Indiana University Mathematics Journal vol:61 issue:3 pages:1129-1156
We classify the stable constant mean curvature spheres in the homogeneous Riemannian 3-manifolds: the Berger spheres, the special linear group and the Heisenberg group. We show that all of them are stable in the last two cases while in some Berger spheres there are unstable ones. Also, we classify the stable compact orientable constant mean curvature surfaces in a certain subfamily of the Berger spheres. This allows to solve the isoperimetric problem in some Berger spheres.