Unstable CMC spheres and outlying CMC spheres in AF 3-manifolds

In this paper, we introduce a non linear ODE method to construct CMC surfaces in Riemannian manifolds with symmetry. As an application we construct unstable CMC spheres and outlying CMC spheres in asymptotically Schwarzschild manifolds with metrics like $g_{ij}=(1+\frac{1}{l}{)}^2{\delta }_{ij}+O(l^{-2})$. The existence of unstable CMC spheres tells us that the stability condition in Qing-Tian’s work [Qing-Tian-CMC] can not be removed generally.