with Zero Scalar Curvature

In this work we generalize the case of scalar curvature zero the results of Simmons (Ann. Math. 88 (1968), 62-105) for minimal cones in R n+1 .I fM n−1 is a compact hypersurface of the sphere S n (1) we represent by C(M)e the truncated cone based on M with center at the origin. It is easy to see that M has zero scalar curvature if and only if the cone base on M also has zero scalar curvature. Hounie and Leite (J. Differential Geom. 41 (1995), 247-258) recently gave the conditions for the ellipticity of the partial differential equation of the scalar curvature. To show that, we have to assume n 4 and the three-curvature of M to be different from zero. For such cones, we prove that, for n 7 there is an e for which the truncate cone C(M)e is not stable. We also show that for n 8 there exist compact, orientable hypersurfaces M n−1 of the sphere with zero scalar curvature and S3 different from zero, for which all truncated cones based on M are stable.