Hilário Alencar, Manfredo do Carmo, Renato Tribuzy

Resumo: We consider surfaces M² immersed in E_c × R, where E_c is a simply connected n-dimensional complete Riemannian manifold with constant sectional curvature c = 0, and assume that the mean curvature vector of the immersion is parallel in the normal bundle. We consider further a Hopf-type complex quadratic form Q on M² , where the complex structure of M² is compatible with the induced metric. It is not hard to check that Q is holomorphic. We will use this fact to give a reasonable description of immersed surfaces in E^{n}_{c} × R that have parallel mean curvature vector.

Pierre Bérard, Philippe, Marcos Cavalcante

Resumo: In this paper, we give a lower bound for the spectrum of the Laplacian on minimal hypersurfaces immersed into Hm × R. As an application, in dimension 2, we prove that a complete minimal surface with finite total extrinsic curvature has finite index. On the other hand, for stable, minimal surfaces in H3 or in H2 × R, we give an upper bound on the infimum of the spectrum of the Laplacian and on the volume growth.

Vinícius Mello, Luiz Velho

Resumo: In this paper we will develop a theory for simplicial diffeomorphims, that is, diffeomorphims that preserve the incidence relations of a simplicial complex, and analyze alternative schemes to construct them with different properties. In combining piecewise linear functions on complexes with simplicial diffeomorphisms, we propose a new representation of curves and surfaces (and hypersurfaces, in general) that is simultaneously implicit and parametric.

Pierre Bérard, Philippe, Marcos Cavalcante

Resumo: In this paper, we determine the maximally stable, rotationally invariant domains on the catenoids Ca (minimal surfaces invariant by rotations) in the Heisenberg group. We show that these catenoids have finite Morse index at least 3 and we bound the index from above in terms of the parameter a. We also study the rotationally symmetric stable domains on the higher dimensional catenoids.