Geometric Analysis & Differential Geometry
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Hipersuperfícies Totalmente Umbílicas de MxR
Lecturer: Ronaldo Freire de Lima (UFRN)
Link: https://meet.google.com/xsy-pcrz-pqj
Date: 11/18/2021
Time: 10h00
More information: https://sites.google.com/mat.ufc.br/gdag
Abstract:
Nesta palestra, apresentaremos os resultados obtidos num recente trabalho em parceria com João P. dos Santos. Nele, caracterizamos as hipersuperfícies totalmente umbílicas de produtos riemannianos MxR, bem como classificamos as superfícies totalmente umbílicas de H^n x R e S^n x R.
Veremos, também, que resultados análogos são válidos nos correspondentes produtos "warped''.
Four-dimensional gradient shrinking Ricci solitons
Complete translating graphs in R^3
Lecturer: Eddygledson Gama (UFERSA)
Place: https://meet.google.com/kvd-bvjt-zaf
Date: 08/26/2021
Time: 10h00
More Informations: https://sites.google.com/mat.ufc.br/gdag
Abstract:
The main goal of this talk is to give a study about the translating graphs for the mean curvature flow in R^3.
In order to do so, we divide our lecture into two topics: structure of the graph and classification.
About the structure part, we introduce a new way to decompose a complete translating graph in a slab. As a consequence of this way of viewing the graphs, we show that the entropy of a complete graph is equal to the number of planes that the graph develops in the “upper” infinity.
On the other hand, in the classification part, we classify graphs under natural assumptions which came from the structure part. More precisely, we classify graphs with one of the following assumptions: a small number of wings, low entropy, low width, or lying in a half-slab.
On r-Trapped Submanifolds Immersed in Lorentzian Spacetimes
Abstract: The behavior of spacelike submanifolds immersed in Lorentzian manifolds is an important object of study which has aroused a lot of interest in recent years, from both the physical and mathematical points of view. Into this branch, the trapped submanifolds appear as an important particular case. The concept of trapped submanifolds, originally formulated by Penrose, is related to the causal orientation of the mean curvature vector field of the submanifold, that is, a spacelike submanifold of a spacetime is said to be trapped if its mean curvature vector field is timelike. Recently, de Lima, dos Santos and Veásquez (2016) obtained rigidity for trapped submanifolds in Lorentzian spaces forms, they condidered assumptions such as parallel mean curvature and pseudo-umbilicity. Later, Alías, Cánovas and Colares (2017), considered codimension two trapped submanifolds immersed in generalized Robertson-Walker spacetimes and obtained results of nonexistence and rigidity.
In this seminar, will we introduce the notion of r-trapped submanifolds immersed Lorentzian spacetimes as generalization of the trapped submanfolds introduced by Penrose. Within this scope, we will present rigidity and nonexistence results for r-trapped in some configurations of generalized Robertson Walker (GRW) spacetimes and, lastly, we provide examples of r-trapped submanifolds, some of them are also simultaneously trapped, but we provided examples proving that the notion of r-trapped submanifolds are different accordingly to the natural number r.
Superfícies capilares: estabilidade, índice e estimativas de curvatura
Lecturer: Artur Saturnino (University of Pennsylvania)
Place: meet.google.com/tji-kbtq-rqg
Date: 08/12/2021
Time: 10h00
More Informations: https://sites.google.com/mat.ufc.br/gdag
Abstract: Superfícies capilares são pontos críticos para variações que preservam volume de um funcional que modela a energia na superfície de um líquido incompressível em um recipiente, ignorando a gravidade. Os índices fraco e forte de uma superfície capilar medem o quão distante essa superfície está de minimizar energia até o segundo grau de aproximação. Nessa palestra investigamos a conexão entre o índice e a geometria e topologia de superfícies capilares. Apresentaremos estimativas para o índice de superfícies capilares compactas em variedades de dimensão 3, também estudamos superfícies capilares não compactas com índice finito e mostramos que, em condições apropriadas de curvatura, tais superfícies são conformes a superficies de Riemann com bordo compactas com um número finito de furos. Usando esse resultado, nós provamos que uma superfície capilar estável imersa em uma meio-espaço de $\mathbb{R}^3$ que é mínima ou tem ângulo de contato menor ou igual a $\pi/2$ deve ser um meio-plano. Usando esse resultado de unicidade nós obtemos estimativas de curvatura para superfícies capilares estáveis imersas em uma variedade de dimensão 3 com geometria limitada.
Essa palestra é baseada em trabalho conjunto com Han Hong.
