Título: On complete Schouten solitons

Resumo: 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.

Título: A two-piece problem for free boundary minimal surfaces in the 3-dimensional ball

Resumo: 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.

Título: L^p Hessian and gradient estimates for solutions of the Poisson equation on complete manifolds

Resumo: 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.

Título: Sharp systolic inequalities for 3-manifolds with boundary

Resumo: Systolic Geometry dates back to the late 1940s, with the work of Loewner and his doctoral student Pu. This branch of differential geometry received more attention after the seminal work of Gromov, where he proved his famous systolic inequality and introduced many important concepts. In this talk I will recall the notion of systole and present some sharp systolic inequalities for free boundary surfaces in 3-manifolds.

Título: Obstruções a curvatura escalar positiva em variedades cônicas

Resumo: 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).

Título: Positive Energy Theorems in Fourth-Order Gravity

Resumo: In this talk, we will analyse a recent notion of energy which is associated to fourth-order gravitational theories and plays an analogous role to that of the classical ADM energy in the context of general relativity. We will first introduce such an energy as conserved quantity canonically associated to perturbations of solutions of the fourth-order space-time equations possessing a time-like Killing field, and then present several of its properties as well as develop some intuitions about it. We will show that, in specific limits, its analysis is tractable and prove positivity and rigidity statements. In particular, in the stationary limit, we will show that the resulting notion of energy is deeply connected to Q-curvature analysis and prove a positive energy theorem which underlies several rigidity phenomena. In particular, we will see that recent positive mass theorems associated to the Paneitz operator appear as a consequence of our analysis.

Título: Gráficos Solitons do Fluxo da Curvatura Média

Resumo: Discutiremos a existência e não existência de gráficos solitons do fluxo da curvatura média com bordo no infinito em variedades com estrutura de produto warped MxR, com ênfase no caso em quem o espaço ambiente é o espaço hiperbólico. Esta palestra é parte de um trabalho em conjunto com o Professor Dr. Luciano Mari (Università degli Studi di Torino).

Título: Superfícies capilares: estabilidade, índice e estimativas de curvatura

Resumo: 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 superficies capilares. Apresentaremos estimativas para o índice de superfícies capilares compactas em variedades de dimensão 3, também estudamos superficies capilares não compactas com índice finito e mostramos que, em condições apropriadas de curvatura, tais superficies 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 superficies capilares estáveis imersas em uma variedade de dimensão 3 com geometria limitada. Essa palestra é baseada em trabalho conjunto com Han Hong.

Título: On r-Trapped Submanifolds Immersed in Lorentzian Spacetimes

Resumo: 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, Santos and Velásquez (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.

Título: Complete translating graphs in R^3

Resumo: 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.

Título: On the intersection of minimal hypersurfaces of S^k

Resumo: It is known since the work of Frankel that two compactly immersed minimal hypersurfaces in a manifold with positive Ricci curvature must have an intersection point. Several generalizations of this result can be found in the literature, for example in the works of Lawson, Petersen and Wilhelm, among others. In the special case of minimal hypersurfaces of S^k, we prove a stronger version of Frankel's theorem. Namely, we show that if two compact minimal hypersurfaces M_1, M_2 of S^k and a point p ∈ S^k are given, then M^1 and M^2 have an intersection point in the hemisphere with respect to p. As a corollary of this result, we give an alternative proof to Ros' two-piece property of minimal surfaces of S^3, for the general dimension case.

Título: Geometric Flows on Hypersurfaces in the Space Forms and some Applications

Resumo: In this talk we discuss geometric flows on hypersurfaces in the space forms and present few applications. First, we give a brief introduction about geometric flows. Second, we consider an inequality conjectured by Ge, Wang and Wu in 2015 for hypersurfaces in hyperbolic space. More precisely, using a geometric flow, which we call the support function flow (SFF), we give a counterexample to the conjectured inequality assuming the initial condition to be zero and that the ambient space is of dimension three. Moreover, we prove an inequality very similar to the conjectured one. Finally, we present some open problems which we believe can be solved by means of geometric flows.

Título: Stable minimal hypersurfaces in R^4

Resumo: I will explain why a two-sided stable minimal hypersurface in R^4 is flat (this is joint work with Chao Li).

Título: On the singular Q-curvature problem

Resumo: The connections between geometry and partial differential equations have been extensively studied in the last decades. In particular, some problems arising in conformal geometry, such asthe classical Yamabe problem, can be reduced to the study of PDEs with critical exponent on manifolds. More recently, the so-called Q-curvature equation, a fourth-order elliptic PDE with critical exponent, is another class of conformal equations that has drawn considerable attention by its relation with a natural concept of curvature. In this talk, I would like to discuss how fixed point methods can be helpful to study the Q-curvature equation in a singular setting, and discuss some interesting problems related to this topic.

Joint work with J.H. Andrade, J. M do ́O, J. Ratzkin and A. Silva Santos.

Título: Four-dimensional gradient shrinking Ricci solitons

Resumo: In this talk, we will discuss 4-dimensional complete (not necessarily compact) gradient shrinking Ricci solitons. We will show that a 4-dimensional complete gradient shrinking Ricci soliton satisfying a pointwise condition involving either the self-dual or anti-self-dual part of the Weyl tensor is either Einstein, or a finite quotient of either the Gaussian shrinking soliton $R^4,$ or $S^3\times R$, or $S^2\times R^2.$ In addition, we will present some curvature estimates for 4-dimensional complete gradient Ricci solitons. Some open problems will also be discussed. This is a joint work with Huai-Dong Cao (Lehigh University) and Detang Zhou (UFF).

Título: Soluções do fluxo pela curvatura média em uma esfera via transições de fase

Resumo: Desde a década de 80, certas equações diferenciais parciais parabólicas que modelam fenômenos de transição de fase são utilizadas como uma regularização do fluxo pela curvatura média para construir e estudar soluções desse fluxo geométrico. Motivados pelas conexões entre os respectivos pontos estacionários e por um trabalho recente de Kyeongsu Choi e Christos Mantoulidis, discutiremos problemas de existência e simetria para tais EDPs e como eles podem ser utilizados para construir fluxos pela curvatura média que conectam superfícies mínimas de menor área na esfera tridimensional.

Este seminário é baseado em um trabalho conjunto com Jingwen Chen (The University of Chicago).

Título: Hipersuperfícies Totalmente Umbílicas de MxR

Resumo: 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''.

Título: On the geometry of higher dimensional black holes

Resumo: In this talk we present some results concerning the geometry and classification of some kinds of static manifolds. Some open problems will be discussed.

Título: TBA

Resumo: TBA