Spherical harmonics generating bounded biharmonics

Let ${H^2}B(R)$ be the family of bounded nonharmonic biharmonic functions on a Riemannian manifold $R$. On the punctured Euclidean $N$-space $E_0^N = \{ x = ({x^1}, \ldots ,{x^N})\vert < \vert x\vert < \infty \}$, ${H^2}B$ is void for $N > 3$, whereas for $N = 2,3$, it is generated by certain fundamental spherical harmonics. It is also known that ${H^2}B$ remains void on the Riemannian manifold $E_\alpha ^N$, $N > 3$, obtained by endowing $E_0^N$ with the non-Euclidean metric $d{s_\alpha } = {r^\alpha }\vert dx\vert$, $\alpha \in R$.

The purpose of the present paper is to show that the fundamental spherical harmonics continue generating ${H^2}B(E_\alpha ^3)$, despite the distorting metric $d{s_\alpha }$. An analogous result holds for $E_\alpha ^2$.