Abstract
We study the stability properties of constant mean curvature (CMC) surfaces of revolution in general simply-connected spherically symmetric 3-spaces, and in the particular case of a positive-definite 3-dimensional slice of Schwarzschild space. We derive their Jacobi operators, and then prove that closed CMC tori of revolution in such spaces are unstable, and finally numerically compute the Morse index of some minimal and closed non-minimal CMC surfaces of revolution in the slice of Schwarzschild space.