用极小曲面表示的\R^4中的所有超共形曲面
All superconformal surfaces in \R^4 in terms of minimal surfaces
摘要：
本文给出了欧氏空间中任意单连通超共形曲面$ϕ\colon M^2\to \R^4$的一个显式构造，它是由一对共轭极小曲面$g,h\colon M^2\to\R^4$表示的。$ϕ$是超共形的意思是它的曲率椭圆在任何点都是圆。我们刻画了共轭极小曲面对$(g,h)$的特征，这些曲面对通过在$\R^4$中的反演而产生全纯曲线的像，并通过球极平面投影而产生球面$\Sf^4$或双曲空间$\Hy^4$中的超极小曲面的像.我们还确定了共轭极小曲面的对$(g,h)$与超共形曲面和它的图像之间的关系，通过反演。特别是，这产生了一个新的极小曲面在$\R^4$的转换。
英文摘要：
We give an explicit construction of any simply-connected superconformal surface $ϕ\colon M^2\to \R^4$ in Euclidean space in terms of a pair of conjugate minimal surfaces $g,h\colon M^2\to\R^4$. That $ϕ$ is superconformal means that its ellipse of curvature is a circle at any point. We characterize the pairs $(g,h)$ of conjugate minimal surfaces that give rise to images of holomorphic curves by an inversion in $\R^4$ and to images of superminimal surfaces in either a sphere $\Sf^4$ or a hyperbolic space $\Hy^4$ by an stereographic projection. We also determine the relation between the pairs $(g,h)$ of conjugate minimal surfaces associated to a superconformal surface and its image by an inversion. In particular, this yields a new transformation for minimal surfaces in $\R^4$.
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