Sobre a estabilidade de cones em R^(n+1) com curvatura escalar nula

Abstract

In this work we will generalize for the case of zero scalar curvature, the results of Simmons [14] for minimal cones in Rn+1. If Mn−1 is a hypersurface of the sphere Sn(1) we represent by C(M)" the truncated cone based on M and center at the origin. It is easy to see that M has zero scalar curvature if and only if the cone with base on M also has zero scalar curvature. Hounie and Leite [10] recently gave conditions for the ellipticity of the partial differential equation of scalar curvature. To show this we have to assume n greater than or equal to 4 and that 3 – curvature of M is non-zero. For such cones, we will prove that for n less than or equal to 7 there is a "for which the truncated cone C(M)" is not stable. We will also show that for n greater than or equal to 8 there are compact hypersurfaces and orientable spheres Mn−1 with zero scalar curvature and nonzero S3 for which all truncated cones based on M are stable.