Abstract: In this talk, I will describe a non-linear ODE method to construct CMC spheres in Riemannian

manifolds with symmetry. As an application we construct unstable CMC spheres and outlying CMC spheres

in asymptotically Schwarzschild manifolds. The existence of unstable CMC spheres reveals that the stability

condition in Qing-Tian’s work about the uniqueness of CMC surfaces can not be removed generally.

The existence of outlying CMC spheres reproves a result of Brendle and Eichmair.