The Geometry Research Group at IME USP


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 The Geometry Seminar at IME-USP is organized by Prof. Marcos Alexandrino and Prof. Paolo Piccione and  supported by Projeto Temático-Fapesp of Prof.  Paolo Piccione ( Fapesp 2016/23746-6/Fapesp: 2011/21362-2)

  SP Geometry Seminar (new!): This is a joint seminar that is organized by the research groups of   IMECC-Unicamp ( Prof. Lino Grama and Prof.  Henrique Sa' Earp),  IME-USP ( Prof. Marcos Alexandrino and Prof. Paolo Piccione), UFABC (Prof.Francisco J. Gozzi and Prof. Marcus Marrocos), UFSCar (Prof.Luiz Hartmann  and Prof. Guillermo Antonio Lobos Villagra), ICMC-USP (Prof.Fernando Manfio), Unifesp (Prof.Llohann Dallagnol Sperança)

Next talks and events:


  • December, 20   2019 at 14h
  • Title: Convex Algebraic Geometry of Curvature Operators
  • Speaker: Prof. Renato Bettiol (  Cuny EUA)
  • Abstract: I will discuss the structure of the set of algebraic curvature operators of n-dimensional Riemannian manifolds satisfying a sectional curvature bound (e.g., nonnegative or nonpositive sectional curvature), under the light of the emerging field of Convex Algebraic Geometry. More precisely, we completely determine in what dimensions n this convex semi-algebraic set is a spectrahedron or a spectrahedral shadow (these are generalizations of polyhedra where linear programing extends to as semidefinite programming, and are of great interest in applied mathematics and optimization). Furthermore, if n=4, we describe this set as an algebraic interior with respect to an explicit irreducible polynomial.
    This is based on joint work with M. Kummer (TU Berlin) and R. Mendes (Univ of Oklahoma).
  • venue:   (IME-USP) 




 
 
    Future talks and events:

  

Previous talks and events:


  • Dezember 05, 2019 (SP Geometry Seminar at Unicamp):  Prof. Llohann Dallagnol Sperança (Unifesp)
  • Dezember 05, 2019 (SP Geometry Seminar at Unicamp): Prof. Alexandre Paiva(UFScar)
  • Dezember 05, 2019 (SP Geometry Seminar at Unicamp):  Hugo Cattarucci Botós (ICMC) -short presentation
  • Dezember 05, 2019 (SP Geometry Seminar at Unicamp): Jackeline Conrado (USP)-short presentation
  • November  29, 2019 at 14h
  • Title: Produtos torcidos com curvatura harmônica sobre superfícies" (trabalho em colaboração com Paolo Piccione)
  • Speaker:  Prof. Andrzej Derdzinski (Ohio-state)
  • Abstract:Uma variedade riemanniana é dita  ter curvatura harmônica se a divergência do seu tensor de curvatura é nula (ou,equivalentemente, o​ seu tensor de Ricci satisfaz à equação de Codazzi). Nós descrevemos a estrutura local das variedades riemannianas de dimensões​ $n>3\,$ que, além de possuir curvatura harmônica, admitem​
    decomposições  não triviais como produto torcido com bases bidimensionais.​ A descrição, expressa em termos da função torção  $\phi$ e​ da métrica  $\,g=\phi^{-\nh2}\nh h\,$ conforme á métrica da base $h$,​ envolve uma dicotomia natural: ou $g$ tem curvatura Gaussiana $K$​ constante, ou $K$ é  não constante e igual a uma função​ específica da função torção $\phi$. Em ambos os casos​ $\phi$ satisfaz a uma equação diferencial quase-linear​ elíptica da segunda ordem.​ Construímos também exemplos ilustrando que ambos os casos acima são​ realizados por variedades compactas. 

