  We prove the existence of branched immersed constant mean curvature 2-spheres
in an arbitrary Riemannian 3-sphere for almost every prescribed mean curvature,
and moreover for all prescribed mean curvatures when the 3-sphere is positively
curved. To achieve this, we develop a min-max scheme for a weighted Dirichlet
energy functional. There are three main ingredients in our approach: a
bi-harmonic approximation procedure to obtain compactness of the new
functional, a derivative estimate of the min-max values to gain energy upper
bounds for min-max sequences for almost every choice of mean curvature, and a
Morse index estimate to obtain another uniform energy bound required to reach
the remaining constant mean curvatures in the presence of positive curvature.
