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| One of the central problems in mathematical relativity is the existence and uniqueness of constant mean curvature surfaces. In proving the uniqueness, the stability condition is usually required. In this paper, we introduce a non linear ODE method to construct CMCsurfaces in Riemannian manifolds with symmetry. As an applicationwe construct unstable CMC spheres and outlying CMC spheres in asymptoticallySchwarzschild manifolds with metrics like $g_{ij}=(1+ rac{1}{l})^{2}delta_{ij}+O(l^{-2})$.The existence of unstable CMC spheres tells us that the stabilitycondition in Qing-Tian's work “On the uniqueness of the foliation of spheres of constant mean curvature in asymptotically flat 3-manifolds” can not be removedgenerally. |
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| Keywords:asymptotically flat manifold, constant mean curvature surface, stability, ordinary differential equation, Riemannian manifold |
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