffusion equations this levied property is also equivalent to the completeness and these these these less two items are related to the infinity capacity and also the elliptic constant of them the function so i mean we have the same characterization proved by Stefano Pigola and Albert Setti this is the same capacity and we have also this kind of minimization problem for liquids okay so just put here the minimizing property of course we also have a corresponding corresponding we have the cause means properties but i avoid to put here because i will use several slides to put this so i would like to do some idea for a proof of this theorem but right now i okay my pen return okay so i think i have some time in five minutes just i would like to to talk about the proof because it's very nice and we will use that leap sheets to make that we talk about so i need to change here my board so just a second please okay so i think you are looking for white board so i would like to to talk implication three what what mean three this fashion you normalize it is this with some subset they close the supremum of you at the bottom so this is three and i would like to prove that three implies why but my pen is very dedicated to the completeness okay so how to prove this using just the ideas that i talk to you have talked to you so the the first the first idea here is the step one the step one it's just a remark we can after some tricks we can we can select the function g because the function g in the item three is our general so we can select the function g satisfy this property g is no net no decreasing and t zero at zero and also we will restrict it to the case where you the the image of you is in the zero one interval so what we will do in this case so we select this kind of properties for the functions and the idea is we pick a point x in the small ball forward ball here and we take exaltions so this is an exhaustion sorry this is only g plus one and two and so on so we take this is an exhaustion for the manifold and consider the solution of the following problem we studied the fintla-plash equation equals to g on the annulus omega g minus b where uj equals function fj on the boundary and the idea here is the following we pick the point and here's the boundary of b and here we have the boundary of omega g and here the boundary of omega g plus one so the function fj it satisfies put one here so it is zero at the boundary of the ball and it is one at the boundary of the exhaustion so we can put here this is fj this is fj plus one so we start this kind of problem the existence of this solution is given by the Perron's method and we have a solution uj uj plus one and so on so we have these solutions for the fintla-plash equation and we have also a leap sheets estimate for these solutions in terms of that what we have so this is the boundary of ag plus two so we have this a main point in the proof that we would not talk about but it's just use the definition of the cones in the definition of this sliding slope is these sequences decreasing so using this we conclude that up to the sequences equilibria and it implies that up to a subsequence we have the sequence converged to an infinite function and this function because of the stability of the solutions satisfies the same equation and further it is zero at the boundary of b so it is that so we just extend the function uinfinity to be zero inside of the ball and we see that this function is a subsolution for the sub equation so we apply our hypothesis to conclude that uinfinity is equals to zero because it is constant but it is zero at the boundary so it is zero okay so we conclude that but we it remains to prove that the matrix the fintla-plash for this complete okay to do that as usual we take a geodesk like this issuing from some point this point x we select a ball around x and we will and do the same construction that we have proved here we have done here and by contradiction we assume that t is not infinite okay but what we will do we just define functions wj defined by uj composition with gamma so we call these functions because they are in composition with uj we have wj equals to one we can extend it by one after okay after some tj of course less than t because uh after some some time we have c the function uj can be done can be defined like that so it is equals to one okay so to conclude we just observe that wj on t wj on s over t minus s is less than or equal to uj gamma t minus uj gamma s t minus s and this is less than or equal to the leap sheets constant of uj on m and since they are globally leap sheets because we know that this is bound from above and here we are using that s is less than t okay so we let letting t small t the big t we have one minus wj s is as we know equal to l t minus s and this gives a contradiction to us because this converts to zero locally uniformly and if you pick s close to t so s close to t so it should be less than one less than half for instance okay so we have a contradiction and that's it we prove that the the space is complete okay so come back to my presentation just to show these are the the the works that the i based my presentation the first one is the the core of the has the core of the the duality where we prove the duality theorem for a general class of super harmonic sub equations and after we we have also deals this work where we treat also some potential theory relating to uh polar sets and this is the main the main work for this this presentation so that's it i i would like to thank you a lot for the attention thank you for your nice talk so are there any questions you can either admit yourself or write in the chat if you prefer so if not i'm going to start but this is not really tall but at the beginning of the presentation you were talking about echelon principle adi was equivalent to the completeness so i would like to ask if the forward completeness is also equivalent or to a yes yes principle or actually i'm sorry because uh in the paper if you look the paper and if you look all the items that we list in the paper uh we put the echelon maximum principle but uh here i decided to just talk about the infinite laplacian and i i mean uh and at the final i decided to put off the echelon but indeed this kind of definition for the echelon that we have done using the real property and the mainly the alphas property it is a somewhat viscosity version of the echelon maximum principle so the completeness is equivalent to the echelon maximum principle this is very general it's not proved by us but for the forward completeness we have the this kind of characterization using the alphas version for the echelon maximum principle or the viscosity version of the echelon also we can do the same for the backward one but instead to consider sub equations we we should consider uh i mean sub equations like uh super solutions i'm sorry sub solutions we we consider super solutions thanks a lot uh are there any other questions if not we thank our speaker again thanks lander thank you very much and we hope to see you in in the next session that is on the 4th of March