  • November  29, 2019 at 15h
  • Title: Formal integrability of geometric structures
  • Speaker:  Francesco Cattafi (Utrecht University)
  • Abstract: A Lie pseudogroup $\Gamma$ is a collection of locally defined diffeomorphisms arising as solutions of a PDE.
    This object gives rise to the notion of $\Gamma$-structure on a manifold $M$:
    it is a maximal atlas whose changes of coordinates take values in $\Gamma$.
    For instance, the set of local symplectomorphisms of the canonical symplectic structure of
    $\mathbb{R}^{2n}$ forms a Lie pseudogroup, and the associated $\Gamma$-structure
    is a symplectic structure on $M$. More generally, any integrable $G$-structure is a special kind of $\Gamma$-structure.
    This leads us to a natural question: what is the counterpart of a non-integrable $G$-structure in the Lie pseudogroup world
    (which we call "almost $\Gamma$-structure")? And when does an almost $\Gamma$-structure come from a $\Gamma$-structure? In this talk we are going to review these notions and provide an answer to the questions sketched above. In particular, we present a new characterisation of formal integrability  in the setting of $\Gamma$-structures; this will be obtained by introducing the concept of principal  Pfaffian bundle and studying its prolongations to higher orders; we draw inspiration from similar results for PDEs on jet bundles and for $G$-structures, which we are going to recover. This is joint work with Marius Crainic.
     

  • Short Lectures: 
  • November  26,27,28,  2019 at 14h
  • Title: An introduction to Lie pseudgroups and Geometric Structures
  • Speaker:  Francesco Cattafi (Utrecht University)
  • venue:165B (IME-USP)
  • Abstract: In this minicourse I aim to present a general theory of geometric structures described by a Lie pseudogroup and to study the related integrability problem. 
  • Lecture 1 - The problem: integrability of almost structures
  • Lecture 2 - The tools: principal Pfaffian bundles
  • Lecture 3 - The solution: prolongations and intrinsic torsions

  • November  22, 2019 at 15:15h
  • Title: The behavior of basic equivariant cohomology of Killing foliations under deformations.
  • Speaker:  Dr. Francisco Caramello (IME-USP)
  • Abstract: Killing foliations form an important class of Riemannian foliations which includes foliations given by orbits of isometric actions and Riemannian foliations on simply connected manifolds.  Such foliations have a built-in action of a Lie algebra of Killing vector fields whose orbits are the closures of the leaves. It is then natural and relevant to study the basic equivariant cohomology (with respect to this action) of those foliations. On the other hand, we’ve obtained recently a deformation technique for Killing foliations which allows one to reduce many aspects of their study to the geometry and topology of orbifolds. In this talk we will see that the ring structure of basic equivariant cohomology is preserved throughout those deformations and, as a consequence, coincides with equivariant cohomology of an orbifold , with respect to a torus action. As an application, we’ll give an algebraic condition for the basic Betti numbers to remain constant thoughout deformations.

  • Preliminary talk of 2nd Workwhop of the  São Paulo Journal of Mathematical Science
  • November  11, 2019 at 14:00h
  • Title: A "promenade" towards energy an volume of unit vector fields.
  • Speaker:  Prof.  Fabiano Brito
  • Abstract: We present results on the subject in the title. The general goal is to tell about "interesting " minimizations of the volume and energy functional for unit flows . Most of the subject is in relation to previous or future papers written or to be written in collaboration with the following co authors:P. Walczak,A. Naveira,P. Chacon,O. Gil Medrano,V. Borrelli R. Mesquita, A. Gomes,G. Nunes ,  I. Gonçalves. A. Nicoli,J. Conrado .This order is  intended to be  chronological according to date of publications( or  expected future ones)
  • venue: 243-A (IME-USP) 


  • Event: 2nd Workwhop of the  São Paulo Journal of Mathematical Science: Jean-Louis Koszul in São Paulo, His Work and Legacy
  •  November 13-14, 2019 
  • https://www.ime.usp.br/~2wspjm/

