PALESTRA: Finite index CMC hypersurfaces in six dimensional manifolds
RESUMO: We address a question of do Carmo in six-dimensional Riemannian manifolds with bounded curvature, extending results from lower dimensions. In particular, we show that every complete, finite-index, non-minimal CMC hypersurface immersed in a closed Riemannian manifold with nonnegative sectional curvature is compact.
We also study the general case of a Riemannian manifold with bounded curvature and derive partial results. In particular, we show that a complete, finite-index CMC hypersurface immersed in the hyperbolic space $\mathbb{H}^6$ with mean curvature $|H|>7$ is compact. This gives a partial answer to a question posed by Chodosh in his survey for the ICM.
PALESTRA: Linking numbers and Biot-Savart kernels
RESUMO: O cálculo do número de entrelaçamento de dois nós em R^3 via uma fórmula integral é bem conhecido desde Gauss. Iremos usar a teoria de Hodge no contexto de correntes (espaços duais a formas diferenciais) para produzir umas fórmulas universais de cálculo deste número quando a variedade ambiente geométrica é uma variedade Riemanniana compacta 3-dimensional. Concomitante, iremos explicar o que isso tem a ver com a fórmula de Biot-Savart do electromagnetismo.
PALESTRA: How to Identify Some Minimal Surfaces in S³(2) with Area-Minimizing Unit Vector Fields on S²∖{N,S}
RESUMO: In this talk, we shall present a correspondence between a class of minimally immersed surfaces in S³(2), introduced by Lawson, and area-minimizing unit vector fields on the antipodally punctured unit sphere S²\{N,S}. As a consequence of this correspondence, we obtain a stability relation for Lawson cylinders.
PALESTRA: Desigualdades geométricas para hipersuperfícies com quinas
RESUMO: Estendemos, para hipersuperfícies compactas com quinas, algumas desigualdades válidas para hipersuperfícies suaves, como a desigualdade de Alexandrov-Fenchel e sua versão com peso. Como aplicação, obtemos desigualdades geométricas para hipersuperfícies com bordo planar.
Jaciane Gonçalves (UFC)
Possui graduação e mestrado em Matemática pela Universidade Federal do Piauí, e doutorado em Matemática pela Universidade Federal do Ceará UFC.
PALESTRA: Remarks on potential functions of noncompact quasi-Einstein manifolds
RESUMO: A good understanding of the growth rate and uniqueness of potential functions provides fundamental information, since many key properties of special classes of manifolds depend on these aspects.
In this work, we study m-quasi-Einstein potentials on noncompact manifolds. We show that the space of all positive potential functions on a three-dimensional quasi-Einstein manifold has dimension at most two; in the case of equality, we obtain a classification of such manifolds. We then investigate the behavior of these potentials under the additional assumption that the manifold is asymptotically flat and show that any asymptotically flat n-dimensional quasi-Einstein manifold without boundary with λ=0 is necessarily Ricci-flat.
Neilha Pinheiro (UFAM)
Possui doutorado em Matemática pela UFC, e pós-doutorados pela UFAM, UFC e UFRGS. É professora da UFAM desde 2019.
PALESTRA: The index of the cosmological horizon and the area-charge-inequality
RESUMO: In this talk, we discuss the index of the MOTS given by a spatial cross section of the cosmological horizon in the Kerr-Newman-de Sitter spacetime. We show that its index is at least one in the symmetrized sense for a small positive parameter $a$, such parameter defines the angular momentum. Assuming a lower bound for the mass, we prove that this MOTS has index one. In addition, we establish an estimate relating the area and the charge of a MOTS with index one in a Cauchy data satisfying the dominant energy condition.
Ernani Ribeiro Jr (UFC)
Graduou-se em Matemática pela Universidade Estadual do Piauí (UESPI) e possui mestrado e doutorado em matemática pela UFC. Realizou estágios de Pós-doutorado na Lehigh University e na UFF. É professor da UFC desde 2011.
PALESTRA: Four-Dimensional Gradient Shrinking Ricci Solitons
RESUMO: In this talk, we discuss the classification problem for four-dimensional complete (not necessarily compact) gradient shrinking Ricci solitons. We show that a four-dimensional complete gradient shrinking Ricci soliton satisfying a pointwise condition involving the self-dual part of the Weyl tensor is either Einstein or a finite quotient of one of the following model spaces: the Gaussian shrinking soliton \mathbb{R}^4, \mathbb{S}^3 \times \mathbb{R}, or \mathbb{S}^2 \times \mathbb{R}^2.
Moreover, we prove that if the quotient of the norm of the self-dual part of the Weyl tensor and the scalar curvature is sufficiently close, in a suitable sense, to the corresponding for a Kähler metric, then the gradient Ricci soliton must be either half-conformally flat or locally Kähler.