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.

我们证明了一个 Liouville 型定理，用于最小化从\({\mathbb {R}}^m\)到欧几里得单位球体\({\mathbb {S}}^n\) 的外在双调和映射：如果\(u \in W^{2,2} _{loc}( {\mathbb {R}}^m, {\mathbb {S}}^{n})\)是一个最小化的外在双调和映射和\(m\le 4\ )那么u是常数。在\(m \ge 5\)的情况下，如果我们假设\( \int _{{\mathbb {R}}^m}\left| \Delta u\right| ^{2} dx\)是有限，我们也证明u是常数。作为直接结果，从\({\mathbb {R}}^4\)到\({\mathbb {S}}^4\)的立体投影的逆不是最小化外在双调和映射。然而，相比之下，我们证明了它对欧几里德单位球\(B^4\) 的限制是最小化外在双调和映射。