Stability of rr-minimal cones in Rn+1Rn+1

The study of minimal cones C(M)C(M) in Rn+1Rn+1 construed over compact minimal hypersurface Mn−1Mn−1 of a unit Euclidean sphere SnSn has an important link with the Bernstein problem, see e.g. Bombieri et al. [E. Bombieri, E. de Giorgi, E. Giusti, Minimal cones and Bernstein problem, Invent. Math. 7 (1969) 243–268]. It was studied by many authors with a remarkable paper due to Simmons [J. Simmons, Minimal varieties in Riemannian manifolds, Ann. of Math. 88 (1968) 62–105]. In a recent work Barbosa and Do Carmo [J.L.M. Barbosa, M.P. Do Carmo, On the stability of cones in Rn+1Rn+1 with zero scalar curvature, Ann. Global Anal. Geom. 28 (2005) 107–122] treated cones in Rn+1Rn+1 with the second function of curvature S2=0S2=0 and S3⁄=0S3⁄=0. In these papers the authors showed the existence of a truncated cone which is unstable as well as truncated cones over Clifford tori that are stable. Here we partially extend such results for cones construed over compact hypersurfaces Mn−1Mn−1 of the unit sphere SnSn with Sr=0Sr=0 and Sr+1⁄=0Sr+1⁄=0 by showing that there exists ε>0ε>0 for which the truncated cone C(M)εC(M)ε is (r−1)(r−1)-unstable provided n≤r+5n≤r+5. Moreover, we also show that for n≥r+6n≥r+6 there exists a Clifford torus Sp(r1)×Sq(r2)⊂SnSp(r1)×Sq(r2)⊂Sn with Sr=0Sr=0 and Sr+1⁄=0Sr+1⁄=0, for which all truncated cones based on such a torus are (r−1)(r−1)-stable.