  • SP Geometry Seminar
  • November  8, 2019 at 14.00h
  • Title: Existence Defects of Geometric Structures
  • Speaker: Prof. Michel Nguiffo Boyom (Alexander Grothendieck Research Institute, University of Montpellier, France.)
  • Abstract:  Let $S) be a type of Geometric structure. A smooth manifold $M$ is called a $S$-manifold when it is equipped with a structure of type $S$. A foliation the leaves of which are smoothly $S$-manifolds is called a foliation of type $S$, ( the notion of $S$-foliation might carry another meaning.) Given a smooth manifold $M$, let $S$ be a type of Geometric structure. Arises the question whether $M$ can support a foliation of type $S$. That is a rather hard problem in the differential topology. The aim is to point out that for many important types of Geometry structures this existence problem is linked with sheaves of associative algebras of solutions of Hessian equations defined by Koszul connections in the tangent bundle $TM$. The talk is devoted to overview a few cases of geometric structures which have significant impacts on other research domains. Instances are Hessian structures in Riemannian manifolds; Hessian structures in locally flat manifolds; symplectic structure in manifolds; left invariant symplectic structure in Lie groups.
  • venue: Auditório Jacy Monteiro (IME-USP) 


  • SP Geometry Seminar
  • November  8, 2019 at 15.00h
  • Title: On the spectrum of warped products and $G$-manifolds
  • Speaker:  Prof. Marcus Marrocos (UFABC)
  • Abstract: see links
  • venue: Auditório Jacy Monteiro   (IME-USP) 


  • SP Geometry Seminar- short presentation
  • November  8, 2019 at 16.30h
  • Title: A semi-local model for singular Riemannian foliations.
  • Speaker:  Marcelo Inagaki (USP)
  • Abstract: In this talk we present a semi-local model for a singular Riemannian foliation $\mathcal{F}$. More precisely, in a distinguish tubular neighbourhood, the first order approximation of $F$ (linearized foliation $\mathcal{F}^\ell$) wich partially describes the dynamic of $\mathcal{F}$ will be given by the action of a Lie groupoid. Moreover $\mathcal{F}$ and $\mathcal{F}^\ell$ will be foliated diffeomorphic to a generalization of the holonomy foliation and his linearization, respectivelly. This talk is based on joint work with prof. Marcos Alexandrino (IME-USP) and prof. Ivan Struchiner (IME-USP).
  • venue: Auditório Jacy Monteiro (IME-USP) 



  • SP Geometry Seminar- short presentation
  • November  8, 2019 at 17:00h
  • Title: Moduli space of contact Instantons in $7$-dimensions
  • Speaker:  Luiz Portilla (IMECC)-
  • Abstract: see link
  • venue: Auditório Jacy Monteiro (IME-USP) 


  • October  18, 2019 at 15.15h
  • Title: Cheeger deformations and Positive Ricci and Scalar curvatures
  • Speaker: Leonardo F. Cavenaghi (IME-USP)
  • Abstract: Let $(M,g)$ be a compact Riemannian manifold with an isometric $G$-action. If a principal orbit has finite fundamental group and the quotient space has a metric with positive Ricci curvature, then Searle and Wilhelm showed that $M$ carries a $G$-invariant metric of positive Ricci curvature. To do so, they make use of a conformal change on the original metric and Cheeger deformations. The question remained whether Cheeger deformations are sufficient to prove the Theorem. In this short talk we discuss how to approach this question studying the tensors associated to Cheeger deformations on singular points of the action. We both construct examples where the Cheeger deformation is not enough and give sufficient (algebraic) conditions where the deformation is enough. Once understood the geometry near singular points, we show that provided $G$ is non-abelian, any $G$--invariant metric develops positive scalar curvature after a finite Cheeger deformation. This is a joint work with Renato J.M. e Silva and Llohann D. Sperança.