Gráficos Solitons do fluxo da curvatura média
Lecturer: Levi Lopes de Lima (UFC)
Place: https://meet.google.com/qhr-posk-tfs
Date: 07/15/2021
Time: 10h00
More informations: https://sites.google.com/mat.ufc.br/gdag
Abstract:
Versões apropriadas da fórmula do índice de Atiyah-Singer para operadores de Dirac serão usadas para estender a variedades spin com singularidades cônicas isoladas as clássicas obstruções à existência de métricas com curvatura escalar positiva. No caso em que a variedade subjacente possui fronteira disjunta da região cônica, variações do método geram obstruções a métricas que, adicionalmente, tornam esta fronteira convexa em média (baseado em arXiv:2104.13882).
Seminário Conjunto: Geometria do Nordeste Webinar (GENE) & Seminário São Paulo de Geometria Diferencial (AmSul).
L^p Hessian and gradient estimates for solutions of the Poisson equation on complete manifolds
Lecturer: Stefano Pigola (Università degli Studi di Milano-Bicocca)
Date: 05/27/2021
Time: 10h00
Abstract: We will give a survey of recent results and techniques, based on different geometric assumptions on the underlying manifold M, to prove the validity and the failure of global inequalities of Calderón-Zygmund type.
A two-piece problem for free boundary minimal surfaces in the 3-dimensional ball
Lecturer: Ana Menezes (Princeton University)
Date: 05/20/2021
Time: 15h00
Abstract: In this talk we will prove that every plane passing through the origin divides an embedded compact free boundary minimal surface of the euclidean 3-ball in exactly two connected surfaces. This result gives evidence to a conjecture by Fraser and Li. This is a joint work with Vanderson Lima from UFRGS.
On complete Schouten solitons
Lecturer: Valter Borges (Universidade Federal do Pará)
Date: 05/13/2021
Time: 10h00
Abstract: In this talk, we investigate the geometry of complete gradient Schouten solitons. These are the self-similar solutions of the Schouten flow, a geometric evolution equation for Riemannian metrics introduced in Bourguignon’s classical paper. These metrics were first investigated by Catino and Mazzieri in 2016, where it was shown that compact Schouten solitons are Einstein. Another classification found in this paper is that of the complete steady Schouten solitons, where it was proved that these metrics are Ricci flat. The results of this talk concern shrinking and expanding complete noncompact Schouten solitons. We present optimal inequalities between the potential function and the norm of its gradient and show that the scalar curvature of such metrics must be bounded.
On the geometry and topology of initial data sets in General Relativity
Lecturer: Gregory J. Galloway (University of Miami)
Place: meet.google.com/ujd-ndsi-hvw
Date: 03/18/2021
Time: 15h00
Abstract:
A theme of long standing interest (to the speaker!) concerns the relationship between the topology of spacetime and the occurrence of singularities (causal geodesic incompleteness). Many results concerning this center around the notion of topological censorship, which has to do with the idea that the region outside of all black holes (and white holes) should be topologically simple. In this talk we present results which provide support for topological censorship at the pure initial data level, thereby circumventing difficult issues of global evolution. The proofs rely on the theory of marginally outer trapped surfaces, which have played an important role in the theory of black holes, and which may be viewed as spacetime analogues of minimal surfaces in Riemannian geometry. The talk will begin with a brief overview of general relativity and topological censorship. The talk is based primarily on joint work with various collaborators: Lars Andersson, Mattias Dahl, Michael Eichmair and Dan Pollack.
Initial data rigidity results
Lecturer: Abraão Mendes do Rêgo
Place: meet.google.com/nkp-jdwx-cwr
Date: 02/19/2021
Time: 14h00
Abstract:
In this lecture we aim to present some rigidity results for initial data sets that are motivated by the spacetime positive mass theorem. A key step is to show that certain marginally outer trapped surfaces (MOTS) are weakly outermost. As a special case, our results include a rigidity result for Riemannian manifolds with a lower bound on their scalar curvature.
Existência de hipersuperfícies mínimas de fronteira livre em domínios quádricos
Place: Via Conferência Web em: http://meet.google.com/yfp-afeh-rwz
Date: Friday, 02/12/2021
Time: 14h00
Estimativa para o volume de ciclos em variedades hiperbólicas com bordo
Lecturer: Moreno Bonutti (UFAL)
Place: http://meet.google.com/wsj-qsrx-rcs
Date: 01/29/2021
Time: 14h00
O índice de Morse e o primeiro número de Betti de subvariedades mínimas
Lecturer: Diego Alves Adauto (UERN/UFAL)
Place: Via Conferência Web em: meet.google.com/rhq-hajy-hvy
Date: 01/14/2021
Time: 10h30
