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Let      H 2 B ( R ) {H^2}B(R)       be the family of bounded nonharmonic biharmonic functions on a Riemannian manifold      R R      . On the punctured Euclidean      N N      -space      E 0 N = { x = ( x 1 , …  , x N ) | 0 &gt; | x | &gt; ∞  } E_0^N = \{ x = ({x^1}, \ldots ,{x^N})|0 &gt; |x| &gt; \infty \}      ,      H 2 B {H^2}B       is void for      N &gt; 3 N &gt; 3      , whereas for      N = 2 , 3 N = 2,3      , it is generated by certain fundamental spherical harmonics. It is also known that      H 2 B {H^2}B       remains void on the Riemannian manifold      E α  N E_\alpha ^N      ,      N &gt; 3 N &gt; 3      , obtained by endowing      E 0 N E_0^N       with the non-Euclidean metric      d s α  = r α  | d x | d{s_\alpha } = {r^\alpha }|dx|      ,      α  ∈  R \alpha \in R      . The purpose of the present paper is to show that the fundamental spherical harmonics continue generating      H 2 B ( E α  3 ) {H^2}B(E_\alpha ^3)      , despite the distorting metric      d s α  d{s_\alpha }      . An analogous result holds for      E α  2 E_\alpha ^2      .

Spherical harmonics generating bounded biharmonics