Levi Lopes de Lima

The ADM mass of asymptotically flat hypersurfaces

We provide integral formulae for the ADM mass of asymptotically flat hypersurfaces in Riemannian manifolds with a certain warped product structure in a neighborhood of infinity, thus extending Lams recent results on Euclidean graphs to this broader context. As applications we exhibit, in any dimension, new classes of manifolds for which versions of the Positive Mass and Riemannian Penrose inequalities hold and discuss a notion of quasi-local mass in this setting. The proof explores a novel connection between the co-vector defining the ADM mass of a hypersurface as above and the Newton tensor associated to its shape operator, which takes place in the presence of an ambient Killing field.

Constant mean curvature one surfaces in hyperbolic 3-space using the Bianchi-Calò method

Nesta nota apresentaremos um metodo para construir superficies de curvatura media constante um no 3-espaco hiperbolico, a partir de funcoes holomorfas. Tal metodo foi introduzido nas Lezioni di Geometria Differenziale de Bianchi em 1927, antecedendo, portanto, em muitos anos, os pontos de vista mais modernos de Bryant, Small e outros. Alem do seu obvio interesse historico, o objetivo da nota e complementar a analise de Bianchi, obtendo formulas explicitas para as superficies de curvatura media constante um, e comparar os varios pontos de vista encontrados na literatura.

Deformations of 2k-Einstein structures

Abstract It is shown that the space of infinitesimal deformations of 2 k -Einstein structures is finite dimensional on compact non-flat space forms. Moreover, spherical space forms are shown to be rigid in the sense that they are isolated in the corresponding moduli space.

Monotonicity inequalities for ther-area and a degeneracy theorem forr-minimal graphs

We establish monotonicity inequalities for the r-area of a complete oriented properly immersed r-minimal hypersurface in Euclidean space under appropriate quasi-positivity assumptions on certain invariants of the immersion. The proofs are based on the corresponding first variational formula. As an application, we derive a degeneracy theorem for an entire r-minimal graph whose defining function ƒ has first and second derivatives decaying fast enough at infinity: Its Hessian operator D2 ƒ has at least n − r null eigenvalues everywhere.

Deforming the Scalar Curvature of the De Sitter–Schwarzschild Space

Building upon the work of Brendle, Marques, and Neves on the construction of counterexamples to Min-Oo’s conjecture, we exhibit deformations of the de Sitter–Schwarzschild space of dimension

Penrose inequalities and a positive mass theorem for charged black holes in higher dimensions

n\ge 3

Positive mass and Penrose type inequalities for asymptotically hyperbolic hypersurfaces

n≥3 satisfying the dominant energy condition and agreeing with the standard metric along the event and cosmological horizons, which remain totally geodesic. Our results hold for generalized Kottler–de Sitter–Schwarzschild spaces whose cross-sections are compact rank one symmetric spaces and indicate that there exists no analogue of the rigidity statement of the Penrose inequality in the case of positive cosmological constant. As an application, we construct solutions of Einstein field equations satisfying the dominant energy condition and being asymptotic to (or agreeing with) the de Sitter–Schwarzschild space–time both at the event horizon and at spatial infinity.

A rigidity result for the graph case of the Penrose inequality

The Christoffel problem, in its classical formulation, asks for a characterization of real functions defined on the unit sphere