On Stability of Cones in Rn+1 with Zero Scalar Curvature

Abstract

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 . If M n−1 is a compact hypersurface of the sphere S n (1) we represent by C(M) ε 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 ε for which the truncate cone C(M) ε 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 S 3 different from zero, for which all truncated cones based on M are stable. (2000): 53C42, 53C40, 49F10, 57R70. Mathematics Subject Classifications

Stability of r-minimal cones in Rn+1

“…A few years later, Barbosa and Do Carmo extended Simmons's result when the second function of curvature is null, see [2]. More specifically, they shown that for orientable compact hypersurface M n−1 ⊂ S n with n ≤ 7, there exists e ∈ (0, 1) such that the truncated cone C (M) ε is 1-unstable provided S 2 = 0 and S 3 = 0.…”

“…Moreover, we follow closely the ideas contained in the work of Simmons [10], Barbosa and Do Carmo [2].…”

“…Before beginning, we remember some basic facts about stability that appear in the literature, essentially in [2,3] and [8]. Let x : Σ n Q n+1 c be an isometric immersion and D ⊂ Σ n be a domain with compact closure in Σ n .…”

Examples of scalar-flat hypersurfaces in $${\mathbb{R}^{n+1}}$$

“…For a detailed explanation on these products of spheres, we refer the reader to [1]. We begin our analysis of the equation (22) by solving first the non-homogeneous linear Dirichlet problem for the Jacobi operator…”

Stability of area-preserving variations in space forms

“…Theorem 5 6. Let M be an n-dimensional connected, compact without boundary and oriented Riemannian manifold, letM(c) be an open hemisphere of the unit sphere S n+1(1) or the hyperbolic space H n+1 (−1) according to c = 1 or −1 respectively.…”