Hipersuperfícies r-mínimas no espaço euclidiano

Abstract

n the first part (chapters 2, 3 and 4) of this Thesis we will study the hypersurfaces of R p+q+2 that are r-minimum (Sr = 0) and invariant by the canonical action of the group O(p + 1) × O(q + 1). We will get a rating complete of all hypersurfaces of R p+q+2 which are O(p + 1) × O(q + 1)-invariant and have the r-'th null mean curvature (2 ≤ r ≤ min{p, q} ), analyzing whether such hypersurfaces are complete, embedded and (r − 1)-stable. With this we will obtain the following existence result: “Let p, q, r ∈ N be such that p + q ≥ r + 5 and 2 ≤ r ≤ min{p, q}, then there is a hypersurface Mp+q+1 ⊂ R p+q+2 complete, layered, with r-´th null mean curvature which is globally (r − 1)-stable”. In chapter 5 we will study the cones C(M) ⊂ R n+1 r-minimum, whose base Mn−1 ⊂ S n is a compact hypersurface such that Sr = 0 and Sr+1 is a non-zero constant. We will prove that: “If r + 2 ≤ n ≤ r + 5, then there is 0 < ε < 1 such that the trunk of cone C(M)ε is not (r − 1)-stable”. Furthermore, we will construct a Clifford Torus with Sr = 0 and Sr+1 6 = 0 to show that this result is not valid when n ≥ r + 6.