The study of minimal and, more generally, constant mean curvature hypersurfaces in Riemannian space forms is a classical topic in differential geometry. As is well known, minimal hypersurfaces are critical points of the variational problem of minimizing area. Similarly, hypersurfaces with constant mean curvature are also solutions to that variational problem, when restricted to volume-preserving variations. In this paper we review about the stability index of both minimal and constant mean curvature hypersurfaces in Euclidean spheres, including some recent progress by the author, jointly with some of his collaborators. One of our main objectives on writing this paper has been to make it comprehensible for a wide audience, trying to be as self-contained as possible.

Neste trabalho tratamos das hipersuperfícies fechadas com curvatura média constante e duas curvaturas principais imersas na esfera euclidiana. No caso de hipersuperfícies mínimas, Perdomo (Perdomo 2004) e Wang (Wang 2003) obtiveram uma desigualdade integral envolvendo o quadrado da norma da segunda forma fundamental de M, onde ocorre a igualdade se e somente se M é o toro de Clifford. Neste trabalho, usando o segunda forma fundamental modificada com traço nulo de M, obtemos uma generalização da fórmula integral acima e uma nova caracterização dos H(r)-toros.

In this paper we consider compact oriented hypersurfaces M with constant mean curvature and two principal curvatures immersed in the Euclidean sphere. In the minimal case, Perdomo (Perdomo 2004) andWang (Wang 2003) obtained an integral inequality involving the square of the norm of the second fundamental form of M, where equality holds only if M is the Clifford torus. In this paper, using the traceless second fundamental form of M, we extend the above integral formula to hypersurfaces with constant mean curvature and give a new characterization of the H(r)-torus.

Nesta nota, mostramos como o clássico teorema de Bernstein sobre as superfícies mínimas no espaço Euclideano pode ser visto como uma consequência do teorema de Calabi-Bernstein sobre as superfícies máximas no espaço de Lorentz-Minkowski (e vice-versa). Isto decorre de uma simples, mas elegante, dualidade entre soluções a suas correspondentes equações diferenciais.

In this note we show how classical Bernstein's theorem on minimal surfaces in the Euclidean space can be seen as a consequence of Calabi-Bernstein's theorem on maximal surfaces in the Lorentz-Minkowski space (and viceversa). This follows from a simple but nice duality between solutions to their corresponding differential equations.