  • October  18, 2019 at 15.40h
  • Title: A topological lower bound for the energy of a unit vector field on a closed euclidean hypersurface
  • Speaker: Adriana Vietmeier Nicoli (IME-USP)
  • Abstract: For a unit vector field on a closed immersed Euclidean hypersurface $M^{2n+1}$ , $n ≥ 1$, we exhibit a nontrivial lower bound for its energy which depends on the degree of the Gauss map of the immersion. When the hypersurface is the unit sphere  $S^{2n+1}$, immersed with degree one, this lower bound corresponds to a well established value from the literature. We introduce a list of functionals $B_k$ on a compact Riemannian manifold $M^m$, $1 ≤ k ≤ m$, and show that, when the underlying manifold is a closed hypersurface, these functionals possess similar properties regarding the degree of the immersion. In addition, we prove that Hopf flows minimize $B_n$ on $S^{2n+1} .
    This is a joint work with Fabiano G. B. Brito and Ícaro Gonçalves.
     

  • October  4, 2019 at 14h
  • Title: On Mean curvature flow of Singular Riemannian foliations: Non compact cases..
  • Speaker: Prof. Marcos Alexandrino (IME-USP)
  • Abstract:In this talk we discuss  the  mean curvature flow (MCF) of a regular leaf of a closed generalized isoparametric foliation as initial datum, generalizing previous results of Radeschi and first author. We show that, under bounded curvature conditions,  any finite time singularity is a singular leaf, and the singularity is of type I. We also discuss the existence of  basins of attraction, how cylinder structures can affect convergence of basic MCF of immersed submanifolds and  make a few remarks  on MCF of non closed leaves of generalized isoparametric foliation.
    This talk is based on a joint work with  Leonardo F. Cavenaghi, Icaro Gonçalves

  • September  20, 2019 at 14h
  • Title: Projectively induced Ricci-flat Kaehler metrics.
  • Speaker: Prof. Michela Zedda
  • Abstract:The first part of this talk gives an overview of the problem of classifying Kaehler-Einstein manifolds which admit a Kaehler immersion into the complex projective space, which leads to the conjecture that the only Ricci-flat projectively induced Kaehler metric is the flat one. In the second part, we give evidence to this conjecture for the Calabi's Ricci-flat metrics on holomorphic line bundles over compact Kaehler-Einstein manifolds.
  • venue: 259-A (IME)

  • August  23, 2019 at 16h
  • Title: The analysis of the Ricci flow on compact manifolds
  • Speaker: Adam Rudnik
  • Abstract:In this talk I will present DeTurck’s proof of the existence and uniqueness of the Ricci flow on compact manifolds.
    I will then briefly talk about a condition to the long time existence of the Ricci flow. In the last part
    I will present what is known in the case of compact, orientable 2-dimensional manifolds.
  • venue: 268-A  (IME)


  • August  30, 2019 at 14h
  • Title: Widths and volumes of three-spheres
  • Speaker: Prof. Lucas Ambrozio
  • Abstract: Simon and Smith have shown that every Riemannian  three-dimensional sphere contains an embedded minimal two-sphere by  implementing a min-max procedure for the area functional in the space  of embedded two-spheres. Such procedure actually defines a geometric  invariant as well, which we call 'width'. In this talk, we will  discuss results that compare the width to other geometric invariants  of Riemannian thee-spheres, e.g. the total volume. In particular, we  
    prove a sharp inequality between widths and volumes that characterise  the canonical metric, and show that equidistribution of minimal  two-spheres of bounded index and area is a phenomenon associated to  local maxima of the width among unit volume metrics in a given  conformal class. This is joint work with Rafael Montezuma.
  • venue: 268-A (IME)

  • August  15, 2019 at 14h
  • Title: Função Transnormal Finsleriana
  • Speaker: Prof. Benigno Alves (UFBA)
  • Abstract: Nesta palestra apresentaremos o conceito e algumas propriedades de função Transnormal Finsleriana que é uma generalização da função distância. Quando tal função é analítica com fibras conexas definida em uma variedade Finsler compacta, seus níveis são subvariedades equidistantes, em particular, constituem uma folheação Finsleriana singular.
    Por fim apresentaremos condições que garantem a existência de uma métrica Riemanniana tal que a função em questão é transnormal Riemanniana. Tal apresentação é baseada em trabalho conjunto com Prof. M. Alexandrino e Profa  H. R. Dehkordi,
  • venue: 101B (IME)


