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In the 1-parameter family of Berger spheres S^3(a), a > 0 (S^3(1) is theround 3-sphere of radius 1) we classify the stable constant mean curvaturespheres, showing that in some Berger spheres (a close to 0) there are unstableconstant mean curvature spheres. Also, we classify the orientable compactstable constant mean curvature surfaces in S^3(a), 1/3 <= a < 1 proving thatthey are spheres or the minimal Clifford torus in S^3(1/3). This allows tosolve the isoperimetric problem in these Berger spheres.

Compact stable constant mean curvature surfaces in the Berger spheres