Bubbles with constant mean curvature, and almost constant mean curvature, in the hyperbolic space

Given a constant k> 1 , let Z be the family of round spheres of radius artanh(k-1) in the hyperbolic space H3, so that any sphere in Z has mean curvature k. We prove a crucial nondegeneracy result involving the manifold Z. As an application, we provide sufficient conditions on a prescribed function ϕ on H3, which ensure the existence of a C1-curve, parametrized by ε≈ 0 , of embedded spheres in H3 having mean curvature k+ εϕ at each point.