Stability of r-minimal cones in Rn+1

Abstract

a b s t r a c t 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.

The halfspace theorem for minimal hypersurfaces in regions bounded by minimal cones

“…It is a remarkable fact that a Simons' type theorem still holds for ‐minimal cones. It was proved in appropriated dimensions by Barbosa and do Carmo in for zero scalar curvature cones with and by Barros and Sousa in for ‐minimal cones assuming that is a non‐null constant. In this setting, we have the following theorem: Theorem Let , , be an unstable ‐minimal cone for some .…”