Abstract
We prove a Liouville type theorem for minimizing extrinsic biharmonic maps from \({\mathbb {R}}^m\) to the Euclidean unit sphere \({\mathbb {S}}^n\) : if \(u \in W^{2,2} _{loc}( {\mathbb {R}}^m, {\mathbb {S}}^{n})\) is a minimizing extrinsic biharmonic map and \(m\le 4\) then u is constant. In the case that \(m \ge 5\), if we suppose that \( \int _{{\mathbb {R}}^m}\left| \Delta u\right| ^{2} dx\) is finite, we also show that u is constant. As a direct consequence, the inverse of the stereographic projection from \({\mathbb {R}}^4\) to \({\mathbb {S}}^4\) is not a minimizing extrinsic biharmonic map. However in contrast, we prove that its restriction to the Euclidean unit ball \(B^4\) is a minimizing extrinsic biharmonic map.
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Communicated by A. Mondino.
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