We classify constant mean curvature surfaces invariant by a 1-parameter group of isometries in the Berger spheres and in the special linear group Sl(2, R). In particular, all constant mean curvature spheres in those spaces are described explicitly, proving that they are not always embedded. Besides new examples of Delaunay-type surfaces are obtained. Finally the relation between the area and volume of these spheres in the Berger spheres is studied, showing that, in some cases, they are not solution to the isoperimetric problem.

In this article, we construct compact, arbitrary Euler characteristic, orientable and non-orientable minimal surfaces in the Berger spheres. Also, we show an interesting family of surfaces that are minimal in every Berger sphere, characterizing them by this property. Finally we construct, via the Daniel correspondence, new examples of constant mean curvature surfaces in S 2 × R, H 2 × R and the Heisenberg group with many symmetries.

Stable compact minimal submanifolds of the product of a sphere and any Riemannian manifold are classified whenever the dimension of the sphere is at least three. The complete classification of the stable compact minimal submanifolds of the product of two spheres is obtained. Also, it is proved that the only stable compact minimal surfaces of the product of a 2-sphere and any Riemann surface are the complex ones.2010 Mathematics Subject Classification. Primary 53C40, 53C42.

We construct non-zero constant mean curvature H surfaces in the product spaces S 2 × R and H 2 × R by using suitable conjugate Plateau constructions. The resulting surfaces are complete, have bounded height, and are invariant under a discrete group of horizontal translations. A 1-parameter family of unduloid-type surfaces is produced in S 2 × R for any H > 0 (some of which are compact), and in H 2 × R for any H > 1/2 (which are shown to be properly embedded bigraphs). Finally, we give a different construction in H 2 × R for H = 1/2 giving surfaces with the symmetries of a tessellation of H 2 by regular polygons.

In the 1-parameter family of Berger spheres {S 3 α , α > 0} (S 3 1 is the round 3-sphere of radius 1) we classify the stable constant mean curvature spheres, showing that in some Berger spheres (α close to 0) there are unstable constant mean curvature spheres. Also, we classify the orientable compact stable constant mean curvature surfaces in S 3 α , 1/3 ≤ α < 1 proving that they are spheres or the minimal Clifford torus in S 3 1/3 . This allows to solve the isoperimetric problem in these Berger spheres.