Stability of [formula omitted]-minimal cones in [formula omitted

Abstract

The study of minimal cones C ( M ) in R n + 1 construed over compact minimal hypersurface M n − 1 of a unit Euclidean sphere S n 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 R n + 1 with zero scalar curvature, Ann. Global Anal. Geom. 28 (2005) 107–122] treated cones in R n + 1 with the second function of curvature S 2 = 0 and S 3 ⁄ = 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 M n − 1 of the unit sphere S n with S r = 0 and S r + 1 ⁄ = 0 by showing that there exists ε > 0 for which the truncated cone C ( M ) ε is ( r − 1 ) -unstable provided n ≤ r + 5 . Moreover, we also show that for n ≥ r + 6 there exists a Clifford torus S p ( r 1 ) × S q ( r 2 ) ⊂ S n with S r = 0 and S r + 1 ⁄ = 0 , for which all truncated cones based on such a torus are ( r − 1 ) -stable.