Minimal surfaces in ${\mathbb{R}}^{4}$ foliated by conic sections and parabolic rotations of holomorphic null curves in ${\mathbb{C}}^{4}$
J. Korean Math. Soc. 2020 Vol. 57, No. 1, 1-19
https://doi.org/10.4134/JKMS.j180363
Published online January 1, 2020
Hojoo Lee
Jeonbuk National University
Abstract : Using the complex parabolic rotations of holomorphic null curves in ${\mathbb{C}}^{4}$, we transform minimal surfaces in Euclidean space ${\mathbb{R}}^{3}$ to a family of degenerate minimal surfaces in Euclidean space ${\mathbb{R}}^{4}$. Applying our deformation to holomorphic null curves in ${\mathbb{C}}^{3}$ induced by helicoids in ${\mathbb{R}}^{3}$, we discover new minimal surfaces in ${\mathbb{R}}^{4}$ foliated by hyperbolas or straight lines. Applying our deformation to holomorphic null curves in ${\mathbb{C}}^{3}$ induced by catenoids in ${\mathbb{R}}^{3}$, we rediscover the Hoffman-Osserman catenoids in ${\mathbb{R}}^{4}$ foliated by ellipses or circles.
Keywords : Minimal surfaces, conic sections, holomorphic null curves
MSC numbers : 53A10, 49Q05
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