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

Abstract

Using the complex parabolic rotations of holomorphic null curves in ${\mathbb{C}}^4$, we transform minimal surfaces in Euclidean space ${\mathbb{R}}^3\subset {\mathbb{R}}^4$ to a family of degenerate minimal surfaces in Euclidean space ${\mathbb{R}}^4$. Applying our deformation to holomorphic null curves in ${\mathbb{C}}^3\subset {\mathbb{C}}^4$ induced by helicoids in ${\mathbb{R}}^3$, we discover new minimal surfaces in ${\mathbb{R}}^4$ foliated by conic sections with eccentricity grater than $1$: hyperbolas or straight lines. Applying our deformation to holomorphic null curves in ${\mathbb{C}}^3$ induced by catenoids in ${\mathbb{R}}^3$, we can rediscover the Hoffman-Osserman catenoids in ${\mathbb{R}}^4$ foliated by conic sections with eccentricity smaller than $1$: ellipses or circles. We prove the existence of minimal surfaces in ${\mathbb{R}}^4$ foliated by ellipses, which converge to circles at infinity. We construct minimal surfaces in ${\mathbb{R}}^4$ foliated by parabolas: conic sections which have eccentricity $1$.

Comments welcome