Rotationally invariant constant Gauss curvature surfaces in Berger spheres

Abstract

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.