  • August  15, 2019 at 15h
  • Title: Submanifold Theory of Finsler manifolds
  • Speaker: Prof. Miguel Angel Javaloyes (Murcia-Spain)
  • Abstract: In this talk, we will describe basic results in the theory of submanifolds of Finsler manifolds. First, we will give some tools needed for this study. In particular, we will show that it is possible to define a Levi-Civita anisotropic connection associated with a Finsler metric. We will also show how to define the derivative of an anisotropic tensor (a tensor which depends on the direction rather than on the points of the manifold) and how to define the curvature tensor of an anisotropic connection, giving some results as Bianchi Identities and formulas relating the curvature of two different anisotropic connections. Next, we will study the curves that minimize the distance to a submanifold, showing that locally orthogonal geodesics are minimizers and that in fact, there exist tubular neighbourhoods of precompact submanifolds. This result is crucial for the study of singular Finsler foliations. Afterwards, we will study the Gauss equation of a submanifold and we will finish the talk speaking briefly about minimal submanifolds and the embedding problem.
  • venue: 101B (IME)

  • August  16, 2019 at 14h
  • Title: Estimativas para o primeiro autovalor do operador Laplaciano Aproximado
  • Speaker: Prof Giovanni Nunes
  • Abstract: see link
  • venue: 101B  (IME)
  • August  16, 2019 at 15h
  • Title: Invariant theory for real reductive representations
  • Speaker: Prof. Leonardo Biliotti (Parma-Italy)
  • Abstract:  Let G be a connected real reductive Lie group acting linearly on a finite dimensional vector space V over R. This action admits a Kempf-Ness function and so we have an associated gradient map. If G is Abelian we explicitly compute the image of G orbits under the gradient map, generalizing a result proved by Kac and Peterson in the complex setting. We also give  a new proof of the Hilbert-Mumford criterion and the Kempf-Ness Theorem  for real reductive Lie groups avoiding any algebraic result. Finally we investigate the natural G action on the projective space P(V) and we discuss some new results wich are still in working in progress.
  • venue: 101B  (IME)
  • USP-UNICAMP Geometry Seminar 321 (IMECC).
  • June 27, 2019 : 14:00 - 15:00
  • Title:  Anosov actions which are affine
  • Speaker: Dr. Uirá Norberto Matos de Almeida (pos-doc IME-USP)
  • Abstract: http://www.ime.unicamp.br/~geodif/index.html

  • USP-UNICAMP Geometry Seminar 321 (IMECC).
  • June 27, 2019 : 15h-15:30h
  • Title:  Bifurcation and Local Rigidity of Homogeneous Solutions to the Yamabe Problem on Maximal Flag Manifolds
  • Speaker: Kennerson Nascimento de Sousa Lima -UNICAMP
  • Abstract: http://www.ime.unicamp.br/~geodif/index.html

  • USP-UNICAMP Geometry Seminar 321 (IMECC).
  • June 27, 2019 : 15:45h-16:45h,
  • Title: Gap de energia em teoria de Yang-Mills para variedades completas com invariante de Yamabe positivo
  • Speaker: Prof Matheus Vieira (UFES)
  • Abstract: http://www.ime.unicamp.br/~geodif/index.html


  • June 28, 2019 at 14h
  • Title:  Lie groupoids and semi-local models of Singular Riemannian foliations
  • Speaker: Marcos Alexandrino (IME-USP)
  • Abstract:  In a previous paper, Radeschi and Alexandrino, proved the conjecture of Molino that for every singular Riemannian foliation (M,F), the partition given by the closures of the leaves of F is again a singular Riemannian foliation. For that they studied the linearized subfoliation of F that roughly speaking describe the semi-local dynamical behavior of F.
    In this talk we define a Lie groupoid whose orbits are the leaves of the linearized subfoliation and review the semi-local models of Singular Riemannian foliations. This talk is based on a joint work with  Marcelo K. Inagaki (IME-USP) and prof. Ivan Struchiner (IME-USP).
  • venue: B-138 (IME)

