Minimal surfaces in ${\mathbb{R}}^{4}$ foliated by conic sections and parabolic rotations of holomorphic null curves in ${\mathbb{C}}^{4}$

Hojoo Lee

doi:10.4134/JKMS.j180363

Journal of the
Korean Mathematical Society JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2020; 57(1): 1-19

Online first article November 8, 2019 Printed January 1, 2020

https://doi.org/10.4134/JKMS.j180363

Minimal surfaces in ${\mathbb{R}}^{4}$ foliated by conic sections and parabolic rotations of holomorphic null curves in ${\mathbb{C}}^{4}$

Hojoo Lee

Jeonbuk National University

Abstract

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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