The classical notion of the Darboux transformation of isothermic surfaces can be generalised to a transformation for conformal immersions. Since a minimal surface is Willmore, we can use the associated $$\mathbb { C}_*$$ C ∗ -family of flat connections of the harmonic conformal Gauss map to construct such transforms, the so-called $$\mu $$ μ -Darboux transforms. We show that a $$\mu $$ μ -Darboux transform of a minimal surface is not minimal but a Willmore surface in 4-space. More precisely, we show that a $$\mu $$ μ -Darboux transform of a minimal surface f is a twistor projection of a holomorphic curve in $$\mathbb { C}\mathbb { P}^3$$ C P 3 which is canonically associated to a minimal surface $$f_{p,q}$$ f p , q in the right-associated family of f . Here we use an extension of the notion of the associated family $$f_{p,q}$$ f p , q of a minimal surface to allow quaternionic parameters. We prove that the pointwise limit of Darboux transforms of f is the associated Willmore surface of f at $$\mu =1$$ μ = 1 . Moreover, the family of Willmore surfaces $$\mu $$ μ -Darboux transforms, $$\mu \in \mathbb { C}_*$$ μ ∈ C ∗ , extends to a $$\mathbb { C}\mathbb { P}^1$$ C P 1 family of Willmore surfaces $$f^\mu : M \rightarrow S^4$$ f μ : M → S 4 where $$\mu \in \mathbb { C}\mathbb { P}^1$$ μ ∈ C P 1 .

中文翻译：

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最小曲面的 $$\mu $$ μ -Darboux 变换

等温面的 Darboux 变换的经典概念可以推广到保形浸没的变换。由于最小曲面是 Willmore，我们可以使用相关的 $$\mathbb { C}_*$$ C ∗ -调和共形高斯图的平面连接族来构造这种变换，即所谓的 $$\mu $ $ μ -Darboux 变换。我们证明了最小曲面的 $$\mu $$ μ -Darboux 变换不是最小的，而是 4 空间中的 Willmore 曲面。更准确地说，我们证明了最小表面 f 的 $$\mu $$ μ -Darboux 变换是 $$\mathbb { C}\mathbb { P}^3$$ CP 3 中全纯曲线的扭曲投影与 f 的右关联族中的最小表面 $$f_{p,q}$$ fp , q 规范相关。这里我们使用关联族 $$f_{p,q}$$ fp 的概念的扩展，q 允许四元数参数的最小表面。我们证明 f 的 Darboux 变换的逐点极限是 f 在 $$\mu =1$$ μ = 1 处的相关 Willmore 曲面。此外，Willmore 曲面族 $$\mu $$ μ -Darboux 变换 $$\mu \in \mathbb { C}_*$$ μ ∈ C ∗ 扩展到 $$\mathbb { C}\mathbb { P}^1$$ CP 1 Willmore 曲面族 $$f^\mu : M \rightarrow S^4$$ f μ : M → S 4 where $$\mu \in \mathbb { C}\mathbb { P}^1$$ μ ∈ CP 1 。