Biharmonic and f-biharmonic maps from a 2-sphere

We study biharmonic maps and f-biharmonic maps from the standard sphere (S2,g0), the latter maps are equivalent to biharmonic maps from Riemann spheres (S2,f−1g0). We proved that for rotationally symmetric maps between rotationally symmetric spaces, both biharmonicity and f-biharmonicity reduce to a 2nd order linear ordinary differential equation. As applications, we give a method to produce biharmonic maps and f-biharmonic maps from given biharmonic maps and we construct many examples of biharmonic and f-biharmonic maps from the standard sphere S2 and between two such spheres. Our examples include non-conformal proper biharmonic maps from Riemann spheres (S2,f−1g0)⟶S2 and (S2,f−1g0)⟶Sn, or non-conformal f-biharmonic maps from the standard spheres (S2,g0)⟶S2 and (S2,g0)⟶Sn with two singular points.