Biharmonic and f-biharmonic maps from a 2-sphere

TL;DR

This paper investigates biharmonic and f-biharmonic maps from a 2-sphere, providing a method to generate such maps, including many explicit examples with applications to maps between round spheres.

Contribution

It reduces the problem to a second order linear ODE for symmetric maps and constructs numerous explicit examples of biharmonic and f-biharmonic maps from the 2-sphere.

Findings

Reduction of biharmonicity conditions to a 2nd order linear ODE for symmetric maps

Construction of many explicit examples of biharmonic and f-biharmonic maps from the 2-sphere

Identification of non-conformal proper biharmonic and f-biharmonic maps with singular points

Abstract

We study biharmonic maps and f-biharmonic maps from a round sphere $(S^2, g_0)$, the latter maps are equivalent to biharmonic maps from Riemann spheres $(S^2, f^{-1}g_0)$. 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 a round sphere $S^2$ and between two round spheres. Our examples include non-conformal proper biharmonic maps $(S^2, f^{-1}g_0)\longrightarrow S^2$ and $(S^2, f^{-1}g_0)\longrightarrow S^n$, or non-conformal f-biharmonic maps $(S^2, g_0)\longrightarrow S^2$ and $(S^2,g_0)\longrightarrow S^n$ from round sphere with two singular points.