O(p + 1) x O(p + 1)-Invariant hypersurfaces with zero scalar curvature in euclidean space

Abstract

We use equivariant geometry methods to study and classify zero scalar curvature O(p + 1) × O(p + 1)-invariant hypersurfaces in R 2p+2 with p > 1.

O(p + 1) × O(q + 1)-invariant (r – 1)-minimal hypersurfaces in Euclidean space

“…On the other hand, according to Proposition (3.1) and Remark (3.1) of [16], the following facts concerning O(p + 1) × O(q + 1)-invariant hypersurfaces M p+q+1 ⊂ R p+q+2 will be used: M is embedded if and only if the associated profile curve is embedded. Moreover, if the orbit of X associated to the profile curve is defined for all t, then the corresponding hypersurface is complete.…”

“…Thus a bilinear symmetric form I r may be defined by In order to show our result concerning to r-stability we will need the next proposition and lemma that appear in [7], [8] and [16]. Now let M be an O(p+1)×O(q +1)-invariant hypersurface in R p+q+2 with H r = 0.…”

“…Years later Alencar et al [2] presented a study of minimal hypersurfaces of R p+q+2 invariant by the action of O(p + 1) × O(q + 1) with p, q > 1. The study of O(p + 1) × O(q + 1)-invariant hypersurfaces in R p+q+2 with zero scalar curvature began with the work due to Palmas [14] when p = q = 1 whereas the case p = q > 1 was generalized by Sato [16]. Finally, Sato and Souza Neto [17] closed the case of zero scalar curvature for p = q.…”

On Hypersurfaces with Zero r-Mean Curvature

“…In [21] and [19], one finds a classification of the hypersurfaces, with zero scalar curvature in R 2m , which are invariant by the action of O(m) × O(m). These results were obtained by studying the generating curves.…”

The complete hyper-surfaces with zero scalar curvature in $$\mathbb{R }^{n+1}$$

“…Hence, it will be equally interesting to consider other elementary symmetric functions of the second fundamental form. In particular, it is natural to ask whether hyper-surfaces with zero scalar curvature have Bernstein type property [2,6,10,17]. However, observe that the equation for the hyper-surfaces with zero mean curvature is elliptic, the analogy for surfaces with zero scalar curvature is only a degenerate elliptic equation.…”