  • June 19, 2019 at 14h
  • Title:   Ações aleatórias na Esfera de Riemann
  • Speaker: Lucas Kaufmann Sacchetto (Paris 6)
  • Abstract:  Os automorfismos da Esfera de Riemann são representados por matrizes em SL_2(C), que agem por meio de transformações de Möbius. Considere o seguinte problema: dadas duas matrizes A e B em SL_2(C), o que podemos dizer sobre a ação de produtos da forma …AABABBBABA onde A e B são escolhidos aleatoriamente e o número de matrizes tende ao infinito? Esse problema e seus variantes remontam aos anos 60 com o trabalho pioneiro de Furstenberg. Mesmo depois de muitos avanços realizados nos anos 70 e 80, o assunto continua a despertar interesse até os dias de hoje e respostas definitivas a algumas questões foram dadas apenas recentemente. O plano dessa palestra será apresentar alguns resultados básicos dessa teoria e mostrar como podemos olhar para o problema sob o ponto de vista da Dinâmica Holomorfa.


  • May 24, 2019 at 15h-16h
  • Title:  Anosov actions which are affine
  • Speaker: Dr Uirá Norberto Matos de Almeida (IME-USP)
  • Abstract:  There exists a standing conjecture that higher ranked abelian Anosov actions are always algebraic.
    This conjecture does not hold true for generic Anosov flows, however, it is known that in the presence
    of some additional geometric conditions, it is possible to show the ”algebraicity” of this flows.
    Motivated by this, we proposed the study of Anosov actions with some additional geometrical conditions.
    In this talk, we present some conditions under which a higher ranked abelian Anosov action is conjugated to
    an affine action of $\mathbb{R}^k$ on a homogeneous space $\Gamma \backslash G \slash H$.
    We put the non-specialized public at ease and assure you that basic concepts will be recalled and this talk is direct to the mathematical public at large.



  • USP-UNICAMP Geometry Seminar (at IME USP)
  • May 17, 2019 at 14h-15h
  • Title: Hipersuperfícies mínimas em $S^5$ com curvatura escalar e função simétrica elementar $\sigma_3$ constantes
  • Speaker: prof. Luiz Amancio Machado de Sousa Junior
  • Abstract: vide link
  • venue: B4 (IME)

  • USP-UNICAMP Geometry Seminar (at IME USP)
  • May 17, 2019 at 15h-16h
  • Title: ACTIONS ON POSITIVELY CURVED MANIFOLDS AND BOUNDARY IN THE ORBIT SPACE
  • Speaker: prof. Claudio Gorodski (IME-USP)
  • Abstract:    We study isometric actions of compact Lie groups on complete orientable positively curved n-manifolds whose orbit space has non-empty boundary in the sense of Alexandrov geometry and prove that for sufficiently large
    n (in terms of the Lie group), either the group has a fixed point, or a normal subgroup thereof, containing all isotropy groups associated to boundary strata, has a fixed point. Among our applications, we classify representations of simple Lie groups whose orbit space has non-empty boundary. (joint work with A. Kollross and B. Wilking).
  • venue: B4 (IME)


  • USP-UNICAMP Geometry Seminar (at IME USP)
  • May 17, 2019 at 16:30h-17h
  • Title: T-duality and mirror symmetry on nilmanifolds (Short Communications)
  • Speaker:Dr Leonardo Soriani (UNICAMP)
  • Abstract: opological T-duality can be undestood as a Courant algebroid isomorphism. This allows one to transport generalized complex structures between T-dual spaces. Sometimes this transport has a mirror symmetric behaviour, that is,
    it transports a symplectic structure to a complex one. We show that this phenomena translates very well into a Lie algebraic setting, yielding the notion of infinitesimal duality, Under reasonable assumptions, infinitesimal duality of nilpotent Lie algebras can be upgraded to actual T-duality on corresponding nilmanifolds. We also show how this machinery is used to transport generalized complex branes between T-duals.
  • venue: B3 (IME)


