Programação
O Workshop de Verão em Geometria será realizado no Auditório Ricardo de Carvalho Ferreira, CCEN/UFPE, Recife, durante os dias 6 e 7 de Fevereiro de 2020.
Mini-curso: Self-shrinkers do Fluxo da Curvatura Média sob a Ótica da Teoria das Subvariedades - Prof. Márcio Batista (UFAL)
Resumo: Neste mini curso introduziremos os conceitos básicos sobre o fluxo da curvature média e focaremos em soluções auto-similares do mesmo. Tais tipos de solução são chamados de self-shrinkers do fluxo da curvatura média, FCM, e podem ser estudados sob a ótica da teoria das subvariedades. Apresentaremos o estado da arte do tema em foco e apresentaremos uma lista de exemplos e estratégias bem sucecidas na caracterização de self-shrinkers do FCM sob hipóteses adequadas.
Quinta-Feira - 06/02/2020
Name: Ernani Ribeiro Jr (UFC)
Title: An overview on four-manifolds with positive curvature
Abstract: A classical topic in Geometry is to classify 4-dimensional manifolds with positive sectional curvature. Very few topological obstructions to positive sectional curvature are known, and many conjectures about this subject remain open. In this talk, we will present some open problems related to the classification of 4-dimensional manifolds with positive sectional curvature. Moreover, we will show some partial answers obtained by different approaches.
Name: Henrique Lima (UFCG)
Title: Rigidity results in certain manifolds with density
Abstract: In this talk, we will apply some maximum principles in order to obtain rigidity results concerning two-sided hypersurfaces immersed in a Killing warped product endowed with a suitable density. A particular study of entire Killing graphs will also be made. The results that will be present in this talk are contained in a joint work with Eraldo Lima Jr. (UFPB), Adriano Medeiros (UFPB) and Márcio Santos (UFPB), recently published in Publicationes Mathematicae Debrecen (DOI:10.5486/PMD.2019.8297).
Name: Feliciano Aguiar (UFAL)
Title: Superfícies CMC em variedades homogêneas tridimensionais
Abstract: Nesta palestra, iremos expor o problema de existência de uma imersão uma variedade produto com curvatura média constante via um sistema de Equações Diferenciais Parciais e classificamos aquelas que possuem curvatura intrínseca constante. Posteriormente, utilizando a chamada "sister correspondence" trataremos as superfícies CMC com curvatura intrínseca constante nas outras variedades de Thurston.
Name: Cícero Aquino (UFPI)
Title: On the angle of complete hypersurfaces in semi-Riemannian product spaces
Abstract: The purpose of this talk is to present some uniqueness results concerning complete hypersurfaces satisfying some pinching curvature condition. Here, we use the generalized maximum principle of Omori-Yau to obtain uniqueness results for complete spacelike hypersurfaces immersed in a Lorentzian product space. In addition, we obtain the analogue results for complete hypersurfaces immersed in a Riemannian product space.
Sexta-Feira - 07/02/2020
Name: Antônio Caminha (UFC)
Title: Maximum principles at infinity with applications to geometric vector fields
Abstract: This talk is an extract of joint work with professors Luís Alías and Francisco Yuri Nascimento. We first derive a new form of maximum principle for a vector field of nonnegative divergence in a connected, oriented, complete noncompact riemannian manifold. We then use it to show that, under a reasonable condition at infinity, a connected, orientable, complete noncompact hypersurface of a riemannian manifold, transversal to a Killing vector field of constant norm and having nonnegative second fundamental form, is totally geodesic. Also, if instead of having nonnegative second fundamental form the hypersurface is of cmc, then we show that it has to be totally geodesic too. We finally consider a complete noncompact spacelike hypersurface of a conformally stationary spacetime, showing that if the future second fundamental form is positive semidefinite, then the hypersurface has to be totally geodesic. I shall also give a glimpse on a second form of maximum principle for vector fields, from which one can easily estimate the size of bounded complete noncompact minimal hypersurfaces in Euclidean spaces.
Name: Marco Antônio Velásquez (UFCG)
Title: On the geometry of complete spacelike submanifolds immersed in the $(n+p)$-dimensional anti-de Sitter space of index $q^*$
Abstract: First, motivated by the nonexistence of compact spacelike hypersurfaces in the anti-de Sitter space $\mathbb H_1^{n+1}$, in this paper we study the geometry of complete spacelike hypersurfaces immersed in $\mathbb H_1^{n+1}$ with either constant mean or scalar curvature. In this setting, under suitable restrictions on the norm of the tangential part of a fixed timelike vector, we show that such hypersurfaces must be isometric to a certain hyperbolic space. After that, we studied the rigidity of complete submanifolds immersed in the anti-de Sitter space $\mathbb{H}^{n+p}_{q}$ of index $q$. In this setting, supposing suitable constraints on the Ricci curvature and second fundamental form, we show that a complete maximal submanifold $M^n$ of $\mathbb{H}^{n+p}_{q}$ must be totally geodesic. Furthermore, we establish sufficient conditions to guarantee that a complete spacelike submanifold with nonzero parallel mean curvature vector in $\mathbb{H}^{n+p}_{p}$ must be pseudo-umbilical, which means that its mean curvature vector is an umbilical direction.
Name: Eraldo Lima (UFPB)
Title: Reduction of codimension for weakly trapped submanifolds in GRW spacetimes
Abstract: We present weakly trapped submanifolds of codimension two in Generalized Robertson-Walker spacetimes. In this setting, we apply some generalized maximum principles in order to investigate their geometry. For instance, we show sufficient conditions to guarantee that such a spacelike submanifold must be a hypersurface of the Riemannian base of the ambient spacetime. As a consequence, we prove non-existence of future (past) trapped submanifolds time bounded in contracting (expanding) GRW spacetimes. We also present the rigidity of stochastically complete weakly trapped submanifolds immersed in such spacetimes.