Rotationally invariant constant Gauss curvature surfaces in Berger spheres

We give a full classification of complete rotationally invariant surfaces with constant Gauss curvature in Berger spheres: they are either Clifford tori, which are flat, or spheres of Gauss curvature K≥K0 for a positive constant K0, which we determine explicitly and depends on the geometry of the ambient Berger sphere. For values of K0≤K≤KP, for a specific constant KP, it was not known until now whether complete constant Gauss curvature K surfaces existed in Berger spheres, so our classification provides the first examples. For K>KP, we prove that the rotationally invariant spheres from our classification are the only topological spheres with constant Gauss curvature in Berger spheres. Each point on the picture represents a compact constant Gauss curvature (cgc) K surface in the Berger sphere Sτ3 (τ2 is on the horizontal axis and K is on the vertical axis, τ=1 corresponds to the round sphere). We show the existence of rotationally invariant cgc spheres in the dark gray region and uniqueness in the striped area (excluding the boundary). The light gray area represents the non-existence region (excluding the case K=0 that corresponds to Hopf tori) for compact cgc surfaces given in Torralbo-Urbano 2010. The existence of compact cgc surfaces in the white area is unknown. •Rotationally invariant constant Gauss curvature surfaces are classified in the Berger spheres.•There exist constant Gauss curvature spheres whose extrinsic curvature is not everywhere positive in the Berger spheres.•Constant Gauss curvature spheres are rotationally invariant if their extrinsic curvature is positive in the Berger spheres.