  • USP-UNICAMP Geometry Seminar (at IME USP)
  • May 17, 2019 at 170h-17:30h
  • Title: A topological lower bound for the energy of a unit vector fieeld on closed Euclidean hypersurfaces (Short Communications)
  • Speaker: Adriana Nicolli (IME-USP)
  • Abstract:    see Link
  • venue: B3 (IME)
 
  • May 10, 2019 at 14h-16h
  • Title: Métricas maximamente torcidas de curvatura harmônica
  • Speaker: Prof. Andrzej Derdzinski (The Ohio State University)
  • Abstract:    Uma variedade riemanniana  é dita de ter curvatura harmônicas e a divergência do seu tensor de curvatura  é nula (ou, o que  é o mesmo, o seu tensor de Ricci satisfaz  `a equação de Codazzi). Nós descrevemos a estrutura local das variedades riemannianas de dimensões n  > 2 que, al ém de possuir curvatura harmônica, admitem o número máximo, em um sentido bem definido, de decomposições produto torcido locais e, ao mesmo tempo, em algum ponto, seu tensor de Ricci tem n autovalores distintos. Mostramos também que em cada dimensão n  > 2 fixa, os tipos de isometria local de tais variedades formam um espaço de módulos de dimens~ao finita, e que um subconjunto aberto não
    vazio deste espaço é realizado por m étricas completas.
    (trabalho em colaboração com Paolo Piccione)

  • March 22, 2019 at 14h
  • Title: Critérios de solubilidade de tipo Serrin para problemas de Dirichlet para equações de curvatura média pré-determinada em variedades
  • Speaker: Dra. Yunelsy Napoles Alvarez (Puc-rio)
  • Abstract:    Neste seminário apresentaremos novos resultados de não existência e de existência de gráficos verticais no produto MxR com curvatura média pré-determinada H=H(x,z), onde M é uma variedade completa de dimensão n. Como consequência de tais resultados se derivam critérios de solubilidade sharp para o problema de Dirichlet para a equação vertical da curvatura média H em MxR. Esses resultados generalizam resultados clássicos de Jenkins-Serrin e de Serrin no ambiente Euclidiano, bem como resultados de Spruck no contexto MxR.
  • venue: 241-A(IME)

  • March 22, 2019 at 14h
  • Title: Critérios de solubilidade de tipo Serrin para problemas de Dirichlet para equações de curvatura média pré-determinada em variedades
  • Speaker: Dra. Yunelsy Napoles Alvarez (Puc-rio)
  • Abstract:    Neste seminário apresentaremos novos resultados de não existência e de existência de gráficos verticais no produto MxR com curvatura média pré-determinada H=H(x,z), onde M é uma variedade completa de dimensão n. Como consequência de tais resultados se derivam critérios de solubilidade sharp para o problema de Dirichlet para a equação vertical da curvatura média H em MxR. Esses resultados generalizam resultados clássicos de Jenkins-Serrin e de Serrin no ambiente Euclidiano, bem como resultados de Spruck no contexto MxR.

  • March 15, 2019 at 14h
  • Title: Genuine infinitesimal bendings of Euclidean submanifolds 
  • Speaker: Miguel Ibieta Jiménez (IMPA)
  • Abstract:  In this talk we focus on a notion of bending of a submanifold. This notion is associated to variations of a submanifold by immersions that preserve lengths ``up to the first order". More precisely, an infinitesimal bending of an isometric immersion $f:M^n \to \mathbb{R}^{n+p}$ is the variational vector field associated to
    a variation of $f=f_0$ by immersions $f_t$  whose induced metrics $g_t$ satisfy $g'_t(0)=0$.  We present some results concerning genuine infinitesimal bendings of submanifolds in low codimension. That an infinitesimal bending is genuine means that it is not determined by an infinitesimal bending of a submanifold of larger dimension. We show that a strong local condition for a submanifold to be genuinely infinitesimally bendable is to be ruled and we estimate the dimension of the rulings. We also describe the situation for infinitesimal bendings of compact submanifolds in codimension $2$. Finally, we give a description of the complete Euclidean hypersurfaces that are infinitesimally bendable. 

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