Resumos

Sobre a geometria de curvas e superfícies no espaço-tempo de Lorentz-Minkowski

Nesta apresentação vamos estudar alguns aspectos de curvas e superfícies no espaçotempo de Lorentz-Minkowski. Inicialmente, para curvas regulares tipo- espaço ou tipo-tempo, estabelecemos as equações de Frenet, as quais descrevem o comportamento local de tal uma curva a través do conhecimento de sua curvatura e torção. Depois, estudamos algumas superfícies regulares definidas como imagem inversa de um valor regular de uma certa função diferençável, estudo que logo depois vai nos permitir definir o espaço-tempo de Willen de Sitter e um modelo para o espaço hiperbólico.

Rigidity of surfaces with constant extrinsic curvature in Riemannian product spaces

In this talk we deal with complete surfaces having constant extrinsic curvature in a Riemannian product space M2(c)×R, where M2(c) is a space form with constant sectional curvature c ∈ {−1,1}. In such setting, we find a Simons-type formula for Cheng-Yau’s operator is used to prove that such surfaces are isometric to a cylinder H1×R, when c =−1 or isometric to a slice S2×{t} for some t ∈ R when c = 1. Finally, we extend the result, when c = −1, for the Weingarten linear case.

Two-sided hypersurfaces, entire Killing graphs and the mean curvature equation in warped products with density

Our purpose is to obtain uniqueness results related to the mean curvature equation for entire Killing graphs constructed over the n-dimensional base P of a warped product of the type P_f ×ρ R with warping function ρ and density f . For this, we establish a suitable f-parabolicity criterion and, under appropriate constraints on the Bakry-Émery-Ricci tensor and on the f-mean curvature, we prove some rigidity results concerning two-sided hypersurfaces immersed in P_f ×ρ R.

Sharp bounds for the norm of the second fundamental form of a class of Weingarten hypersurfaces

We provide sharp bounds for the squared norm of the second fundamental form of a wide class of Weingarten hypersurfaces in Euclidean space satisfying Hr = aH + b for constants a,b ∈ R, where Hr stands for the r-th mean curvature and H the mean curvature of the hypersurface. Moreover, we are able to characterize those hypersurfaces where these bounds are attained by showing that it must be a cylinder of the type R×S^{n−1} (ρ).

Characterizing horospheres of the hyperbolic space via higher order mean curvatures

In this talk, our aim is to present new characterization results concerning horospheres of the hyperbolic space under certain appropriate constraints in the behavior of the higher order mean curvatures. Our approach will be based on a suitable maximum principle for complete Riemannian manifolds which is an extension of the classical Hopf maximum principle.

Espaços Homogêneos e Métricas Riemannianas Invariantes

Os espaços homogêneos exercem um papel relevante em Geometria Diferencial, sobretudo porque muitas de suas propriedades geométricas podem ser deduzidas através da aplicação de ferramentas algébricas tais como Teoria de Representação. Neste seminário, serão introduzidos o conceito de espaço homogêneo e exemplos importantes desses espaços no caso compacto. Em seguida, construiremos um tipo de métrica Riemanniana especial sobre espaços homogêneos, a saber, métricas que são invariantes por um certo grupo de isometrias.

On the linear Weingarten spacelike submanifolds immersed in a locally symmetric semi-Riemannian space

Let M be an n-dimensional complete linear Weingarten spacelike submanifold immersed with parallel normalized mean curvature vector field and flat normal bundle in a (n+p)-dimensional locally symmetric semi-Riemannian space L of index p, which obeys standard curvature constraints (such an ambient space can be regarded as an extension of a semi-Riemannian space form). In this setting, our purpose is to establish sufficient conditions guaranteeing that such a spacelike submanifold M be either totally umbilical or isometric to an isoparametric hypersurface of a (n+1)-dimensional totally geodesic submanifold N of L, with two distinct principal curvatures, one of which is simple. Our approach is based on a suitable Simons type formula jointly with a version of the Omori-Yau’s generalized maximum principle for a Cheng-Yau’s modified operator.

Characterizing horospheres of the hyperbolic space via higher order mean curvatures

Submanifolds immersed in a warped product with density

We study n-dimensional complete submanifolds immersed in a weighted warped product of the type I ×f M n+p ϕ , whose warping function f has convex logarithm and weight function ϕ does not depend on the real parameter t ∈ I. Assuming the constancy of an appropriate support function involving the ϕ-mean curvature vector field of such a n-submanifold Σ jointly with suitable constraints on the Bakry-Émery-Ricci tensor of Σ , we prove that it must be contained in a slice of the ambient space. As applications, we obtain codimension reductions and Bernstein-type results for complete ϕ-minimal bounded multi graphs constructed over the n-dimensional Gaussian space. Our approach is based on the weak Omori-Yau’s generalized maximum principle and Liouville-type results for the drift Laplacian.

(r,k,a,b)-stability of hypersurfaces in space forms

In a Riemannian space form, we define the (r,k,a,b)-stability concerning closed hypersurfaces, where r and k are entire numbers satisfying the inequality 0 ≤ k < r ≤ n−2 and a and b are real numbers (at least one nonzero). In this context, when b = 0, we provide a characterization of the geodesic spheres as critical points of the Jacobi functional associated with the notion of (r, k,a,0)-stability. Moreover, in the case b is nonzero, by supposing that a hypersurface Σ is contained either in an open hemisphere of the Euclidean sphere or in the Euclidean space or in the hyperbolic space, and considering some appropriate restrictions on the constants a and b, we are able to show that Σ is (r,k,a,b)-stable if, and only if, Σ is a geodesic sphere.

Linear Weingarten submanifolds in the hyperbolic space

We use suitable maximum principles in order to obtain characterization results concerning n-dimensional linear Weingarten submanifolds immersed with globally flat normal bundle and parallel normalized mean curvature vector field in the n + p-dimensional hyperbolic space. In particular, when p = 2, we present complete descriptions of these submanifold.

Gráficos de Killing, conceito e definição

Nosso objetivo é detalhar a definição de gráficos de Killing. Para tal, definiremos fluxo de um campo de vetores, distribuição integrável, campos paralelos, campos de Killing, campos de Killing conformes e, por fim, gráficos de Killing. Ao final comentaremos alguns resultados envolvendo gráficos de Killing conformes inteiros.

Uma desigualdade integral para hipersuperfícies com curvatura escalar constante

Neste trabalho de dissertação, estudamos a rigidez de hipersuperfícies compactas com curvatura escalar constante imersas isometricamente em uma forma espacil Riemanniana de curvatura seccional constante. Nesta configuração estabelecemos uma formula do tipo Simons e aplicamos ela para obeter uma desigualdade intergal envonvendo a norma da segunda forma fundamental sem traço e uma certa função. Mostramos que esta desigualdade é atingida em hipersuperfícies totalmente umbilicas e em certas familias de toros de Clifford na esfera Euclidiana (n+1)-dimensional.